*5.3. Conclusions*

In this Section, we have reviewed the development and applications of an effective algebraic scheme of constructing density operator and density functional representations for the local quantum current algebra and its application to quantum Hamiltonian and symmetry operators reconstruction. We analyzed the corresponding factorization structure for quantum Hamiltonian operators, spatially governing many- and one-dimensional integrable dynamical systems. The quantum generalized oscillatory and Calogero–Moser– Sutherland models of spin-less bose-particles were analyzed in detail. The central vector of the density operator current algebra representation proved to be the ground vector state of the corresponding completely integrable factorized quantum Hamiltonian system in the classical Bethe anzatz form. The latter makes it possible to quantum classify completely integrable Hamiltonian systems a priori, allowing the factorized form and those whose groundstate is of the Bethe anzatz from. These and related aspects of the factorized and completely integrable quantum Hamiltonians systems are planned to be studied in other places.

#### **6. The Quantum Current Algebra Quasi-Classical Representations and the Collective Variable Approach in Equilibrium Statistical Physics** *IntroductoryNotes*

We consider a large system of *N* ∈ N (one-atomic and spinless) bose-particles with a fixed density *ρ*¯ := *N*/Λ in a volume Λ ⊂ R3, which is specified by a quantum-mechanical Hamiltonian operator *H*ˆ : *L*(*sym*) 2(R3*N*; C) → *L*(*sym*) 2(R3*N*; C) of the form:

$$\hat{H} := -\frac{\hbar^2}{2m} \sum\_{j=1}^{N} \nabla\_j^2 + \sum\_{j$$

where ∇*j* := *<sup>∂</sup>*/*∂xj*, *j* = 1, *N*, *h*¯—the Planck constant, *m* ∈ R+—a particle mass and *<sup>V</sup>*(*x* − *y*) := *<sup>V</sup>*(|*x* − *y*|), *x*, *y* ∈ Λ,—a two-particle potential energy, allowing a partition *V* = *V*(*l*) + *<sup>V</sup>*(*s*), where *V*(*s*)—a short range potential of the Lennard–Johns type and *V*(*l*)—a long range potential of the Coulomb type. Making use of the second quantization representation [13,22,56,59,63,140,141], the Hamiltonian (233) as Λ → R<sup>3</sup> and *N* → ∞ can be written as a sum H = H0 + V, where

$$\begin{aligned} \mathcal{H}\_0 &:= -\frac{\hbar^2}{2m} \int\_{\mathbb{R}^3} d^3 \mathbf{x} \psi^+ \nabla\_x^2 \psi, \\ \mathcal{V} &:= \frac{1}{2} \int\_{\mathbb{R}^3} d^3 \mathbf{x} \int\_{\mathbb{R}^3} d^3 \mathbf{y} V(\mathbf{x} - \mathbf{y}) \psi^+(\mathbf{x}) \psi^+(\mathbf{y}) \psi(\mathbf{y}) \psi(\mathbf{x}), \end{aligned} \tag{234}$$

and the operator H : Φ*F* → Φ*F* acts on the corresponding Fock space Φ*F* and *ψ*+(*x*), *ψ*(*y*): Φ*F* → Φ*F* are the creation and annihilation operators at points *x* ∈ R<sup>3</sup> and *y* ∈ R3. Assume now that our particle system is in a thermodynamically equilibrium state at an "*inverse*" temperature R+ *β* → ∞. Assume also that this equilibrium state is compatible with the respectively constructed quantum current algebra G representation in a separable Hilbert space <sup>Φ</sup>*μ* [4,6,56,68,141], whose generating cyclic vector Ω ∈ <sup>Φ</sup>*μ* realizes the ground state of the Hamiltonian operator H : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ*. Then, the corresponding *n*-particles distribution functions can be written down [56,86,142] as

$$f\_n(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) := (\Omega | : \rho(\mathbf{x}\_1)\rho(\mathbf{x}\_1)...\rho(\mathbf{x}\_n):\Omega),\tag{235}$$

where *n* ∈ N, *ρ*(*x*) : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ*, *x* ∈ R3—the density operator acting on the Hilbert space <sup>Φ</sup>*μ* and : · :—the related Wick normal ordering, naturally ensued from that defined over the creation and annihilation operators, and Ω ∈ <sup>Φ</sup>*μ* is the ground state of the Hamiltonian (234) at the temperature *β* → <sup>∞</sup>, normed by the stability condition (Ω|Ω) = 1. Having introduced the corresponding Bogolubov generating functional

$$\mathcal{L}(\mathbf{f}) := \left(\Omega \middle| \exp[i\rho(\mathbf{f})] \Omega\right) \tag{236}$$

for any "test" Schwartz function f ∈ *F* - S( R3; <sup>R</sup>), where *ρ*(f) :<sup>=</sup> R<sup>3</sup> *<sup>d</sup>*3*x*f(*x*)*ρ*(*x*), then for the *n*-particle distribution functions (235) one can ge<sup>t</sup> the expression

$$f\_{\boldsymbol{\theta}}(\mathbf{x}\_{1}, \mathbf{x}\_{2}, \dots, \mathbf{x}\_{n}) =: \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(\mathbf{x}\_{1})} \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(\mathbf{x}\_{2})} \dots \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(\mathbf{x}\_{n})} : \mathcal{L}(\mathbf{f})|\_{\mathbf{f}=\mathbf{0}}.\tag{237}$$

Here *xj* ∈ R3, *j* = 1, *n*, *n* ∈ N, and the symbol ": 1*i δ <sup>δ</sup>*f(*<sup>x</sup>*1) 1*i δ <sup>δ</sup>*f(*<sup>x</sup>*2)... 1*i δ <sup>δ</sup>*f(*xn*) :" imitates the normal ordering symbol ": :" action on operator expressions *ρ*(*<sup>x</sup>*1)*ρ*(*<sup>x</sup>*1)...*ρ*(*xn*), that is

$$\begin{split} \left. \begin{split} \text{s.} \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(\mathbf{x}\_{1})} := \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(\mathbf{x}\_{1})} , \end{split} \right. \\ \left. \begin{split} \text{s.} \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(\mathbf{x}\_{1})} \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(\mathbf{x}\_{2})} := \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(\mathbf{x}\_{1})} [\frac{1}{i} \frac{\delta}{\delta \mathbf{f}(\mathbf{x}\_{2})} - \delta(\mathbf{x}\_{1} - \mathbf{x}\_{2})] , \end{split} \end{split} \tag{238}$$

and so on. Consider now the expression (236) at some *β* ∈ R+, making use of the statistical operator *P* : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ* and the "*shifted*" 'Hamiltonian H(*λ*) := <sup>H</sup>−*<sup>λ</sup>* R<sup>3</sup> *<sup>d</sup>*3*xρ*(*x*) with *λ* ∈ R being a suitable "*chemical*" potential:

$$\mathcal{L}(\mathbf{f}) := \text{Tr}(P \exp[i\rho(\mathbf{f})]), \quad P := \frac{\exp(-\beta \mathcal{H}^{(\mu)})}{\text{Tr}\exp(-\beta \mathcal{H}^{(\mu)})},\tag{239}$$

where "Tr" means the operator trace-operation on the Hilbert space <sup>Φ</sup>*μ*. Keeping in mind within the task of studying distribution functions (235) in the classical statistical mechanics case, we need to calculate the trace in (239) as *h*¯ → 0. The latter gives rise to the following expressions:

$$\mathcal{L}(\mathbf{f}) = Z(\mathbf{f}) / Z(\mathbf{0}), \quad Z(\mathbf{f}) := \exp[-\beta \mathbf{V}(\delta)] \mathcal{L}\_0(\mathbf{f}), \tag{240}$$

$$\mathcal{L}\_0(\mathbf{f}) = \exp(\varsigma \int\_{\mathbb{R}^3} d^3 \mathbf{x} \{\exp[i \mathbf{f}(\mathbf{x})] - 1\}),$$

where *ς* := exp(*βλ*)(<sup>2</sup>*πh*¯ <sup>2</sup>*βm*)−3/2 is the system "*activity*" [56,142], and

$$\mathcal{V}(\delta) := \frac{1}{2} \int\_{\mathbb{R}^3} d^3 \mathbf{x} \int\_{\mathbb{R}^3} d^3 y V(\mathbf{x} - y) : \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(\mathbf{x})} \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(y)} : \, . \tag{241}$$

Based on expressions (240) and (241) we can formulate the following proposition.

**Proposition 8.** *The functional (236) satisfies [20,56,86] the following functional Bogolubov type equation:*

$$\begin{split} \left[\nabla\_{\mathbf{x}} - i\nabla\_{\mathbf{x}}\mathbf{f}(\mathbf{x})\right] \frac{1}{i} \frac{\delta \mathcal{L}(\mathbf{f})}{\delta \mathbf{f}(\mathbf{x})} \\ = -\beta \int\_{\mathbb{R}^3} d^3 y \nabla\_{\mathbf{x}} V(\mathbf{x} - \mathbf{y}) : \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(\mathbf{x})} \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(\mathbf{y})} : \mathcal{L}(\mathbf{f}), \end{split} \tag{242}$$

*with the expression (240) being its exact functional-analytic solution.*

Below, we will proceed to constructing effective analytic tools allowing the exact functional-analytic solutions to the Bogolubov functional Equation (242) to be found, describing equilibrium many-particle dynamical systems, as well as generalizing the obtained results for the case of non-equilibrium dynamical many particle systems.

#### **7. The Bogolubov-Zubarev "Collective" Variables Transform**

Taking into account the two-particle potential energy partition V = V(*s*) + <sup>V</sup>(*l*), owing to the representation (240) one can easily write down the following expression for generating functional *<sup>Z</sup>*(f), f ∈ *F*:

$$Z(\mathbf{f}) = \exp[-\beta \mathbf{V}^{(s)}(\delta)] \mathcal{L}^{(l)}(\mathbf{f}), \quad \mathcal{L}^{(l)}(\mathbf{f}) := \exp[-\beta \mathbf{V}^{(l)}(\delta)] \mathcal{L}\_0(\mathbf{f}), \tag{243}$$

where we put

$$\begin{split} \mathcal{V}^{(l)}(\delta) &:= \frac{1}{2} \int\_{\mathbb{R}^3} d^3 \mathbf{x} \int\_{\mathbb{R}^3} d^3 y V^{(l)}(\mathbf{x} - \mathbf{y}) : \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(\mathbf{x})} \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(\mathbf{y})} : \\ \mathcal{V}^{(s)}(\delta) &:= \frac{1}{2} \int\_{\mathbb{R}^3} d^3 \mathbf{x} \int\_{\mathbb{R}^3} d^3 y V^{(s)}(\mathbf{x} - \mathbf{y}) : \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(\mathbf{x})} \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(\mathbf{y})} : \end{split} \tag{244}$$

Needing to calculate the functional <sup>L</sup>(*l*)(f), f ∈ *F*, corresponding to the long range part V(*l*) of the full potential energy V : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ*, we will apply the analogue of Bogolubov– Zubarev [143,144] "collective" variables transform within the grand canonical ensemble, suggested before in [20,68,145,146]. Namely, denote by L(*l*) (*n*)(f), *n* ∈ N,—a partial solution to the functional Equation (242), possessing exactly *n* ∈ N particles. Then, owing to the results of [86], for L(*l*) (*n*)(f), *n* ∈ N, there holds the following exact expression:

$$\mathcal{L}\_{(n)}^{(l)}(\mathbf{f}) = \int\_{\mathbb{R}^3} d^3 \mathbf{x}\_1 \int\_{\mathbb{R}^3} d^3 \mathbf{x}\_2 \dots \int\_{\mathbb{R}^3} d^3 \mathbf{x}\_n \prod\_{j=1}^n \exp[i \mathbf{f}(\mathbf{x}\_j)] \exp(-\beta V\_n^{(l)}),\tag{245}$$

where *V*(*l*) *n* —the long term part potential energy of an *<sup>n</sup>*−particle group of the system. Then we ge<sup>t</sup> that

$$\mathcal{L}^{(l)}(\mathbf{f}) := \sum\_{n \in \mathbb{Z}\_+} \frac{z^n}{n!} \mathcal{L}^{(l)}\_{(n)}(\mathbf{f}) \mathcal{Q}^{-1}\_{0,} \qquad \mathcal{Q}\_0 := (\sum\_{n \in \mathbb{Z}\_+} \frac{z^n}{n!} \mathcal{L}^{(l)}\_{(n)}(0))^{-1}. \tag{246}$$

The sum in (246) can be calculated exactly, taking into account the expression

$$\mathcal{L}\_{(n)}^{(l)}(\mathbf{f}) = \int \mathcal{D}(\omega) \{ z \int\_{\mathbb{R}^3} d^3 \mathbf{x} \exp[i \mathbf{f}(\mathbf{x})] \mathcal{g}(\mathbf{x}; \omega) \} \,^n I(\omega), \tag{247}$$

where <sup>D</sup>(*ω*) := ∏ *k*∈R<sup>3</sup> *i* 2 (*dω*<sup>∗</sup>*k* ∧ *d<sup>ω</sup>k*), *<sup>ω</sup>*<sup>∗</sup>*k* := *ω*−*k* ∈ C, *k* ∈ R3,

$$g(\mathbf{x};\omega) := \exp\left[-2\pi i(\int\_{\mathbb{R}^3} d^3k \omega\_k \exp(ik\mathbf{x}) + \frac{\beta}{2} \int\_{\mathbb{R}^3} d^3k\nu(k)\right],$$

$$J(\omega) := \exp\left[-\int\_{\mathbb{R}^3} d^3k \frac{2\pi^2}{\beta\nu(k)} \omega\_k \omega\_{-k} + \int\_{\mathbb{R}^3} d^3k \ln \frac{\pi}{\beta\nu(k)}\right] \tag{248}$$

and *ν*(*k*) := (<sup>2</sup>*π*)−<sup>3</sup> R<sup>3</sup> *<sup>d</sup>*3*xV*(*l*)(*x*) exp(−*ikx*), *k* ∈ R3. Now, from (246)–(248) one easily finds that

$$\mathcal{L}^{(l)}(\mathbf{f}) = \int \mathcal{D}(\omega) \exp(\mathbf{z} \int\_{\mathbb{R}^3} d^3 \mathbf{x} \{\exp[\mathrm{if}(\mathbf{x})] - 1\} \mathbf{g}(\mathbf{x}; \omega)) f^{(l)}(\omega) Q^{-1}, \tag{249}$$

where *ς*¯ := *ς* exp( *β*2 R<sup>3</sup> *<sup>d</sup>*3*kν*(*k*)) = *ς* exp[ *β*2 *V*(*l*)(0)] and the function *J*(*l*)(*ω*), *ω* ∈ R3, allows the following series expansion:

$$J^{(l)}(\omega) := J(\omega) \exp\left[\int\_{\mathbb{R}^3} d^3 \mathbf{x} \, g(\mathbf{x}; \omega)\right] = J(\omega) \exp\left[-\frac{(2\pi)^2}{2!} (2\pi)^3 \int\_{\mathbb{R}^3} d^3 k \omega\_k \omega\_{-k} \qquad \text{(250)}$$

$$+ \sum\_{n \neq 2} \frac{(-2\pi i)^n}{n!} (2\pi)^3 \int\_{\mathbb{R}^3} d^3 k\_1 \int\_{\mathbb{R}^3} d^3 k\_2 \dots \int\_{\mathbb{R}^3} d^3 k\_n \prod\_{j=1}^n \omega\_{k\_j} \delta\left(\sum\_{j=1}^N k\_j\right)\right].$$

The expression (249) can now be represented [22,115,117,118] in the following cluster Ursell form:

$$\mathcal{L}^{(l)}(\mathbf{f}) = \exp\left(\sum\_{n=1}^{\infty} \frac{2^n}{n!} \int\_{\mathbb{R}^3} d^3 \mathbf{x}\_1 \int\_{\mathbb{R}^3} d^3 \mathbf{x}\_2 \dots \int\_{\mathbb{R}^3} d^3 \mathbf{x}\_n \prod\_{j=1}^n \{\exp[i\mathbf{f}(\mathbf{x})] - 1\} g\_n(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n)\right). \tag{251}$$

Here for any *n* ∈ Z+

$$g\_{\boldsymbol{\pi}}(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) := \sum\_{\boldsymbol{\sigma}[n]} (-1)^{m+1} (m-1)! \prod\_{j=1}^m R\_{\boldsymbol{\sigma}[j]}(\mathbf{x}\_k \in \boldsymbol{\sigma}[j]),$$

$$R\_{\boldsymbol{\pi}}(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) := \sum\_{\boldsymbol{\sigma}[n]} \prod\_{j=1}^m g\_{\boldsymbol{\sigma}[j]}(\mathbf{x}\_k \in \boldsymbol{\sigma}[j]),\tag{252}$$

where *gn*(*<sup>x</sup>*1, *x*2, ..., *xn*), *n* ∈ N, are called the *<sup>n</sup>*−particle Ursell cluster functions, *Rn*(*<sup>x</sup>*1, *x*2, ..., *xn*), *n* ∈ N, are suitable "correlation" functions [20,56,68] and *σ*[*n*] denotes a partition of the set {1, 2, ..., *n*} into non-intersecting subsets {*σ*[*j*] : *j* = 1, *<sup>m</sup>*}, that is *σ*[*j*] ∩ *σ*[*k*] = ∅ for *j* = *k* = 1, *m*, and *σ*[*n*] = <sup>∪</sup>*mj*=1*<sup>σ</sup>*[*j*]. Having separated from the function *J*(*l*)(*ω*), *ω* ∈ C3, the natural "Gaussian" part *J*(*l*) 0 (*ω*), *ω* ∈ C3, one can write down that

$$\mathcal{g}\_1(\mathbf{x}\_1) = G(\mathfrak{f}\_k^{(1)}) / G(0), \ \mathcal{g}\_2(\mathbf{x}\_1, \mathbf{x}\_2) = G(\mathfrak{f}\_k^{(2)}) / G(0) - \mathcal{g}\_1(\mathbf{x}\_1)\mathcal{g}\_1(\mathbf{x}\_2), \dots \tag{253}$$

where *ξ*(*n*) *k* := −2*πi* ∑*ns*=<sup>1</sup> exp(*ikxs*), *k* ∈ R3, *n* ∈ N,

$$\mathbb{G}(\xi\_{\mathbf{k}}^{(n)}) := \exp[\mathbf{M}(\xi\_{\mathbf{k}}^{(n)})] \int D(\omega) g^{(l)}(\xi\_{\mathbf{k}}^{(n)}; \omega) \, ]0(\omega),$$

$$\mathbf{M}(\xi\_{\mathbf{k}}^{(n)}) := \sum\_{m \neq 2} \frac{(-2\pi i)^{m}}{m!} (2\pi)^{3} \int\_{\mathbb{R}^{3}} d^{3}k\_{1} \int\_{\mathbb{R}^{3}} d^{3}k\_{2} \dots \int\_{\mathbb{R}^{3}} d^{3}k\_{m} \delta\left(\sum\_{s=1}^{m} k\_{s}\right) \prod\_{s=1}^{m} \frac{\delta}{\delta\xi\_{k\_{s}}^{(n)}},$$

$$\mathbf{g}^{(l)}(\xi\_{\mathbf{k}}^{(n)}; \omega) := \prod\_{j=1}^{n} \mathbf{g}(x\_{j}; \omega). \tag{254}$$

Since the integrals <sup>D</sup>(*ω*)*g*(*l*)(*ξ*(*n*) *k* ; *<sup>ω</sup>*)*J*(*l*)(*ω*), *n* ∈ N, one can calculate exactly, the formulae (251) and (253) are sources of the so called "virial" variables for Ursell–Mayer "cluster" correlation functions *gn*(*<sup>x</sup>*1, *x*2, ..., *xn*), *n* ∈ N, having important applications. In particular, from the function *J*(*l*)(*ω*), *ω* ∈ C3, one gets right away that the cluster expansion for the functions *gn*(*<sup>x</sup>*1, *x*2, ...*xn*), *n* ∈ N, are fulfilled by means of the "screened" potential function *V* ¯ (*l*)(*x* − *y*), *x*, *y* ∈ R3, where

$$\mathcal{V}^{(l)}(\mathbf{x} - \mathbf{y}) := \int\_{\mathbb{R}^3} d^3k \frac{\nu(k) \exp[ik(\mathbf{x} - \mathbf{y})]}{1 + \nu(k)\beta\overline{\varepsilon}(2\pi)^3}. \tag{255}$$

In particular, from (237) and (251) one easily finds that

*f*1(*<sup>x</sup>*1) = *z* <sup>D</sup>(*ω*)*g*(*x*; *ω*)*J*(*l*)(*ω*) <sup>D</sup>(*ω*)*J*(*l*)(*ω*)−<sup>1</sup> = = *ρ* ¯ - *z* ¯ exp *β*2 *d*3*k βν*<sup>2</sup>(*k*)(<sup>2</sup>*π*)<sup>3</sup>*z*¯ 1 + *ν*(*k*)*βz*¯(<sup>2</sup>*π*)<sup>3</sup> , *f*2(*<sup>x</sup>*1, *<sup>x</sup>*2) = *z*2 <sup>D</sup>(*ω*)*g*(*x*1; *ω*)*g*(*x*2; *ω*)*J*(*l*)(*ω*) <sup>D</sup>(*ω*)*J*(*l*)(*ω*)−<sup>1</sup> - - *ρ* ¯2 exp[−*βV*¯ (*l*)(*<sup>x</sup>*2 − *<sup>x</sup>*1)]'<sup>1</sup> + *ρ*¯ R3 *<sup>d</sup>*3*x*3[exp−*βV*¯ (*l*)(*<sup>x</sup>*1 − *x*3) − 1 + *βV* ¯ (*l*)(*<sup>x</sup>*1 − *<sup>x</sup>*3)][exp−*βV*¯ (*l*)(*<sup>x</sup>*2 − *x*3) − 1 + *βV*¯ (*l*)(*<sup>x</sup>*2 − *<sup>x</sup>*3] + *ρ*¯ R3 *<sup>d</sup>*3*x*3[−*βV*¯ (*l*)(*<sup>x</sup>*1 − *<sup>x</sup>*3)][exp(−*βV*¯ (*l*)(*<sup>x</sup>*2 − *<sup>x</sup>*3)) − 1 + *βV*¯ (*l*)(*<sup>x</sup>*2 − *<sup>x</sup>*3)] +*ρ*¯ R3 *<sup>d</sup>*3*x*3[−*βV*¯ (*l*)(*<sup>x</sup>*2 − *<sup>x</sup>*3)][exp(−*βV*¯ (*l*)(*<sup>x</sup>*1 − *<sup>x</sup>*3)) − 1 + *βV*(*l*)(*<sup>x</sup>*1 − *<sup>x</sup>*3)]+(... (256)

and so on. The result, presented above, can be obtained by means of slightly formal calculations, based on generalized functions and operator theories [22,115,118,147]. Really, as ¯*h* → 0 one has that

$$\mathcal{L}^{(l)}(\mathbf{f}) = \exp[-\beta \mathbf{V}^{(l)}(\boldsymbol{\delta})] \mathcal{L}\_0(\mathbf{f}) \mathbf{Q}^{-1} = \tag{257}$$

$$\begin{split} &= \operatorname{tr}\left\{\exp(-\beta\mathcal{H}\_{0}^{(\mu)})\exp\left[-\frac{\beta}{2}\int\_{\mathbb{R}^{3}}d^{3}k\nu(k):\rho\_{k}\rho\_{-k}:\right]\exp[i\rho(\mathbf{f})]\right\} \\ &= \operatorname{tr}\left\{\exp(-\beta\mathcal{H}\_{0}^{(\mu)})\exp\left[\frac{\beta}{2}\int\_{\mathbb{R}^{3}}d^{3}k\nu(k)\int\_{\mathbb{R}^{3}}d^{3}x\rho(x)\right] \\ &\times\int\mathcal{D}(\omega)\exp\left[-\int\_{\mathbb{R}^{3}}d^{3}k\frac{2\pi^{2}}{\beta\nu(k)}\omega\_{k}\omega\_{-k}-\int\_{\mathbb{R}^{3}}d^{3}k2\pi i\omega\_{k}\rho\_{k}\right]\exp[i\rho(\mathbf{f})]\right\}Q^{-1} \\ &= \int\mathcal{D}(\omega)f(\omega)\operatorname{tr}\left\{\exp(-\beta\mathcal{H}\_{0}^{(\mu)})\exp\left[i\left(\rho,\mathfrak{f}-2\pi\int\_{\mathbb{R}^{3}}d^{3}k\omega\_{k}\exp(ikx)-\frac{i\beta}{2}\int\_{\mathbb{R}^{3}}d^{3}k\nu(k)\right)\right]\right\}Q^{-1} \\ &= \int\mathcal{D}(\omega)f(\omega)\mathcal{L}\_{0}(\mathfrak{f}-2\pi\int\_{\mathbb{R}^{3}}d^{3}k\omega\_{k}\exp(ikx)-\frac{i\beta}{2}\int\_{\mathbb{R}^{3}}d^{3}k\nu(k))Q^{-1} \\ &= \int\mathcal{D}(\omega)f^{(l)}(\omega)\exp\left(\int\_{\mathbb{R}^{3}}d^{3}k\{\exp[i\mathfrak{f}(x)]-1\}g(x,\omega)\right). \end{split}$$

where H(*μ*) 0 := H0 − *<sup>λ</sup>* R<sup>3</sup> *<sup>d</sup>*3*xρ*(*x*), *ρk* :<sup>=</sup> R<sup>3</sup> *<sup>d</sup>*3*xρ*(*x*) exp(*ikx*), *k* ∈ R3. The expression (257) coincides exactly with that of (251), thereby proving the validity of our expressions (240) and (243) for the N.N. Bogolubov type generating functional L(f), f ∈ *F*, satisfying the functional Equation (242) of Proposition (8).

#### **8. The Functional-Analytic Solution and Its Ursell–Mayer Type Diagram Expansion**

Having considered (243) and (249) as starting expressions with just known functions *gn*(*<sup>x</sup>*1, *x*2, ...*xn*), *n* ∈ N, for the functional L(f), f ∈ *F*, one can obtain the following expansion:

$$\begin{split} \mathcal{L}(\mathbf{f}) &= Z(\mathbf{f})/Z(\mathbf{0}), \; Z(\mathbf{f}) = \exp[-\beta \mathbf{V}^{(\varepsilon)}(\delta)] \mathcal{L}^{(l)}(\mathbf{f}) \\ &= \exp[-\beta V^{(\varepsilon)}(\delta)] \exp\left[\sum\_{n=1}^{\infty} \frac{z^n}{n!} \int\_{\mathbb{R}^3} d^3 \mathbf{x}\_1 \int\_{\mathbb{R}^3} d^3 \mathbf{x}\_2 \dots \int\_{\mathbb{R}^3} d^3 \mathbf{x}\_n \\ &\times \prod\_{j=1}^n \{\exp[i \mathbf{f}(\mathbf{x}\_j)] - 1\} \mathcal{g}\_n(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) \right] \\ &= \exp\left[\sum\_{N=1}^{\infty} \frac{1}{N!} W(\mathcal{G}\_N^{(c)})\right], \end{split} \tag{258}$$

where functionals *W*(*G*(*c*) *N* ), *N* ∈ N, are calculated via the following rule. Denote by *G*(*c*) *N* , *N* ∈ N, such a connected graph that: it consists of exactly *N* generalized vertices of [*γ*(*nj*)] type, *j* = 1, *N*, and <sup>∑</sup>*Nj*=<sup>1</sup> *nj* ordinary vertices of [*α*] type. Moreover, each vertex [*y*(*n*)] is necessarily connected with *n* vertices of type [*α*] by means of dashed lines each to other, and [*α*] vertices can be connected arbitrarily by means of uniform lines. If now, to attribute to each generalized [*γ*(*n*)]−vertex—the factor *gn*(*<sup>x</sup>*1, *x*2, ...*xn*), to each simple [*α*]−vertex—the factor *<sup>ς</sup>* R<sup>3</sup> *d*3*x* exp[*i*f(*x*)], and to the line connecting them—the factor {*exp*[−*βV*(*s*)(*xl*1 − *xl*2 )] − <sup>1</sup>}, then the obtained resulting expression will be exactly equal to the functional *W*(*G*(*c*) *N* ). The final summing up over all such connected graphs gives rise to the expression (257), where the factor 1/*N*! counts the symmetry order of the graph *G*(*c*) *N* under the generalized vertices permutations. It is evident that, by representing the factor exp[*i*f(*x*)], entering the vertex [*α*], as {exp[*i*f(*x*)] − 1} + 1, the expression (257) can easily be resumed into Ursell–Mayer type expressions but already with other suitable *gn*—functions, replacing the former ones, giving rise to expansions similar to (256), based already on the "screened" potential (255).

Thereby, we can formulate, taking into account the results of [20,68], the next proposition, characterizing the Bogolubov type generating functional L(f), f ∈ *F*, satisfying the functional Equation (242).

**Proposition 9.** *Let the Bogolubov type generating functional* L(f), f ∈ S( R3; <sup>R</sup>), *represented analytically as a series (258) of graph-generated functionals, satisfy the following conditions:*

*(i) continuity with respect to the natural topology on F*, |L(f)| ≤ 1, f ∈ *F*;

*(ii) positivity:* <sup>∑</sup>*nj*,*k*=<sup>1</sup> *cjc*<sup>∗</sup>*k*L(f*<sup>j</sup>* − f*k*) ≥ 0 *for any* f*j* ∈ *F and all cj* ∈ C, *j* = 1, *n*, *n* ∈ N;

*iii) symmetry and normalization conditions:* L∗(f) = L(−f) *for all* f ∈ *F and* L(0) = 1;

*(iv) translational-invariance:* L(f) = L(f*a*), *where* f*a*(*x*) := f(*x* − *<sup>a</sup>*), *x*, *a* ∈ R3, *for any* f ∈ S( R3; <sup>R</sup>);

*(v) cluster condition or, equivalently, the Bogolubov correlation decay:* lim*λ*→∞[L(f + g*λa*) − L(f*a*)L(g*λa*)] = 0, *a* ∈ R3, *for any* f, g ∈ *F*;

*(vi) density condition:* 1*i δ*L(f) *<sup>δ</sup>*f(*x*) |<sup>f</sup>=<sup>0</sup> = *ρ*¯ ∈ R+.

*Then the functional (258) solves the Bogolubov type functional equation (242), allowing the positive measure dμ*¯, *whose Fourier representation on the adjoint tempered generalized functions space F is exactly*

$$\mathcal{L}(\mathbf{f}) = \int\_{F'} d\boldsymbol{\upmu}(\boldsymbol{\upxi}) \exp[i\boldsymbol{\upxi}(\mathbf{f})],\tag{259}$$

*where convolution ξ*(f) :<sup>=</sup> R<sup>3</sup> *<sup>d</sup>*3*xξ*(*x*)f(*x*) *for ξ* ∈ *F and* f ∈ *F*.

The obtained result makes it possible to find the many-particle distribution functions (237) and apply them to constructing different thermodynamic functions important [56,65] for applications.

Below, following the Bogolubov method [86], we obtain, based on the expression (245), the important Kirkwood–Saltzbourg–Simansic functional equation for the Bogolubov generating functional L(f), f ∈ *F*. Namely, making use of the expression (245) we can write down the following relationship:

$$\frac{1}{i}\frac{\delta \mathcal{L}\_{(N+1)}(\mathbf{f})}{\delta \mathbf{f}(\mathbf{x})} = \exp[i\mathbf{f}(\mathbf{x})] \frac{(N+1)Z\_N}{Z\_{N+1}} \mathcal{L}\_{(N)}(\mathbf{f}(\cdot) + i\beta V(\cdot - \mathbf{x})) \tag{260}$$

for any *x* ∈ R3, where *Z N* := R3*<sup>N</sup> dx*1*dx*2...*dx N* exp(−*βVN*), *N* ∈ N.

Since, by definition, lim *N* → ∞ <sup>L</sup>(*N*)(f) = L(f), f ∈ *F*, lim *N* → ∞ (*<sup>N</sup>*+<sup>1</sup>)*ZN ZN*+<sup>1</sup> := *ς* ∈ R+, from (260) one gets right away that

$$\exp\left[-i\mathbf{f}(\mathbf{x})\right]\frac{1}{i}\frac{\delta\mathcal{L}(\mathbf{f})}{\delta\mathbf{f}(\mathbf{x})} = \pounds\mathcal{L}(\mathbf{f}(\cdot) + i\beta V(\cdot - \mathbf{x})),\tag{261}$$

which is called the Kirkwood–Saltzburg–Symanzik functional equation, being very important for proving the Proposition (9) by means of the classical Leray–Schauder fixed point theorem [56,141,148] in some suitably defined Banach space. In particular, at f = 0 from (261) one finds the following important relationship:

$$
\bar{\rho} = \emptyset \mathcal{L}(i\beta V(\cdot - x))\tag{262}
$$

for any *x* ∈ R3.

*Conclusions*

In the article, we have showed that the N.N. Bogolubov generating functional method is a very effective tool for studying distribution functions of both equilibrium and non equilibrium states of classical many-particle dynamical systems. In some cases, the N.N. Bogolubov generating functionals can be represented by means of infinite Ursell–Mayer diagram expansions, whose convergence holds under some additional constraints on a statistical system. We also have shown that the Bogolubov idea [56] to use the Wigner density operator transformation to study the non equilibrium distribution functions proved to be very effective, having proposed a new analytic form of non-stationary solutions to the classical N.N. Bogolubov evolution functional equation.

#### **9. The Wigner Type Current Algebra Representation and Its Application to Non-Equilibrium Classical Statistical Mechanics**

#### *9.1. Many-Particle Distribution Functions Space and Its Poissonian Structure*

In the case of non-stationary (non-equilibrium) states of the many-particle dynamical systems, the Bogolubov's generating functional (236) does not possess all needed information. To specify this case, we introduce the generating representation functional:

$$\mathcal{L}(\mathbf{f}, \mathbf{g}) = (\Omega | \exp[i\rho(\mathbf{f})] \exp[iI(\mathbf{g})] \Omega) = \text{Tr}(P \exp[i\rho(\mathbf{f})] \exp[i\, J(\mathbf{g})]),\tag{263}$$

where Ω ∈ <sup>Φ</sup>*μ* is a cyclic vector of the representation of the current group *G*, satisfying the following additional conditions:

$$\text{T}\rho(\text{f})\text{T}^{-1} = \rho(\text{f}), \quad \text{T}\Omega = \Omega^\*, \text{ T}\text{f}(\text{g})\text{T}^{-1} = -\text{f}(\text{g}), \quad \text{T}\text{HT}^{-1} = \text{H}\_2$$

with the mapping *T* : R *t* → −*t* ∈ R being the operator of time inversion, and *f* ∈ J (R3; <sup>R</sup>), g ∈ J (R3; R<sup>3</sup>) taken arbitrary. In the *<sup>N</sup>*-particle representation of the current Lie algebra G (29) for any finite *N* ∈ N the functional L(*f* , g) (263) allows the following [5–7] standard finite-particle form:

$$\mathcal{L}(\mathbf{f}, \mathbf{g}) = \int\_{\mathbb{R}^3} d\mathbf{x}\_1 \dots \int\_{\mathbb{R}^3} d\mathbf{x}\_N \Omega^\*(\mathbf{x}\_1, \dots, \mathbf{x}\_N) \prod\_{j=1}^N \exp[i \mathbf{f}(\mathbf{x}\_j)] \times \tag{264}$$
 
$$\times \exp[i \xi(\mathbf{x}\_j, \mathbf{g})] \Omega(\mathbf{x}\_1, \dots, \mathbf{x}\_N),$$

where *ξ*(*<sup>x</sup>*, g) = 12*i* [g(*x*)∇*x* + ∇*x*g(*x*)], *x* ∈ R3, and Ω ∈ *<sup>L</sup>*2(R3*N*; C) is a cyclic state. The operator exp[*<sup>i</sup>ξ*(*<sup>x</sup>*, g)] acts on any function *ωN* ∈ *<sup>L</sup>*2(R3*N*; C) by the rule:

$$\exp\left[i\xi(\mathbf{x},\mathbf{g})\right]\omega\_N(\mathbf{x}\_1,\dots,\mathbf{x}\_N) = (\phi^\*\omega\_N)(\mathbf{x}\_1,\dots,\mathbf{x}\_N)\left(\det\left|\left|\left|\frac{\partial\phi(\mathbf{x})}{\partial\mathbf{x}}\right|\right|\right|\right)^{1/2}$$

where *φ* ∈ Di*f f*(R<sup>3</sup>) is a diffeomorphism of R3, corresponding to the vector field g ∈ J (R3; <sup>R</sup><sup>3</sup>), that is *φ*(*x*) = *φ*g*t* , where *ddtφ*<sup>g</sup>*t* = <sup>g</sup>(*φ*<sup>g</sup>*t* (*x*)), *x* ∈ R3. For *N* → ∞ the expression (264) becomes

$$\mathcal{L}(\mathbf{f}, \mathbf{g}) = \sum\_{n \in \mathbb{Z}\_+} \frac{1}{n!} \int d\mathbf{x}\_1 \dots \int d\mathbf{x}\_n \int d\mathbf{y}\_1 \dots \int d\mathbf{y}\_n \prod\_{j=1}^n [\delta(\mathbf{x}\_j - \mathbf{y}\_j) \times \tag{265}]$$

$$\times \left\{ \exp[i\mathbf{f}(\mathbf{x}\_j)] \exp[i\mathbf{f}(\mathbf{x}\_j, \mathbf{g})] - 1 \right\} f\_\mathbb{R}(y\_1, \dots, y\_n; \mathbf{x}\_1, \dots, \mathbf{x}\_n) \right\}$$

where for all *n* ∈ N Bogolubov's quantum distribution functions [56] are

$$f\_n(y\_1, \ldots, y\_n; \mathbf{x}\_1, \ldots, \mathbf{x}\_n) = (\Omega | \psi^+(y\_n) \ldots \psi^+(y\_1) \psi(\mathbf{x}\_1) \ldots \psi(\mathbf{x}\_n) \Omega) \tag{266}$$

and satisfy the compatibility conditions

$$f\_n(\mathbf{x}\_1, \dots, \mathbf{x}\_n) := f\_n(\mathbf{x}\_1, \dots, \mathbf{x}\_n; \mathbf{x}\_1, \dots, \mathbf{x}\_n),\tag{267}$$

where *xj* = R3, *j* = 1, *n*, *n* ∈ N.

To proceed with the further study of the classical distribution functions of the manyparticle dynamical system, when the inverse temperature *β* → 0, and the Planck constant → 0. Let us introduce [21,22,149,150] the following quantized selfadjoint Wigner operator *<sup>w</sup>*(*<sup>x</sup>*, *p*) : Φ*W* → Φ*W*, (*<sup>x</sup>*, *p*) ∈ *T*∗(R<sup>3</sup>)

$$w(\mathbf{x}, p) = \frac{1}{(2\pi)^n} \int\_{\mathbb{R}^3} d\mathbf{a} \exp(i \langle a|p\rangle) \psi^+ \left(\mathbf{x} + \frac{\hbar a}{2}\right) \Psi\left(\mathbf{x} - \frac{\hbar a}{2}\right),\tag{268}$$

where, by definition, Φ*W* := lim*β*<sup>→</sup>∞ <sup>Φ</sup>*μ* is the corresponding Hilbert space for the constructed Wigner type current algebra representation with the generating cyclic vector Ω ∈ Φ*W*. Performing transformation (268) in the expression (263), we can find that

$$\begin{split} \mathcal{L}(\mathbf{f}, \mathbf{g}) &\to \mathcal{L}(\widehat{\mathbf{f}}) = \sum\_{n \in \mathbb{Z}\_{+}} \frac{1}{n!} \int\_{\mathbb{R}^{3} \times \mathbb{R}^{3}} d\mathbf{x}\_{1} d\mathbf{p}\_{1} \dots \times \\ &\times \int\_{\mathbb{R}^{3} \times \mathbb{R}^{3}} d\mathbf{x}\_{n} d\mathbf{p}\_{n} \prod\_{j=1}^{n} \{ \exp(i\mathbf{\bar{f}}(\mathbf{x}\_{j}, \mathbf{p}\_{j})) - 1 \} f\_{\mathbf{n}}(\mathbf{x}\_{1}, \mathbf{p}\_{1}; \dots; \mathbf{x}\_{n}, \mathbf{p}\_{n}), \end{split} \tag{269}$$

for some functions ˜ f ∈ J (R<sup>3</sup> × R3; <sup>R</sup><sup>3</sup>). From the expression (269) it also follows that

$$\mathcal{L}(\mathbf{f}) = (\Omega | \exp[iw(\mathbf{f})] \Omega) = \text{Tr}(P \exp[iw(\mathbf{f})]) \tag{270}$$

where *w*(f) = *T*∗(R<sup>3</sup>) *dxdpw*(*<sup>x</sup>*, *p*)f(*<sup>x</sup>*, *p*), ˜ f ∈ J (R<sup>3</sup> × R3; <sup>R</sup><sup>3</sup>), *P* : Φ*W* → Φ*W* is the Gibbs

statistical operator and Tr: End(<sup>Φ</sup>*W*) → C is the corresponding trace-operator, defined on the space B(<sup>Φ</sup>*W*) of the nuclear operators on the corresponding Hilbert space representation Φ*W*. The corresponding quantum current Lie algebra G suitably transforms [55,56] into the Abelian Lie algebra G*W* of the operator functionals {*w*(f) ∈ G : f ∈ J (R<sup>3</sup> × R; <sup>R</sup>)}.

Consider now a quantum dynamical system of many identical particles with the average nonvanishing density *ρ*¯ = limΛ!R<sup>3</sup>(*N*/*A*) ∈ <sup>R</sup>1+\{0} as *N* → ∞ and Λ ! R<sup>3</sup> in the Van Hove's sense [150,151]. Then, according to [1,4,59], the Hamiltonian operator (234) in the Wigner representation (268) looks as

$$\mathrm{IH} = \int\limits\_{T^\*(\mathbb{R}^3)} dz \frac{p^2}{2m} w(z) + \int\limits\_{T^\*(\mathbb{R}^3)} dz \int\limits\_{T^\*(\mathbb{R}^3)} dz' V(x - y'): w(z)w(z'):,\tag{271}$$

where *z* = (*<sup>x</sup>*, *p*) ∈ *<sup>T</sup>*<sup>∗</sup>(R<sup>3</sup>), *z* = (*y*, *q*) ∈ *T*∗(R<sup>3</sup>) and *dz* = *dxdp*, *dz* = *dydq* are the standard phase space measures in *T*∗(R<sup>3</sup>) and the ordering : : operation is naturally inherited from (238). According to the Heisenberg's principle [56], the evolution equation with respect to temporal variable *t* ∈ R+ for an arbitrary observable operator quantity A : Φ*W* → Φ*W* in the Wigner type representation space Φ*W* is

$$d\mathbf{A}/dt = \frac{i}{\hbar}[\mathbf{H}, \mathbf{A}],\tag{272}$$

where [·, ·] is a usual operator commutator, naturally ensued from that on the Hilbert space <sup>Φ</sup>*μ*. Following [20,56,152–154], one can state, that for → 0 in the weak sense the following theorem is true.

**Theorem 5.** *Let us denote* M *as an algebra of the self-adjoint operators with A*(G) *in the Wigner representation. Then, the operator bracket* [·, ·]0 = lim→<sup>0</sup>[·, ·] *on the algebra* M *in the weak sense is equivalent to*

$$\left[a\_j, a\_n\right]\_0 = \sum\_{k=1}^{\min\{j, n\}} \int\_{T^\*(\mathbb{R}^3)} dz\_1 \dots \int\_{T^\*(\mathbb{R}^3)} dz\_k : w(z\_1) \dots w(z\_k) \times \tag{273}$$

$$\times \left\{ \frac{\delta^k a\_j}{\delta w(z\_1) \dots \delta w(z\_k)}, \frac{\delta^k a\_n}{\delta w(z\_1) \dots \delta w(z\_k)} \right\}^{(k)} ; \rho$$

*where* {·, ·}(*k*) *is a standard canonical Poisson bracket on the phase space of k* ∈ N *particles.*

The statement (273) could be proved by means of the next general Bohr–Dirac correspondence principle in the quasi-classical approach:

$$\lim\_{\hbar \to 0} \frac{i}{\hbar} [a, b] = \{a, b\}^{(N)},\tag{274}$$

where *N* ∈ N is a maximal number of the particle in the system and *a*, *b* ∈ *A*(G) are operators in *<sup>N</sup>*-particle Hilbert space representation Φ*N* = *<sup>L</sup>*2(R3*N*; C), *F* = *N* ∑ *j*=1 *<sup>δ</sup>*(*x* − *xj*).

Here, it is worth making the following corollary.

*Corollary.* Algebra of the operators of the observable quantities *A*(G) for → 0 allows "*hierarchical*" representation

$$A(\mathcal{G}) = \sum\_{j \in \mathbb{Z}\_+} A\_j(\mathcal{G}) \Rightarrow \mathcal{M} = \underset{j \rightharpoonup \mathbb{Z}\_+}{\oplus} A\_j(\mathcal{G}) \tag{275}$$

along with Lie bracket [[·, ·]], which is inducted by the bracket [·, ·]0 (273):

$$\left[\left[a,b\right]\right] = \underset{l\in\mathbb{Z}\_+}{\oplus} \sum\_{j,k\in\mathbb{Z}\_+} \left[a\_{j'},b\_k\right]\_{0}^{(l)}\,.\tag{276}$$

where *a*, *b* ∈ M in the Wigner representation and the following expansions hold

$$a = \sum\_{j \in \mathbb{Z}\_+} a\_{j\prime} \quad b = \sum\_{j \in \mathbb{Z}\_+} b\_{j\prime} \quad \left[ a\_{j\prime} b\_k \right]\_0 = \sum\_{l \in \mathbb{Z}\_+} \left[ a\_{j\prime} b\_k \right]\_0^{(l)}.\tag{277}$$

Consider now the following linear mapping *α* : M → *<sup>A</sup>*(G), where

$$a(\underset{j \in \mathbb{Z}\_+}{\oplus} a\_j) = \sum\_{j = \mathbb{Z}\_+} a\_j \in A(\mathcal{G}),\tag{278}$$

and the Lie bracket [[·, ·]] is defined in M, and the corresponding Lie bracket [·, ·]*α* (273) in *<sup>A</sup>*(G). Let us consider the dual to (278) mapping *α*<sup>∗</sup> : *A*(G)∗ → M∗, where

$$\mathcal{M}^\* = \bigoplus\_{l \in \mathbb{Z}\_+} \mathcal{M}^\*\_{j'}, \quad \mathcal{M} = \bigoplus\_{l \in \mathbb{Z}\_+} \mathcal{M}\_{j'} \tag{279}$$

$$\mathcal{M}^\* = \sum\_{j \in \mathbb{Z}\_+} \{ P \in A\_j(\mathcal{G})^\* \, : \, F(a) = \text{Tr}(Pa), \,\, a \in A(\mathcal{G}) \}.$$

Here *P* : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ* is statistic operator of the initial dynamical system (271), which satisfy the Heisenberg–Liouville equation

$$dP/dt = \frac{i}{\hbar}[P, \mathcal{H}] \tag{280}$$

for all *t* ∈ R+. The expression (280), according to (274), transforms into the quasi-classical Liouville equation in the Wigner representation.

It is easy to check that for element *F* ∈ *A*(M)∗ the expression

$$
\mathfrak{a}^\* F = (f\_1, \dots, f\_{\mathfrak{f}'}, \dots) = \mathcal{F} \in \mathcal{M}^\* \tag{281}
$$

defines the representation on the space M∗ of the distribution functions

$$f\_{\rangle}(z\_1, \ldots, z\_{\rangle}) = \text{Tr}(P : w(z\_1) \ldots w(z\_{\rangle}) : ), \tag{282}$$

where *zj* ∈ *<sup>T</sup>*<sup>∗</sup>(R<sup>3</sup>), *j* ∈ Z+, and for any *a* ∈ M

$$a(\mathcal{F}) = \sum\_{j \in \mathbb{Z}\_+} \int\_{T^\*(\mathbb{R}^3)} dz\_1 \dots \int\_{T^\*(\mathbb{R}^3)} dz\_j f\_j(z\_1, \dots, z\_j) a\_j(z\_1, \dots, z\_j) \,. \tag{283}$$

Let *b*(*F*), *c*(*F*) ∈ *D*(*A*(G)∗) be linear functionals on *<sup>A</sup>*(G)<sup>∗</sup>, then on *D*(*A*(G)∗ ) the standard [59] Lie-Poisson bracket {·, ·}0 is defined via the rule

$$\{b(F), c(F)\}\_0 = F([b, c]\_0),\tag{284}$$

where *b*, *c* ∈ *A*(G) are such that *F*(*b*) = *b*(*F*), *<sup>F</sup>*(*c*) = *<sup>c</sup>*(*F*), *F* ∈ *<sup>A</sup>*<sup>∗</sup>(G). In the same way, the dual Lie–Poisson bracket {{·, ·}} is defined on the set of functionals *D*∗(M) over the adjoint space M (279)

$$\{\{b(\mathcal{F}), \mathfrak{c}(\mathcal{F})\}\} = F(\left[b, \mathfrak{c}\right]),\tag{285}$$

where *F*(*b*) = *b*(F), *<sup>F</sup>*(*c*) = *<sup>c</sup>*(F), *F* ∈ M∗.

**Definition 16.** *It is said that mapping of the Lie algebras α* : M → *A*(G) *is canonical (or Poissonian [59]), if for all b*(F) *and c*(F) *the following equality holds*

$$\{a^\*\{b(\mathcal{F}), c(\mathcal{F})\}\}\_0 = \{\{a^\*b(\mathcal{F}), a^\*c(\mathcal{F})\}\},\tag{286}$$

*where* F = *α*<sup>∗</sup>*F* ∈ M∗*.* From reasonings presented above we can formulated the following proposition.

**Proposition 10.** *Let A and* M *be two arbitrary Lie algebras and α* : M → *A be a linear mapping. Then dual mapping α*<sup>∗</sup> : *D*(*A*(G)∗ ) → *D*(M∗) *is canonical if α* : M → *A is Lie algebras homomorphism.*

As a consequence of the statement above, one derives the next theorem.

**Theorem 6.** *Dual mapping α*<sup>∗</sup> : *D*(*A*(G)∗ ) → *<sup>D</sup>*(M∗)*, which was built by means of the hierarchical Lie algebra of the operators* M, *is canonical.*

Let us consider the generating functional L(f), f ∈ J (*T*<sup>∗</sup>(R<sup>3</sup>); <sup>R</sup>), defined by expression (270) in Wigner representation, and apply the developed above algebraic technique to the calculation of the following quantity:

$$\frac{d}{dt}\mathcal{L}(\mathbf{f}) = \lim\_{\hbar \to 0} \frac{i}{\hbar} \text{Tr}(P[\mathbf{H}\_\prime \text{: } \exp[iw(e^{\mathrm{if}} - 1)]) \text{ :]}) \tag{287}$$

for the evolution with respect to the temporal parameter *t* ∈ R. From (270) one can easily obtain that

$$\frac{d}{dt}\mathcal{L}(\mathbf{f})(\mathcal{F}) = \text{Tr}(P[\mathbf{H}, :\exp(iw(e^{\mathbf{i}\mathbf{f}} - 1)): ]) = a^\* \{\mathcal{H}(\mathcal{F}), \mathcal{L}(\mathbf{f})(\mathcal{F})\}\_{0'} \tag{288}$$

where for all F ∈ *A*(G)∗ the Hamiltonian functional H(F) ∈ *D*(*A*(G)∗ ) is given as

$$\mathcal{H}(\mathcal{F}) = \text{Tr}(\text{PH}) = \int\limits\_{T^\*(\mathbb{R}^3)} dz T(p) f\_1(z) + \tag{289}$$

$$+ \frac{1}{2} \int\limits\_{T^\*(\mathbb{R}^3)} dz\_1 \int\limits\_{T^\*(\mathbb{R}^3)} dz\_2 V(x\_1 - x\_2) f\_2(z\_1, z\_2).$$

Based on (288) and Theorem 6, we immediately obtain the Hamiltonian evolution equation

$$\frac{d}{dt}\mathcal{L}(\mathbf{f})(\mathcal{F}) = \{\{\mathcal{L}(\mathbf{f})(\mathcal{F}), \mathcal{H}(\mathcal{F})\}\},\tag{290}$$

where *t* ∈ R, L(f)(F) = *<sup>α</sup>*∗L(f)(*F*), H(F) = *α*<sup>∗</sup>*H*(*F*) and F∈M∗ is arbitrary. Thus, the following theorem is stated.

**Theorem 7.** *The generating Wigner type representation functional* L(f)(F) *(270) on the phase space D*(M) *satisfies the Hamiltonian dynamical system (290) with respect to the Lie–Poisson bracket (285) and Hamiltonian function (289), taken as a smooth functional on* M∗.

Using Equation (290) and formulae (273), (276), we finally ge<sup>t</sup> the following nonequilibrium functional Bogolubov's equation [143]

$$\frac{d}{dt}\mathcal{L}(\mathbf{f}) = \int\limits\_{T^\*(\mathbb{R}^3)} dz \{\frac{1}{i} \frac{\delta \mathcal{L}(\mathbf{f})}{\delta \mathbf{f}(z)}, T(p) \}^{(1)} + \\ \tag{291}$$

$$+ \frac{1}{2} \int\limits\_{T^\*(\mathbb{R}^3)} dz\_1 \int\limits\_{T^\*(\mathbb{R}^3)} dz\_2 \{\div \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(z\_1)} \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(z\_2)} \ast V(\mathbf{x}\_1 - \mathbf{x}\_2) \}^{(2)} \mathcal{L}(\mathbf{f}),$$

where for any *n* ∈ N

$$\frac{1}{i} : \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(z\_1)} \dots \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(z\_n)} := \prod\_{j=1}^n \left[ \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(z\_j)} - \sum\_{k=1}^j \delta (z\_j - z\_k) \right] \tag{292}$$

and, by definition, {·, ·}(*j*) denotes the standard canonical Poisson bracket on the phase space *T*∗(R<sup>3</sup>)*<sup>j</sup>* for all *j* ∈ Z+.

Taking into account that for functional L(f), f ∈ J (*T*<sup>∗</sup>(R<sup>3</sup>); <sup>R</sup>), there exists the unlimited expansion (269):

$$\mathcal{L}(\mathbf{f}) = \sum\_{n \in \mathbb{Z}} \frac{1}{n!} \int\_{T^\*(\mathbb{R}^3)} dz\_1 \dots \int\_{T^\*(\mathbb{R}^3)} dz\_n \prod\_{i=1}^n \{ \exp[\mathrm{if}(z\_j) - 1] \} f\_n(z\_1, \dots, z\_n), \tag{293}$$

from (291), we obtain the kinetic equations for the hierarchy of the Bogolubov distribution functions [143]:

$$\frac{\partial}{\partial t} f\_n(z\_1, \dots, z\_n) = \{f\_n(z\_1, \dots, z\_n), H\_n(z\_1, \dots, z\_n)\}^{(n)} + \tag{294}$$

$$\begin{split} \frac{1}{T^\*(\mathbb{R}^3)} \quad \quad \frac{1}{T^\*(\mathbb{R}^3)} \quad \quad \frac{1}{T^\*(\mathbb{R}^3)} \end{split} \tag{295}$$

where *zj* ∈ R3, *j* = 1, ..., *n*, are the coefficients of the *n*-particle cluster in R3, *Hn*(*<sup>z</sup>*1, ..., *zn*) denotes its corresponding energy:

$$H\_{\rm ll}(z\_1, \ldots, z\_{\rm ll}) = \sum\_{j=1}^{n} \frac{p\_j^2}{2m} + \frac{1}{2} \sum\_{j \neq k=1}^{n} V(x\_j - x\_k) \tag{295}$$

Thus, the problem of the construction of the kinetic theory by Bogolubov is reduced to finding the special solutions of the unlimited hierarchy of the Equations (294), where the selection criterion is based on Bogolubov's fundamental weakening correlation principle:

$$\lim\_{\|\langle n\rangle - \langle m\rangle\| \to \infty} |f\_{n+m}(z\_1, \dots, z\_{n+m}) - f\_n(z\_1, \dots, z\_n)f\_m(z\_{n+1}, \dots, z\_{n+m})| \to 0 \tag{296}$$

where *n* − *m* = *dist*({*zi* ∈ *T*∗(R<sup>3</sup>) : *i* = 1, ..., *<sup>n</sup>*}, {*zi*+*n* ∈ *T*∗(R<sup>3</sup>) : *i* = 1, ..., *m*}) is a distance between two clusters with *n* ∈ Z+ and *m* ∈ Z+ numbers of the particles. If a special solution of the hierarchy (289) exists in the functional form

$$f\_n(z, \ldots, z\_n; t) = f\_n(z\_1, \ldots, z\_n; f\_1(z; t))\tag{297}$$

for all *t* ∈ R+ and *n* ∈ Z+, then the corresponding equation for one-particle distribution function of the system in the external field *V*0 :R<sup>3</sup> → is the following

$$\frac{\partial}{\partial t} f\_1(z;t) + \langle p/m | \nabla\_x f\_1(z;t) \rangle + \langle \nabla\_x V\_0(x) | \nabla\_p f\_1(z;t) \rangle = f(f\_1(z;t)),\tag{298}$$

where *J*(*f*1(*<sup>z</sup>*; *t*)) is the so called "collision integral" [56,143,149,150,155], and is called the kinetic Boltzmann equation [56,149,156]. Below, we will focus on the such special solutions of the Bogolubov's hierarchy of the Equations (294), using the above developed algebraic method of Bogolubov's generating functional.

*9.2. Generating Representation Functional and Its Solution Space Structure*

Let us consider Bogolubov's functional Equation (291)

$$\frac{d}{dt}\mathcal{L}(\mathbf{f}) = \int\_{T^\*(\mathbb{R}^3)} dz \left\{ \frac{1}{i} \frac{\delta \mathcal{L}(\mathbf{f})}{\delta \mathbf{f}(z)}, T(p) \right\}^{(1)} + \tag{299}$$

$$+ \frac{1}{2} \int\_{T^\*(\mathbb{R}^3)} dz\_1 \int\_{T^\*(\mathbb{R}^3)} dz\_2 \left\{ : \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(z\_1)} \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(z\_2)} : \mathcal{L}(\mathbf{f}), \mathcal{V}(x\_1 - x\_2) \right\}^{(2)},$$

generated by the statistical operator evolution

$$P(t, t\_0) = \exp\left[\frac{i}{\hbar}(t\_0 - t)\mathbb{H}\right] \bar{P} \exp\left[\frac{i}{\hbar}(t - t\_0)\mathbb{H}\right] \tag{300}$$

for *t*, *t*0 ∈ R, solving the Heisenberg evolution equation

$$\frac{dP}{dt} = \frac{i}{\hbar}[P, H]\_{\prime} \quad P\bigg|\_{t=t\_0} = P \tag{301}$$

for the statistical Gibbs operator *P* : Φ*W* → Φ*W* with Tr*P* ¯ = 1.

When → 0 in the Wigner representation, the expression (300), as an explicit solution of the (301), allows the following expansion

$$\begin{split} \mathcal{L}(\mathbf{f}) &= \text{Tr}\left( \exp\left[ \frac{i}{\hbar}(t\_0 - t)\mathbf{H} \right] \bar{\mathbf{P}} \exp\left[ \frac{i}{\hbar}(t - t\_0)\mathbf{H} \right] \exp[iw(\mathbf{f})] \right) = \\ &= \text{Tr}\left( \exp\left[ \frac{i}{\hbar}(t\_0 - t)(\mathbf{H}\_0 + \mathbf{V}) \right] \bar{\mathbf{P}} \exp\left[ \frac{i}{\hbar}(t - t\_0)(\mathbf{H}\_0 + \mathbf{V}) \right] \exp[iw(\mathbf{f})] \right) \Big|\_{\mathbf{H} \to \mathbf{0}} \\ &= \text{Tr}\left( \exp\left[ \frac{i}{\hbar}(t - t)\mathbf{H}\_0 \right] \bar{\mathbf{P}} \exp\left[ \frac{i}{\hbar}(t - t\_0)\mathbf{H}\_0 \right] \exp[\pi(t, t\_0)] \exp[iw(\mathbf{f})] \right). \end{split}$$

where we denoted H = H0 + V,

$$\mathcal{H}\_0 = \int \, dz \frac{p^2}{2m} w(z), \mathcal{V} = \int \, dz\_1 \, \int \, dz\_2 \, V(x\_1 - x\_2) : w(z\_1) w(z\_2) \, ; \,\tag{303}$$

$$\exp[\pi(t, t\_0)] = P\_0(t\_0, t) P(t, t\_0), \quad P\_0(t\_0, t) = \exp\left[\frac{i}{\hbar}(t\_0 - t) \mathcal{H}\_0\right] \bar{\mathcal{P}} \exp\left[\frac{i}{\hbar}(t - t\_0) \mathcal{H}\_0\right].$$

The operator *<sup>π</sup>*(*<sup>t</sup>*, *<sup>t</sup>*0), *t*, *t*0 ∈ R, in (303) is called a "cluster operator" and allows the next expansion into the unlimited series:

$$\pi(t\_0, t) = \sum\_{n \in \mathbb{Z}\_+} \frac{1}{n!} \int\limits\_{T^\*(\mathbb{R}^3)} dz\_1 \dots \int\limits\_{T^\*(\mathbb{R}^3)} dz\_n \,\pi\_n(z\_1, \dots, z\_n; t, t\_0) \times \\ \tag{304}$$

$$\times: w(z\_1)...w(z\_n) := \pi(t, t\_0; w),$$

where the functions *<sup>π</sup>n*(*<sup>z</sup>*1, ..., *zn*; *t*, *<sup>t</sup>*0), *n* ∈ N, can be defined uniquely form the representation (303) under the condition that the Gibbs operator *P*¯ : Φ*W* → Φ*W* is defined explicitly

in the Wigner representation. Thus, from (302)–(304) we obtain the following expressions for the Bogolubov's generating functional

$$\mathcal{L}(f) = \text{Tr}(P\_0 \exp[\pi(t, t\_0; w)] \exp(iw))\Big|\_{\hbar \to 0} = \tag{305}$$

$$= \exp\left[\pi\left(t, t\_0; \frac{1}{i}\frac{\delta}{\delta \mathbf{f}}\right)\right] \text{Tr}(P\_0 \exp[iw(\mathbf{f})]) = \exp\left[\pi\left(t, t\_0; \frac{1}{i}\frac{\delta}{\delta \mathbf{f}}\right)\right] \mathcal{L}\_0(\mathbf{f}),$$

where L0(f), f ∈ J (*T*<sup>∗</sup>(R<sup>3</sup>); <sup>R</sup>), is a generating functional if the initial dynamical system of the particles under absent of interaction, that is

$$\begin{aligned} \mathcal{L}\_0(\mathbf{f})(t, t\_0) &= \sum\_{n \in \mathbb{Z}\_+} \frac{1}{n!} \int \limits\_{T^\*(\mathbb{R}^3)} dz\_1 \dots \int \limits\_{T^\*(\mathbb{R}^3)} dz\_n \times \\ \times f\_n\left(\mathbf{x}\_1 + \frac{p\_1}{m}(t\_0 - t), p\_1; \dots; \mathbf{x}\_n + \frac{p\_n}{m}(t\_0 - t), p\_1\right) \prod\_{j=1}^n \left\{ \exp\left[i\mathbf{f}(z\_j)\right] - 1 \right\}. \end{aligned} \tag{306}$$

Applying to (306) when *t*0 → −∞ Bogolubov's correlation weakening (296), we obtain that for all *t* ∈ R+

$$\mathcal{L}o(\mathbf{f})(t) = \exp\left[\int\_{T^\*(\mathbb{R}^3)} dz f\_1(\mathbf{x} - \frac{p}{m}t; p) \{\exp[i\mathbf{f}(z)] - 1\}\right],\tag{307}$$

where L0(f)(*t*) = lim *t*0→−<sup>∞</sup> L0(f)(*<sup>t</sup>*, *<sup>t</sup>*0). Now, according to (305) and (307), we find that

$$\mathcal{L}(\mathbf{f})(t) = \exp\left[\pi \left(t, t\_0; \frac{1}{i} \frac{\delta}{\delta \mathbf{f}}\right) \mathcal{L}\_0(\mathbf{f})(t)\right] \tag{308}$$

is a solution of Bogolubov's functional Equation (299), where

$$\pi\left(t; \frac{1}{i}\frac{\delta}{\delta\mathbf{f}}\right) = \lim\_{t\_0 \to -\infty} \pi\left(t, t\_0; \frac{1}{i}\frac{\delta}{\delta t}\right) \tag{309}$$

for all *t* ∈ R+. To specify the form of the operators (309), we note that operator *ξ*(*<sup>t</sup>*, *t*0; *w*) = exp[*π*(*<sup>t</sup>*, *t*0; *w*)] for all *t*, *t*0 ∈ R+ satisfies under → 0 the following differential evolution relationship:

$$\frac{d\xi^{\circ}}{dt} = \frac{1}{i\hbar} [\xi, \mathcal{H}]\_0 + \lim\_{\hbar \to 0} \frac{1}{\hbar} \left(\mathcal{V} - P\_0^{-1} \mathcal{V} P\_0\right) \xi,\tag{310}$$

where all operators are assumed to be given in the Wigner representation. Expanding the operator *ξ*(*<sup>t</sup>*, *t*0, *w*) into the sum of *n*-particles components, *n* ∈ N, we find

$$\zeta\_{\sharp}^{\eta}(t, t\_0; w) = \sum\_{n \in \mathbb{Z}\_+} \frac{1}{n!} \int \limits\_{T^\*(\mathbb{R}^3)} dz\_1 \dots \int \limits\_{T^\*(\mathbb{R}^3)} dz\_n \, \mathbb{I}\_{\mathbb{R}^3}(z\_1, \dots, z\_n; t, t\_0) : w(z\_1) \dots w(z\_n) \, :\_{\prime} \tag{311}$$

and there is mutually unambiguous correspondence [56,59] between coefficient functions *ξn*(*<sup>z</sup>*1, ..., *zn*; *t*, *<sup>t</sup>*0) in (311) and coefficient functions in the expansion (304)

$$\pi\_n(z\_1, \ldots, z\_n) = \sum\_{\sigma \in \Sigma\_n} (-1)^{n+i} (\sigma - 1) \prod\_{j=1}^{\infty} \xi\_{\sigma}(z\_{\langle k \rangle} \in \sigma\_j), \tag{312}$$

$$\xi\_n(z\_1, \ldots, z\_n) = \sum\_{\sigma \in \Sigma\_n} \prod\_{j=1}^{\infty} \pi\_{\sigma\_j}(z\_{\langle k \rangle} \in \sigma\_j).$$

Here, *σ* ∈ Σ*n* is an arbitrary partition of the symmetry group Σ*n* of all permutations of the set of numbers {1, 2, ..., *n*} on the subsets {*<sup>σ</sup>j* : *j* = 1, ...,*<sup>s</sup>*}, which are not intersect, that is 2*n j*=1 *σj* = {1, ..., *n*} and *ξσj* and *πσj* , *j* = 1, ...,*s*, are the corresponding to this partition coefficient functions. In particular,

$$
\xi\_1^\mathfrak{z}\_1(z\_1) = \pi\_1(z\_1), \quad \pi\_2(z\_1, z\_2) = \xi\_2(z\_1, z\_2) - \xi\_1(z\_1)\xi\_1(z\_2)
$$

and so on. Thus, on the base of the defined operator series (304) or (311), the problem of the explicit calculations of the distribution functions become very simple. Below we will analyze these series by means of the language of Bogolubov's generating functional L(f), f ∈ J (*T*<sup>∗</sup>(R<sup>3</sup>); <sup>R</sup>), using Bogolubov's functional hypothesis [56,143,149,150,155].

#### *9.3. Bogolubov–Boltzmann Kinetic Equation in the Frame of Functional Hypothesis*

The generating functional, as it was stated above, is given by the expression

$$\mathcal{L}(\mathbf{f})(t) = \exp\left[\pi \left(t\_0; \frac{1}{i} \frac{\delta}{\delta \mathbf{f}}\right)\right] \mathcal{L}\_0(\mathbf{f})(t). \tag{313}$$

Here, L0(f)(*t*), *t* ∈ R+, is a generating functional of the system of non-interacting particles, which is equal to the expression (307) when *t*0 → − ∞. From (313), it follows that for all *t* ∈ R+ for the *n*-particle distribution function *fn*(*<sup>z</sup>*1, ..., *zn*; *t*) the general functional relationship holds

$$f\_n(z\_1, \ldots, z\_n; t) := f\_n(z\_1, \ldots, z\_n; f\_1(z; t)). \tag{314}$$

Respectively, the generating functional (313) satisfies, according to (290) when *t* = 0, the following dynamic equation:

$$\frac{d}{dt}\mathcal{L}(\mathbf{f}) = \int\_{T^\*(\mathbb{R}^3)} dz \left\{ \frac{1}{i} \frac{\delta \mathcal{L}(\mathbf{f})}{\delta \mathbf{f}(z)}, T(p) \right\}^{(1)} + \tag{315}$$

$$+ \frac{1}{2} \int\_{T^\*(\mathbb{R}^3)} dz\_1 \int\_{T^\*(\mathbb{R}^3)} dz\_2 \left\{ : \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(z\_1)} \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(z\_2)} : \mathcal{L}(\mathbf{f}), \mathcal{V}(\mathbf{x}\_1 - \mathbf{x}\_2) \right\}^{(2)},$$

Let us put *f*1(*z*) → *f*1(*<sup>z</sup>*; *<sup>τ</sup>*), where *τ* ∈ R− and that

$$\frac{\partial f\_1(z;\tau)}{\partial \tau} = \left\{ f\_1(z;\tau), T(p) \right\}^{(1)} + \int\_{T^\*(\mathbb{R}^3)} dz\_1 \{ f\_1(z\_1) f\_1(z), \mathcal{V}(x\_1 - x) \}^{(2)}.\tag{316}$$

Then from (315), we also obtain that

$$\frac{d}{d\tau}\mathcal{L}(\mathbf{f}) = \left\{ \{ \mathcal{L}(\mathbf{f}), \mathcal{H}(\mathcal{F}) \} \right\} = \int\_{T^\*(\mathbb{R}^3)} dz \left\{ \frac{1}{i} \frac{\delta \mathcal{L}(\mathbf{f})}{\delta \mathbf{f}(\mathbf{z})}, T(p) + \int\_{T^\*(\mathbb{R}^3)} dz\_1 f\_1(\mathbf{z}\_1), \mathcal{V}(\mathbf{x}\_1 - \mathbf{x}) \right\}^{(1)} \tag{317}$$

for all *τ* ∈ R−. Then Equation (317) can be rewritten in the following way:

$$\frac{d}{d\tau}\mathcal{L}(\mathbf{f}) = \left\{ \left\{ \mathcal{L}(\mathbf{f}), \,\,\bar{\mathcal{H}}(\mathcal{F}) \right\} \right\} = \int\_{T^\*(\mathbb{R}^3)} dz \left\{ \frac{1}{i} \frac{\delta \mathcal{L}(\mathbf{f})}{\delta \mathbf{f}(z)}, \bar{\mathcal{H}}(f\_1) \right\}^{(1)},\tag{318}$$

where, by definition, *H*˜ (*f*1) := *δ δ f*1 H˜(F) and

$$\mathcal{H}(\mathcal{F}) = \int\_{T^\*(\mathbb{R}^3)} dz \frac{p^2}{2m} f\_1(z) + \frac{1}{2} \int\_{T^\*(\mathbb{R}^3)} dz\_1 \int\_{T^\*(\mathbb{R}^3)} dz\_2 f\_1(z\_1) f\_1(z\_2) \mathcal{V}(x\_1 - x\_2),\tag{319}$$

is the Vlasov-type Hamiltonian of the self-consistent particles interaction. Let us define the following mapping on the phase space of *n* ∈ Z+ particles:

$$S\_n(\tau)x\_{\dot{j}} = x\_{\dot{j}}(\tau), \quad S\_n(\tau)p\_{\dot{j}} = p\_{\dot{j}}(\tau), \tag{320}$$

where for all *τ* ∈ R−, *j* = 1, *n*,

$$\begin{split} \frac{dx(\tau)}{dt} &= \left\{\hat{H}, x(\tau)\right\}^{(1)}, \quad \frac{dp(\tau)}{dt} = \left\{\hat{H}, p(\tau)\right\}^{(1)},\\ \hat{H} &= \sum\_{j=1}^{n} \frac{p\_j^2}{2m} + \frac{1}{2} \sum\_{j=k}^{n} \mathbf{V}(\mathbf{x}\_j - \mathbf{x}\_k). \end{split} \tag{321}$$

It easy to see that the system of Equations (321) gives the exact solution [157] for the dual Equation (316) in the form of the sum of *δ*-functions of *n* ∈ Z+ particles:

$$f\_1(z) = \sum\_{j=1}^n \delta(z - z\_j)\_\prime \tag{322}$$

where *zj* ∈ R3, *j* = 1, ..., *n*, are the coordinates of the cluster. Using (320) from (318) we obtain that for all *τ* ∈ R

$$\frac{d}{d\tau}\mathcal{L}(\mathbf{f})(\tau) = \left\{ \left\{ \mathcal{L}(\mathbf{f})(\tau), \tilde{\mathcal{H}}(\mathcal{F}) \right\} \right\} + \frac{1}{2} \int\_{\mathbb{R}^3 \times \mathbb{R}^3} dz\_1 \int\_{T^\*(\mathbb{R}^3)} dz\_2 \times \\ \tag{323}$$

$$\times \left\{ : \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(z\_1)} \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(z\_2)} : \mathcal{L}(\mathbf{f}), \mathbb{V}(x\_1 - x\_2)(\tau) \right\}^{(2)} ,$$

where we denoted

$$\begin{aligned} \mathcal{L}(\mathbf{f})(\tau) &= S(\tau)\mathcal{L}(\mathbf{f}|S(-\tau)f\_1), \\ \mathcal{V}(\mathbf{x}\_1 - \mathbf{x}\_2)(\tau) &= S(\tau)\mathcal{V}(S(-\tau)(\mathbf{x}\_1 - \mathbf{x}\_2)), \\ f\_2(z, z\_1)(\tau) &= S(\tau)f\_2(z\_1, z|S(-\tau)f\_1) \end{aligned} \tag{324}$$

Integrating the Equation (323) in limits *τ* ∈ (−∞, <sup>0</sup>), we obtain that

$$\begin{split} \mathcal{L}(\mathbf{f})|\_{\mathsf{T}=0} &= \lim\_{\mathsf{T}\to-\infty} S(\mathsf{r}) \mathcal{L}(\mathbf{f}|S(-\mathsf{r})f\_{1}) + \int\_{-\infty}^{0} d\mathsf{r} \Big\{ \frac{1}{i} \frac{\delta}{\delta\hat{f}(\mathbf{z})} \mathcal{L}(\mathbf{f})(\mathsf{r}), \vec{\mathcal{H}}(\mathcal{F}(\mathbf{r})) \Big\}^{(1)} + \\ &+ \frac{1}{2} \int\_{-\infty}^{0} d\mathsf{r} \int\_{\mathbb{R}^{3}\times\mathbb{R}^{3}} dz\_{1} \int\_{\begin{subarray}{c} \mathsf{T}^{\*}(\mathbb{R}^{3}) \\ T^{\*}(\mathbb{R}^{3}) \end{subarray}} dz\_{2} \Big\{ :\frac{1}{i} \frac{\delta}{\delta\hat{\mathbf{f}}(\mathbf{z}\_{1})} \frac{1}{i} \frac{\delta}{\delta\hat{\mathbf{f}}(\mathbf{z}\_{2})} : \mathcal{L}(\mathbf{f}), \mathsf{V}(\mathbf{x}\_{1} - \mathbf{x}\_{2})(\mathsf{r}) \Big\}^{(2)} \Big] \end{split} \tag{325}$$

We should also note here, that due to the Bogolubov's principle of correlations weakening (296) and using (307) the first item in (325) can be represented in the form

$$\begin{split} \lim\_{\tau \to -\infty} S(\tau) \mathcal{L}(\mathbf{f} | S(-\tau) f\_1) &= \lim\_{\tau \to \infty} \exp \left[ \int\_{T^\*(\mathbb{R}^3)} dz S(\tau) f\_1(z)(\tau) \{\exp[\mathrm{if}(z)] - 1\} \right] = \\ &= \exp \left[ \int\_{T^\*(\mathbb{R}^3)} dz f\_1(z) \{\exp[\mathrm{if} \, \mathbf{f}(z)] - 1\} \right]. \end{split} \tag{326}$$

Applying to the expression (325) the different variants of the successive approximations method [56,143,149,150,155], we can ge<sup>t</sup> the generating functional L(f) in explicit form and then, using formula

$$f\_n(z\_1, \ldots, z\_n) =: \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(z\_1)} \ldots \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(z\_n)} : \mathcal{L}(\mathbf{f}) \Big|\_{\mathbf{f}=0} \tag{327}$$

for all *n* ∈ Z+ obtain distribution function for any order of perturbation theory. In particular, choosing expansion by the particle density in container *A* ∈ R<sup>3</sup> as a small parameter, it is easy to ge<sup>t</sup> the modified kinetic Bogolubov–Boltzmann equation for one-particle distribution function *f*1(*<sup>z</sup>*; *t*), *z* ∈ *<sup>T</sup>*<sup>∗</sup>(R<sup>3</sup>), *t* ∈ R+:

$$\frac{\partial f\_1(z\_1;t)}{\partial t} + \langle \frac{p}{m} | \nabla\_x f\_1(z\_1;t) \rangle = \int\_{T^\*(\mathbb{R}^3)} dz\_2 \{f\_2(z\_1, z\_2; t), \mathcal{V}(x\_1 - x\_2)\}^{(2)},\tag{328}$$

where the function ˜ *f*2(*<sup>z</sup>*2, *z*1; *t*) is defined according to (325) and (326) by the following expression:

$$\begin{aligned} \tilde{f}\_2(z\_1, z\_2; t) &= f\_1(\tilde{z}\_1; t) f\_1(\tilde{z}\_2; t), \\ \tilde{z}\_j &= \lim\_{\tau \to \infty} S\_2(\tau) S\_1(-\tau) z\_j \Rightarrow \begin{cases} \quad \vec{x}\_j = \lim\_{\tau \to \infty} S\_2(-\tau) x\_j + \tau \frac{p\_j}{m}, \\\ \vec{p}\_j = \lim\_{\tau \to \infty} S\_2(-\tau) p\_j, \end{cases} \end{aligned} \tag{329}$$

for *j* = 1, 2. Taking into account that the Poisson bracket {·, ·}(*n*) is invariant with respect to the mappings *Sn*(*τ*), *n* ∈ Z+, from (329) it is easy to find that

$$\begin{split} \left\{ \bar{f}\_{2}(z\_{1}, z\_{2}; t), \mathcal{V}(x\_{2} - x\_{1}) \right\}^{(2)} &= \frac{|p\_{2} - p\_{1}|}{m} \frac{\partial}{\partial \bar{t}} \left( f\_{1}(\bar{z}\_{1}; t) f\_{1}(\bar{z}\_{2}; t) \right) - \\ - \langle \frac{(p\_{2} - p\_{1})}{m} | \nabla\_{x\_{1}} f\_{1}(\bar{z}\_{1}; t) \rangle f\_{1}(\bar{z}\_{2}; t) + \langle \frac{(p\_{2} - p\_{1})}{m} | \nabla\_{x\_{2}} f\_{1}(\bar{z}\_{2}; t) \rangle f\_{1}(\bar{z}\_{1}; t), \end{split} \tag{330}$$

where *ξ* ∈ R<sup>1</sup> is a parameter of the axis in a cylindrical coordination system which is directed along the vector (*p*2 − *p*1) ∈ E<sup>3</sup> and beginning at the point *x*1 ∈ R3. After substituting (330) into (328), we can ge<sup>t</sup> the kinetic Bogolubov–Boltzmann equation [56,143,149,150,155] in the form of (298) with the explicitly defined collision integral *J*(*f*1), obtained from (330) via integration by *ξ* ∈ R. Choosing in (326) other approximations of the generating functional L(f), f ∈ J (*T*<sup>∗</sup>(R<sup>3</sup>); <sup>R</sup>), one can find other forms of Bogolubov–Boltzmann kinetic Equations (298).

We can also make a remark concerning the nature of the operator-functional expression (309) or (304). Namely, it is easy to see that generating functional L(f)(*<sup>t</sup>*, *<sup>t</sup>*0), f ∈ J (*T*<sup>∗</sup>(R<sup>3</sup>); <sup>R</sup>), allows the following operator-functional representation for all *t*, *t*0 ∈ R:

$$
\mathcal{L}(\mathbf{f})(t, t\_0) = \exp\left[\frac{1}{2} \int\_{\begin{subarray}{c} T^\*(\mathbb{R}^3) \\ T^\*(\mathbb{R}^3) \end{subarray}} dz\_1 \int\_{T^\*(\mathbb{R}^3)} dz\_2 \left\{ : \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(z\_1)} \times \\ \qquad \times \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(z\_2)} : \mathcal{V}(x\_1 - x\_2) \right\}^{(2)}(t - t\_0) \right] \mathcal{L}\_0(\mathbf{f})(t, t\_0).
$$

Comparing the expressions (331) and (305), we find that for arbitrary *t*, *t*0

$$
\pi \left( t, t\_0; \frac{1}{i} \frac{\delta}{\delta t} \right) = \frac{1}{2} (t - t\_0) \times \\
$$

$$
\times \begin{array}{c}
\int \limits\_{T^\*(\mathbb{R}^3)} dz\_1 \quad \int \limits\_{T^\*(\mathbb{R}^3)} dz\_2 \Big\{ : \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(z\_1)} \frac{1}{i} \frac{\delta}{\delta \mathbf{f}(z\_2)} : \mathsf{V}(\mathbf{x}\_1 - \mathbf{x}\_2) \Big\} \Big| \tag{332}
$$

since the functional L0(f)(*<sup>t</sup>*, *<sup>t</sup>*0), f ∈ J (*T*<sup>∗</sup>(R<sup>3</sup>); <sup>R</sup>), is arbitrary. It is easy to see from (332), that operator *<sup>π</sup><sup>t</sup>*, *t*0; 1*i δδ*f is not poly-local with respect to the functional derivatives 1*i δδ*f, which corresponds to the singularity in the operator expansion (304). Thus, using the expression (331), the arbitrariness of the initial state and the classical Bogolubov weakening correlation condition gives a possibility to find many types of the solutions via the method of successive approximations, which follows from (331) and the Bogolubov functional hypothesis subject to the generating representation functional of distribution functions.

Having analyzed the Bogolubov generating functional (331) within the quasi-classical Wigner density operator representation (287), one can obtain an exact functional-operator solution to the evolution Bogolubov functional Equation (323):

$$\mathcal{L}(f) = Z(f)/Z(0), \qquad Z(f) = \exp[\tilde{V}(\delta)]\mathcal{L}o(f) \tag{333}$$

for *f* ∈ J (*T*<sup>∗</sup>(R<sup>3</sup>); <sup>R</sup>). Here we denoted

$$\mathcal{V}(\boldsymbol{\delta}) = \sum\_{n \in \mathbb{Z}\_+} \frac{1}{n!} \int\_{T(\mathbb{R}^3)} dz\_1 \int\_{T(\mathbb{R}^3)} dz\_2 \dots \tag{334}$$

$$\times \int\_{T(\mathbb{R}^3)} dz\_n \Phi\_n(z\_1, z\_2, \dots, z\_n | t) : \frac{1}{i} \frac{\delta}{\delta f(z\_1)} \frac{1}{i} \frac{\delta}{\delta f(z\_2)} \dots \frac{1}{i} \frac{\delta}{\delta f(z\_n)} : \ ,$$

$$\mathcal{L}\_0(f) = \sum\_{n \in \mathbb{Z}\_+} \frac{1}{n!} \int\_{T(\mathbb{R}^3)} z\_1 \int\_{T(\mathbb{R}^3)} dz\_2 \dots \int\_{T(\mathbb{R}^3)} dz\_n$$

$$\times \bar{f}\_n(\mathbf{x}\_1 - \frac{p\_1}{m} t\_\prime, \mathbf{x}\_2 - \frac{p\_2}{m} t\_\prime, \dots, \mathbf{x}\_n - \frac{p\_n}{m} t\_\prime p\_1, p\_2, \dots, p\_n) \prod\_{j=1}^n \{\exp[if(z\_j)] - 1\}\_{\mathbf{x}\_j}$$

where ¯ *fn*( *z*1, *z*2, ..., *zn*), *n* ∈ N, are given *<sup>n</sup>*−particle distribution functions at *t* = 0, that is, owing to the definition (237),

$$\vec{f}\_n(z\_1, z\_2, \ldots, z\_n) := \operatorname{tr}(\vec{P} : w(z\_1) w z\_2) \ldots w(z\_n) :)$$

$$\vec{f} = \colon \frac{1}{i} \frac{\delta}{\delta f(z\_1)} \frac{1}{i} \frac{\delta}{\delta f(z\_2)} \ldots \frac{1}{i} \frac{\delta}{\delta f(z\_n)} : \mathcal{L}(f)|\_{t = 0, f = 0 \prime} \tag{335}$$

and <sup>Φ</sup>*n*(*<sup>x</sup>*1, *x*2, ..., *xn*; *p*1, *p*2, ..., *pn*|*t*), *n* ∈ Z+, are so-called cluster potential functions, determined recursively by means of the following functional-operator relationships:

$$\begin{split} \log(P\_0^{-1}P) &:= \Sigma\_{\mathbb{R} \in \mathbb{Z}\_+} \frac{1}{n!} \int\_{T(\mathbb{R}^3)} dz\_1 \int\_{T(\mathbb{R}^3)} dz\_2 \dots \int\_{T(\mathbb{R}^3)} dz\_n \\ &\times \tilde{V}\_{\mathbb{R}}(z\_1, z\_2, \dots, z\_n | t) : w(z\_1) w(z\_2) \dots w(z\_n) : \end{split} \tag{336}$$

with

$$P\_0 = \exp(-\frac{it}{\hbar}\mathcal{H}\_0)\vec{P}\exp(\frac{it}{\hbar}\mathcal{H}\_0) \tag{337}$$

being the statistical operator of the non-interacting particle system.

If the initial distribution at *t* = 0 is "chaotic", that is for all *n* ∈ N, the following relationships

$$f\_n(z\_1, z\_2, \ldots, z\_n) = \prod\_{j=1}^n \bar{f}\_1(z\_j) \tag{338}$$

hold, one easily gets from (334) and (338) that

$$\mathcal{L}\_0(f) = \exp\left(\int\_{T(\mathbb{R}^3)} dz f\_1(\mathbf{x} - \frac{p}{m}t; p) \{\exp[if(z)] - 1\}\right). \tag{339}$$

If the "chaotic" condition is not fulfilled, we can proceed to the usual cluster Ursell–Mayer type representation [20,22,115,118] for the Bogolubov generating functional (333), where

$$\mathcal{L}\_0(f) = \exp\left(\sum\_{n \in \mathbb{Z}\_+} \frac{1}{n!} \int\_{T(\mathbb{R}^3)} dz\_1 \int\_{T(\mathbb{R}^3)} dz\_2 \dots \int\_{T(\mathbb{R}^3)} dz\_n \times \tag{340} $$

$$\times \mathbb{g}\_n \left(\mathbf{x}\_1 - \frac{p\_1}{m} t\_\cdot \mathbf{x}\_2 - \frac{p\_2}{m} t\_\cdot \dots \mathbf{x}\_n - \frac{p\_n}{m} t\_\cdot p\_1, p\_2, \dots, p\_n\right) \prod\_{j=1}^n \left(\exp[if(z\_j)] - 1\right)\right),$$

where "*cluster*" distribution functions *g*¯*n*(*<sup>z</sup>*1, *z*2, ..., *zn*), *n* ∈ N, have the form

$$\mathfrak{g}\_n(z\_1, z\_2, \dots, z\_n) := \sum\_{\sigma[n]} (-1)^{m+1} (m-1)! \prod\_{j=1}^m F\_{\sigma[j]}(z\_k \in \sigma[j]),$$

$$\vec{f}\_n(z\_1, z\_2, \dots, z\_n) := \sum\_{\sigma[n]} \prod\_{j=1}^m \mathfrak{g}\_{\sigma[j]}(z\_k \in \sigma[j]),\tag{341}$$

and *σ*[*n*] denotes a partition of the set {1, 2, ..., *n*} into non-intersecting subsets {*σ*[*j*] : *j* = 1, *<sup>m</sup>*}, that is *σ*[*j*] ∩ *σ*[*k*] = ∅ for *j* = *k* = 1, *m*, and *σ*[*n*] = <sup>∪</sup>*mj*=1*<sup>σ</sup>*[*j*]. In particular,

$$\begin{aligned} \mathfrak{g}\_1(z\_1) &= \bar{f}\_1(z\_1), \\ \mathfrak{g}\_2(z\_1, z\_2) &= \bar{f}\_2(z\_1, z\_2) - \bar{f}\_1(z\_1)\bar{f}\_1(z\_2), \dots \end{aligned} \tag{342}$$

and so on. The classical Bogolubov generating functional (333), owing to (334) and (340), allows a natural infinite series expansion, whose coefficients can be represented as above, by means of the usual Ursell–Mayer type diagram expressions, which can be effectively used for studying the kinetic properties of our many-particle statistical system.

#### *9.4. The Kinetic Equations for Many-Particle Distribution Functions, Their Lie-Algebraic Structure and Invariant Reductions*

It is well known that the classical Bogolubov–Boltzmann kinetic equations under the condition of many-particle correlations [56,86,142,149–151,155,157–161] at weak short range interaction potentials describe long waves in a dense gas medium. In general, based on the Liouville equations of a finite number of particles in a fixed volume, it is easy to ge<sup>t</sup> for these distribution functions a finite chain of the corresponding kinetic equations, within which one can formally proceed to the statistical mechanics limit and ge<sup>t</sup> a chain of equations for the limiting distribution functions. There will be strong difficulties here if we try to mathematically justify the correctness of this limiting transition in a chain of multi-particle kinetic equations. If we do not pay attention to this complex problem, and consider a fairly weak interaction between particles under appropriate initial conditions, one can obtain the related Boltzmann equation, characterizing the process of establishing statistical equilibrium. Many of the problems related to these limiting distribution functions can be omitted if the infinite particle statistical physics ensemble is worked with from the very beginning, making use of the secondly quantized representation [56,76,151,162] of the particle states in the corresponding Fock type space.

Relating to the Boltzmann kinetic equation, the same equation, called the Vlasov equation, as it was shown by N. Bogolubov [157], also describes exact microscopic solutions of the infinite Bogolubov chain [86] for the many-particle distribution functions, which was widely studied, making use of both classical approaches in [20,21,56,76,77,142,161–179], and making use of the generating Bogolubov functional method and the related quantum current algebra representations.

A.A. Vlasov proposed his kinetic equation [180] for electron-ion plasma, based on general physical reasonings that in contrast to the short range interaction forces between neutral gas atoms, interaction forces between charged particles slowly decrease with

distance, and therefore the motion of each such particle is determined not only by its pairwise interaction with either particle, but also by the interaction with the whole ensemble of charged particles. In this case, the Bogolubov equation for distribution functions in a domain Λ ⊂ R<sup>3</sup>

$$\frac{\partial f\_1(z;t)}{\partial t} + \langle \frac{p}{m} | \nabla\_x f\_1(z;t) \rangle = \int\_{T^\*(\Lambda)} dz' \{f\_2(z, z'; t), V(x - x')\}^{(2)},\tag{343}$$

where *z* := (*<sup>x</sup>*, *p*) ∈ *<sup>T</sup>*<sup>∗</sup>(Λ), *t* ∈ R+ is the temporal evolution parameter, {·, ·}(*m*) denotes the canonical Poisson bracket [56,122,181] on the product *<sup>T</sup>*<sup>∗</sup>(Λ)*<sup>m</sup>*, *m* ∈ N, and *<sup>V</sup>*(*x* − *<sup>x</sup>*), *x*, *x* ∈ Λ, is an interparticle interaction potential, - reduces to the Vlasov equation if to put in (343)

$$f\_2(z, z'; t) = f\_1(z; t) f\_1(z'; t),\tag{344}$$

that is to assume that the two-particle correlation function [142,158,161,179] vanishes:

$$g\_2(z, z'; t) = f\_2(z, z'; t) - f\_1(z; t)f\_1(z'; t) = 0\tag{345}$$

for all *z*, *z* ∈ *T*∗(Λ) and *t* ∈ R+. Then one easily obtains from (343) that

$$\frac{\partial f\_1(z;t)}{\partial t} + \langle \frac{p}{m} | \nabla\_x f\_1(z;t) \rangle = \langle \frac{\partial f\_1(z;t)}{\partial p} | \nabla\_x \int\_{T^\*(\Lambda)} dz' \, V(x - x') f\_1(z';t) \rangle \tag{346}$$

for all *z* ∈ *T*∗(Λ) and *t* ∈ R+. We remark here that the Equation (346) is reversible under the time reflection R− −*t t* ∈ R+, thus it is obvious that it can not describe thermodynamically stable limiting states of the particle system in contrast to the classical Bogolubov–Boltzmann kinetic equations [20,56,86,142,149,151,161,166], being *a priori* time non-reversible owing to the choice of boundary conditions in the correlation weakening form. This means that in spite of the Hamiltonicity of the Bogolubov chain for the distribution functions, the Bogolubov–Boltzmann equation *a priori* is not reversible. It is also evident that the condition (345) does not break the Hamiltonicity—the Equation (346) is Hamiltonian with respect to the following Lie–Poisson–Vlasov bracket:

$$\{\{a(f), b(f)\}\} := \int\_{T^\*(\Lambda)} dz \, f(z) \{\text{grad}\, a(f)(z), \text{grad}\, b(f)(z)\}^{(1)},\tag{347}$$

where grad(·) := *<sup>δ</sup>*(·)/*<sup>δ</sup> f* , *f* ∈ *D*(*T*∗(Λ)) := *Mf*1 , respectively *a*, *b* ∈ *D*( *Mf*1 ) are smooth functionals on the functional manifold *Mf*1 , consisting of functions fast decreasing at the boundary *∂*Λ of the domain Λ ⊂ R3. The statement above easily ensues from the following proposition.

**Proposition 11.** *Let M*F *denote a set of many-particle distribution functions. Then the classical Bogolubov–Poisson bracket [20,21,86,172] on the functional space <sup>D</sup>*(*<sup>M</sup>*F ) *reduces invariantly on the subspace D*( *Mf*1 ) ⊂ *<sup>D</sup>*(*<sup>M</sup>*F ) *to the Lie–Poisson–Vlasov bracket (347).*

Concerning the general case when we work with an infinite Bogolubov chain of kinetic equations on the many-particle distribution functions and are forced to break it at some place, numbered by some natural number *N* ∈ N, the usual approaches always give rise to the resulting inconsistency [155,158] of the chain and, as a result, to the nonphysical solutions. The most successful approach to obtaining the Boltzmann kinetic equation for the one-particle distribution function was suggested many years ago by N. Bogolubov [86,149], based on the effective application of the so called weak correlation condition. So far, regretfully, this approach, being conjugated with the complex problem of solving functional equations, also gives rise to inconsistency of the higher order kinetic equations. Nonetheless, being inspired by former studies [20,76,162] of these problems, based on the geometrical

interpretation of the Bogolubov kinetic equations chain, we devised a new functional analytic approach [23] to constructing its compatible reduction a priori free of any nonphysical consequences. We also succeeded in constructing a reduced set of kinetic equations, based on a suitably devised Dirac type invariant reduction scheme of the corresponding many-particle Lie–Poisson phase space. The approach to solving this problem and its different consequences will be analyzed in more detail in sections to follow below.

#### *9.5. The Classical Lie-Poisson-Vlasov Bracket and Kinetic Equation For The One-Particle Distribution Function*

The bracket expression (347) allows a slightly different Lie-algebraic interpretation, based on considering the functional space *<sup>D</sup>*(*Mf*1 ) as a Poisson manifold, related with the canonical symplectic structure on the diffeomorphism group Diff(Λ) of the domain Λ ⊂ R3, first described [31,32] in 1887 by Sophus Lie. Namely, the following classical theorem holds.

**Theorem 8.** *The Lie-Poisson bracket at point* (*μ*; *η*) ∈ *T*∗*η* (Diff(Λ)) *on the coadjoint space T*∗*η* (Diff(Λ)), *η* ∈ Diff(Λ), *is equal to the expression*

$$\{f,\emptyset\}(\mu) = (\mu|[\delta\lg(\mu)/\delta\mu, \delta f(\mu)/\delta\mu])\_{\mathbb{C}} \tag{348}$$

*for any smooth right-invariant functionals f* , g ∈ *<sup>C</sup>*<sup>∞</sup>(*T*<sup>∗</sup>*η* (Diff(Λ)); <sup>R</sup>).

**Proof.** By classical definition [31,32,122,131,181] of the Poisson bracket of smooth functions (*μ*|*a*)*<sup>c</sup>*,(*μ*|*b*)*<sup>c</sup>* ∈ *<sup>C</sup>*<sup>∞</sup>(*T*<sup>∗</sup>*η* (Diff(Λ)); <sup>R</sup>), *a*, *b* ∈ diff(Λ) - *<sup>T</sup>η*(Diff(Λ)) on the symplectic space *T*∗*η* (Diff(Λ)), it is easy to calculate that

$$\begin{array}{lcl}\{\mu(a),\mu(b)\} := \delta a(X\_{a},X\_{b}) =\\ = X\_{a}(a|X\_{b})\_{c} - X\_{b}(a|X\_{a})\_{c} - (a|[X\_{a},X\_{b}])\_{c},\end{array} \tag{349}$$

where *Xa* := *<sup>δ</sup>*(*μ*|*a*)*c*/*δμ* = *a* ∈ diff(Λ), *Xb* := *<sup>δ</sup>*(*μ*|*b*)*c*/*δμ* = *b* ∈ diff(Λ). Since the expressions *Xa*(*α*|*Xb*)*c* = 0 and *Xb*(*α*|*Xa*)*c* = 0 owing the right-invariance of the vector fields *Xa*, *Xb* ∈ *<sup>T</sup>η*(Diff(Λ)), the Poisson bracket (349) transforms into

$$\begin{array}{lcl} \{\,(\mu|a)\_{\mathfrak{c}}, (\mu|b)\_{\mathfrak{c}}\} = -(a|[X\_{\mathfrak{a}}, X\_{\mathfrak{b}}])\_{\mathfrak{c}} = \\ = (\mu|[b, a])\_{\mathfrak{c}} = (\mu|[\delta(\mu|b)\_{\mathfrak{c}}/\delta\mu, \delta(\mu|a)\_{\mathfrak{c}}/\delta\mu])\_{\mathfrak{c}} \end{array} \tag{350}$$

for all (*μ*; *η*) ∈ *T*∗*η* (Diff(Λ)) - diff<sup>∗</sup>(Λ), and any *a*, *b* ∈ diff(Λ). The Poisson bracket (350) is easily generalized to

$$\{f, \emptyset\}(\mu) = (\mu|[\delta \lg(\mu)/\delta \mu, \delta f(\mu)/\delta \mu])\_c \tag{351}$$

for any smooth functionals *f* , *g* ∈ *<sup>C</sup>*<sup>∞</sup>(diff<sup>∗</sup>(Λ); <sup>R</sup>), finishing the proof.

Concerning our special problem of describing evolution equations for one-particle distribution functions, we will consider the one particle cotangent space *T*∗(Λ) over a domain Λ ⊂ R<sup>3</sup> and the canonical Poisson bracket {·, ·} := {·, ·}(1) on *<sup>T</sup>*<sup>∗</sup>(Λ), for which, by definition, for any *f* , *g* ∈ *Mf*1

$$\{f, g\}(z) := \langle \frac{\partial f}{\partial p} | \frac{\partial g}{\partial x} \rangle - \langle \frac{\partial g}{\partial p} | \frac{\partial f}{\partial x} \rangle,\tag{352}$$

where *z* = (*<sup>x</sup>*, *p*) ∈ *<sup>T</sup>*<sup>∗</sup>(Λ). We denote now by G := ( *Mf*1 ; {·, ·}) the related functional Lie algebra and G∗ its adjoint space with respect to the standard bilinear symmetric form (·|·) : *Mf*1 × *Mf*1 → R on the product *Mf*1 × *Mf*1 , where

$$f(f|\mathcal{g}) := \int\_{T^\*(\Lambda)} f(z)\mathcal{g}(z)dz. \tag{353}$$

The constructed Lie algebra G with respect to the bilinear symmetric form (353) proves to be metrized, that is G-G∗ and

$$(\{f, \emptyset\}|h) = (f|\{\emptyset, h\})\tag{354}$$

for any *f* , *g* and *h* ∈ G. *I f γ* ∈ *D*(G∗) is a smooth functional on G∗, its gradient grad *γ*(*f*) ∈ G at point *f* ∈ G∗ is naturally defined via the limiting expression

$$(g|\operatorname{grad}\gamma(f)) := \left. \frac{d}{d\varepsilon}\gamma(f+\varepsilon g) \right|\_{\varepsilon=0} \tag{355}$$

for arbitrary element *g* ∈ G∗. Now we define the Poisson structure {{·, ·}} : G∗ × G∗ → R by means of the standard Lie–Poisson [31,32,59,122,159,181–184] expression:

$$\{\{\gamma,\mu\}\} := (f|\{\text{grad}\,\gamma(f), \text{grad}\,\gamma(f)\})\tag{356}$$

for arbitrary functionals *γ*, *μ* ∈ *<sup>D</sup>*(G∗). It is evident that the expression (356) identically coincides with the Poisson bracket (347).

Consider a functional *γ* ∈ *D*(G∗) and the related coadjoint action of the element grad *γ*(*f*) ∈ G at a fixed element *f* := *f*1 ∈ G∗:

$$\partial f\_1/\partial \mathbf{t} := \operatorname{ad}^\*\_{\text{grad } \gamma(f\_1)} f\_{1\prime} \tag{357}$$

where *t* ∈ R is the corresponding evolution parameter. It is easy observe that

$$
\partial f\_1 / \partial t = \{\{\gamma\_\prime f\_1\}\} \tag{358}
$$

is a Hamiltonian equation with the functional *γ* ∈ *D*(G∗) taken as its Hamiltonian, being simultaneously equivalent to the following canonical Hamiltonian flow:

$$
\partial f\_1 / \partial t = \{f\_1, \text{grad } \gamma(f\_1)\},
\tag{359}
$$

if to choose as a Hamiltonian the following functional

$$\gamma(f\_1) := \int\_{T^\*(\Lambda)} dz\_1 \frac{p\_1^2}{2m} f\_1(z\_1) + \frac{1}{2} \int\_{T^\*(\Lambda)^2} dz\_1 dz\_2 V(x\_1 - x\_2) f\_1(z\_1) f\_1(z\_2),\tag{360}$$

where *<sup>V</sup>*(*<sup>x</sup>*1 − *<sup>x</sup>*2) is a two-particle interaction potential, *x*1, *x*2 ∈ Λ. It is easy to observe here that the Hamiltonian (360) is obtained from the corresponding classical Hamiltonian expression

$$\mathcal{H}(\mathcal{F}) := \int\_{T^\*(\Lambda)} dz\_1 \frac{p\_1^2}{2m} f\_1(z\_1) + \frac{1}{2} \int\_{T^\*(\Lambda)^2} dz\_1 dz\_2 V(x\_1 - x\_2) f\_2(z\_1, z\_2),\tag{361}$$

where F = (*f*1, *f*2, ...) ∈ *M*F denotes an infinite vector from the space *M*F := ∏ *j*∈N*Mfj* of

multiparticle distribution functions, and if to impose on it the constraint (344). Thus, we have stated the following proposition.

**Proposition 12.** *The Boltzmann–Vlasov kinetic Equation (346) is a Hamiltonian system on the functional manifold* G∗ - G = (*Mf* ; {·, ·}) *with respect to the canonical Lie–Poisson structure (356) with Hamiltonian (360). As a consequence, the flow (346) is time reversible.*

#### *9.6. Boltzmann–Vlasov Kinetic Equations and Microscopic Exact Solutions*

Proposition 11, stated above, claims that the Boltzmann–Vlasov Equation (346) is a suitable reduction of the whole Bogolubov chain upon the invariant functional subspace *Mf*1⊂ *M*F . Moreover, this invariance in no way should be compatible

*a priori* [20,21,24,157,166,171,173,174] with the other kinetic equations from the Bogolubov chain, and can even be contradictory. Nonetheless, as it was stated [157] by N. Bogolubov, namely owing to this invariance of the subspace *Mf*1 ⊂ *M*F the Boltzmann–Vlasov Equation (346) in the case of the Boltzmann–Enskog hard sphere approximation of the inter-particle potential possesses exact microscopical solutions which are compatible with the whole hierarchy of the Bogolubov kinetic equations. The latter is, obviously, equivalent to its Hamiltonicity on the manifold *Mf*1 with respect to the Lie–Poisson bracket (356). The Boltzmann–Enskog kinetic equation [151,157,158,161,179] equals

$$\begin{aligned} \frac{\partial f\_1(z;t)}{\partial t} + \langle \begin{array}{c} p \\ m \end{array} \vert \nabla\_x f\_1(z;t) \rangle &= \\ \frac{\partial f\_2(z;t)}{\partial t} \langle m \vert \int\_{\mathbb{R}^3} d\mathbf{p}' \int\_{\mathbb{R}} p' \vert n \rangle \langle \frac{\mathbf{p}'}{m} \vert n \rangle [f\_2(\mathbf{x}, \mathbf{p}; \mathbf{x} + a n\_\prime \mathbf{p}'; t) - f\_2(\mathbf{x}, \mathbf{p}; \mathbf{x} - a n\_\prime \mathbf{p}'; t)] \end{aligned} \tag{362}$$

where *p*˜ := *p* + *n p* − *p*|*n* , *p*˜ := *p* − *n p* − *p*|*n* , *a* > 0− a particle diameter, *n* ∈ S<sup>2</sup> − a unit vector, *n*|*n* = 1, and, by definition, *f*2(*<sup>z</sup>*, *z*; *t*) = 0 for all *z*, *z* ∈ *<sup>T</sup>*<sup>∗</sup>(Λ), *t* ∈ R, satisfying the condition ||*z* − *z*|| < *a*. The Equation (362) easily reduces to the Vlasov– Enskog equation

$$\begin{aligned} \frac{\partial f\_1(z;t)}{\partial t} + \langle \ \frac{p}{m} | \nabla\_x f\_1(z;t) \rangle &= f\_{V-E}(f), \\\\ f\_{V-E}(f) &= a^2 \int\_{\mathbb{S}^2} dn \int\_{\mathbb{E}^3} dp' \, \langle p'|n \rangle \langle \frac{p'}{m}| n \rangle \times \\\\ \times [f\_1(\mathbf{x}, \vec{p}; t) f\_1(\mathbf{x} + an, \vec{p}'; t) - f\_1(\mathbf{x}, p; t) f\_1(\mathbf{x} - an, p'; t)] \end{aligned} \tag{363}$$

for all (*z*; *t*) ∈ *T*∗(Λ) × R owing to its Hamiltonicity on the space *Mf*1 ⊂ *M*F . If, in addition, there exists a nontrivial interparticle potential, the equation above is naturally generalized to the kinetic equation

$$\begin{split} \frac{\partial f\_1(z;t)}{\partial t} + \langle \begin{array}{c} \frac{p}{m} \middle| \nabla\_x f\_1(z;t) \rangle = f\_{V-E}(f) + \\\\ + \quad \int\_{T^\*(\Lambda)} dz' \{f\_1(z;t) f\_1(z';t), V(x-x')\}^{(2)}, \end{split} \tag{364}$$

which remains to be Hamiltonian on *Mf*1 and possesses, in particular, the following exact singular solution:

$$f\_1(z;t) = \sum\_{j=\overline{1,N}} \delta(z - z\_j(t)),\tag{365}$$

where *zj*(*t*) ∈ *<sup>T</sup>*<sup>∗</sup>(Λ), *j* = 1, *<sup>N</sup>*—phase space coordinates in *T*∗(Λ)*<sup>N</sup>* of *N* ∈ N interacting particles in the domain Λ ⊂ R3. Specified above the Hamiltonicity problem and the existence of exact solutions to the Boltzmann–Vlasov kinetic Equation (364) is deeply related to that of describing correlation functions [142,161,179], suitably breaking the infinite Bogolubov chain [20,76,77,86,142,161] of many-particle distribution functions. Namely, if to introduce many-particle correlation functions [142,161,179] for related Bogolubov distribution functions as

$$\begin{aligned} g\_1(z\_1) &= 0, \mathcal{g}\_2(z\_1, z\_2) = f\_2(z\_1, z\_2) - f\_1(z\_1)f\_1(z\_2), \\ g\_3(z\_1, z\_2, z\_3) &= f\_3(z\_1, z\_2, z\_3) - f\_1(z\_1)f\_1(z\_2)f\_1(z\_3) - f\_1(z\_1)g\_2(z\_2, z\_3) - \\ &- f\_1(z\_2)g\_2(z\_3, z\_1) - f\_1(z\_3)g\_2(z\_1, z\_2), \dots \end{aligned} \tag{366}$$

where *zj* ∈ *<sup>T</sup>*<sup>∗</sup>(Λ), *j* ∈ *N*, then the Vlasov Equation (364) is obtained from the Bogolubov hierarchy at *n* = 1 and *g*2(*<sup>z</sup>*1, *<sup>z</sup>*2) = 0 for all *z*1, *z*2 ∈ *<sup>T</sup>*<sup>∗</sup>(Λ).

As it was mentioned above, the constraint imposed on the infinite Bogolubov hierarchy is compatible with its Hamiltonicity. Yet in many practical cases, this closedness procedure by means of imposing the conditions like

$$\partial\_{\lambda}g\_{m+1}(z\_1, z\_2, \dots, z\_{m+1}) = 0 \tag{367}$$

for all *zs* ∈ *<sup>T</sup>*<sup>∗</sup>(Λ),*<sup>s</sup>* = 1, *m* + 1 at some fixed *m* ≥ 2 gives rise to some serious dynamical problems related to its mathematical correctness. Namely, if to close the infinite Bogolubov chain of kinetic equations on many-particle distribution functions in this way, one easily checks that the imposed constraint (367) does not persist in time subject to the evolution of the distribution functions *fj*(*<sup>z</sup>*1, *z*2, ... , *zj*), *zj* ∈ *<sup>T</sup>*<sup>∗</sup>(Λ), *j* = 1, *m*. This means that these naively reduced kinetic equations are written down somehow incorrectly, as the reduced functional submanifold *M*(*m*) F := {F ∈ *M*F : *gm*+<sup>1</sup> = 0} should remain invariant in time. To dissolve this problem, we are forced to consider the whole Bogolubov hierarchy of kinetic equations on multiparticle distribution functions as a Hamiltonian system on the functional manifold *M*F and correctly reduce it on the constructed above functional submanifold *M*(*m*) F ⊂ *M*F via the classical Dirac type [11,59,63,122,181] procedure. The kinetic equations obtained this way by means of the reduced Lie–Poisson–Bogolubov structure will evidently differ from those naively obtained by means of the direct substitution of the imposed constraint (367) into the Bogolubov chain of kinetic equations, and in due course will conserve the functional submanifold *M*(*m*) F⊂ *M*F invariant.

#### *9.7. The Invariant Reduction of the Bogolubov Distribution Functions Chain*

Consider the constructed before Hamiltonian functional H(F) ∈ *<sup>D</sup>*(*<sup>M</sup>*F ) (361)

$$\mathcal{H}(\mathcal{F}) = \int\_{T^\*(\Lambda)} dz\_1 \frac{p\_1^2}{2m} f\_1(z\_1) + \frac{1}{2} \int\_{T^\*(\Lambda)^2} dz\_1 dz\_2 V(x\_1 - x\_2) f\_2(z\_1, z\_2) \tag{368}$$

and calculate the evolution of the distribution functions vector F ∈ *M*F under the simplest constraint (367) at *m* = 1, that is

$$
\log\_2(z\_1, z\_2) = f\_2(z\_1, z\_2) - f\_1(z\_1)f\_1(z\_2) = 0 \tag{369}
$$

for all *z*1, *z*2 ∈ *<sup>T</sup>*<sup>∗</sup>(Λ). To perform this reduction on *M*(1) F ⊂ *M*F , we need [11,59,63] to constraint the *λ*-extended Hamiltonian expression

$$\mathcal{H}\_{\Lambda}(\mathcal{F}) := \mathcal{H}(\mathcal{F}) + \frac{1}{2} \int\_{T^\*(\Lambda)^2} dz\_1 dz\_2 \lambda(z\_1, z\_2) [f\_2(z\_1, z\_2) - f\_1(z\_1) f\_1(z\_2)] \tag{370}$$

for some smooth function *λ* ∈ *D*(*T*∗(Λ)<sup>2</sup>) and next to determine it from the submanifold *M*(1) Finvariance condition

$$\begin{split} \frac{\frac{\partial f\_2(z\_1, z\_2)}{\partial t}}{\partial t} &= \{ \{ \mathcal{H}\_\lambda(\mathcal{F}), \mathcal{g}\_2(z\_1, z\_2) \} \} = \\\\ \frac{\frac{\partial f\_2(z\_1, z\_2)}{\partial t} - \frac{\partial f\_1(z\_1)}{\partial t} f\_1(z\_2) - f\_1(z\_1) \frac{\partial f\_1(z\_2)}{\partial t} &= 0 \end{split} \tag{371}$$

for all *z*1, *z*2 ∈ *T*∗(Λ) and *t* ∈ R. To effectively calculate the condition (371), let us first calculate the evolutions for distribution functions *f*1 and *f*2 ∈ *M*F :

$$\begin{split} \frac{\partial f\_1(z\_1)}{\partial t} &= \{ \{ \mathcal{H}\_\lambda(\mathcal{F}), f\_1(z\_1) \} \} = \left\{ f\_1(z\_1), \frac{\delta \mathcal{H}\_\lambda(\mathcal{F})}{\delta f\_1(z\_1)} \right\}^{(1)} + \\ &+ \int\_{T^\*(\Lambda)} dz\_2 \left\{ f\_2(z\_1, z\_2), \frac{\delta \mathcal{H}\_\lambda(\mathcal{F})}{\delta f\_2(z\_1, z\_2)} \right\}^{(1)}, \end{split} \tag{372}$$

and

$$\begin{split} \frac{\partial f\_{2}(z\_{1}, z\_{2})}{\partial l} &= \left\{ \left\{ \mathcal{H}\_{\lambda}(\varmathcal{F}), f\_{2}(z\_{1}, z\_{2}) \right\} \right\} = \left\{ f\_{2}(z\_{1}, z\_{2}), \frac{\delta \mathcal{H}\_{\lambda}(\varmathcal{F})}{\delta f\_{1}(z\_{1})} + \frac{\delta \mathcal{H}\_{\lambda}(\varmathcal{F})}{\delta f\_{1}(z\_{2})} \right\}^{(2)} + \\ &+ \left\{ f\_{2}(z\_{1}, z\_{2}), \frac{\delta \mathcal{H}\_{\lambda}(\varmathcal{F})}{\delta f\_{2}(z\_{1}, z\_{2})} \right\}^{(2)} + \int\_{T^{\*}(\varLambda)} dz\_{3} \left\{ f\_{3}(z\_{1}, z\_{2}, z\_{3}), \frac{\delta \mathcal{H}\_{\lambda}(\varmathcal{F})}{\delta f\_{2}(z\_{1}, z\_{3})} + \frac{\delta \mathcal{H}\_{\lambda}(\varmathcal{F})}{\delta f\_{2}(z\_{2}, z\_{3})} \right\}^{(2)}, \end{split} \tag{373}$$

which can be rewritten equivalently as follows:

$$\begin{split} \frac{\partial f\_1(z\_1)}{\partial t} &= -\langle \frac{\partial f\_1(z\_1)}{\partial p\_1} \vert \int\_{T^\*(\Lambda)} dz\_2 \frac{\partial \lambda(z\_1, z\_2)}{\partial x\_1} f\_1(z\_2) - \\ &- \langle \frac{p\_1}{m} - \int\_{T^\*(\Lambda)} dz\_2 \frac{\partial \lambda(z\_1, z\_2)}{\partial p\_1} f\_1(z\_2) \vert \frac{\partial f\_1(z\_1)}{\partial x\_1} \rangle + \\ &+ \frac{1}{2} \int\_{T^\*(\Lambda)} dz\_2 \langle \frac{\partial}{\partial x\_1} [V(x\_1 - x\_2) + \lambda(z\_1, z\_2)] \vert \frac{\partial f\_2(z\_1, z\_2)}{\partial p\_1} \rangle - \\ &- \frac{1}{2} \int\_{T^\*(\Lambda)} dz\_2 \langle \frac{\partial \lambda(z\_1, z\_2)}{\partial p\_1} \vert \frac{\partial f\_2(z\_1, z\_2)}{\partial x\_1} \rangle \end{split} \tag{3.74}$$

and

*∂ f*2(*<sup>z</sup>*1, *<sup>z</sup>*2) *∂t* = − *∂ f*2(*<sup>z</sup>*1, *<sup>z</sup>*2) *∂p*1 | *T*∗(Λ) *dz*2 *∂λ*(*<sup>z</sup>*1, *<sup>z</sup>*2) *∂<sup>x</sup>*1 *f*1(*<sup>z</sup>*2) − (375) − *∂ f*2(*<sup>z</sup>*1, *<sup>z</sup>*2) *∂p*2 | *T*∗(Λ) *dz*1 *∂λ*(*<sup>z</sup>*1, *<sup>z</sup>*2) *∂<sup>x</sup>*2 *f*1(*<sup>z</sup>*1) − − *∂ f*2(*<sup>z</sup>*1, *<sup>z</sup>*2) *∂<sup>x</sup>*1 | *p*1*m* − *T*∗(Λ) *dz*2 *∂λ*(*<sup>z</sup>*1, *<sup>z</sup>*2) *∂p*1 *f*1(*<sup>z</sup>*2) − − *∂ f*2(*<sup>z</sup>*1, *<sup>z</sup>*2) *∂<sup>x</sup>*2 | *p*2*m* − *T*∗(Λ) *dz*1 *∂λ*(*<sup>z</sup>*1, *<sup>z</sup>*2) *∂p*2 *f*1(*<sup>z</sup>*1) + + 1 2 *∂ f*2(*<sup>z</sup>*1, *<sup>z</sup>*2) *∂p*1 | *∂∂x*1 [*V*(*<sup>x</sup>*1 − *<sup>x</sup>*2) + *<sup>λ</sup>*(*<sup>z</sup>*1, *<sup>z</sup>*2)] + + 1 2 *∂ f*2(*<sup>z</sup>*1, *<sup>z</sup>*2) *∂p*2 | *∂∂x*2 [*V*(*<sup>x</sup>*1 − *<sup>x</sup>*2) + *<sup>λ</sup>*(*<sup>z</sup>*1, *<sup>z</sup>*2)] − − 1 2 *∂ f*2(*<sup>z</sup>*1, *<sup>z</sup>*2) *∂<sup>x</sup>*1 | *∂λ*(*<sup>z</sup>*1, *<sup>z</sup>*2) *∂p*1 −−12 *∂ f*2(*<sup>z</sup>*1, *<sup>z</sup>*2) *∂<sup>x</sup>*2 | *∂λ*(*<sup>z</sup>*1, *<sup>z</sup>*2) *∂p*2 + + 1 2 *T*∗(Λ) *dz*3 *∂ f*3(*<sup>z</sup>*1, *z*2, *<sup>z</sup>*3) *∂p*1 | *∂∂x*1 [*V*(*<sup>x</sup>*1 − *<sup>x</sup>*3) + *<sup>λ</sup>*(*<sup>z</sup>*1, *<sup>z</sup>*3)] + + 1 2 *T*∗(Λ) *dz*3 *∂ f*3(*<sup>z</sup>*1, *z*2, *<sup>z</sup>*3) *∂p*2 | *∂∂x*2 [*V*(*<sup>x</sup>*2 − *<sup>x</sup>*3) + *<sup>λ</sup>*(*<sup>z</sup>*2, *<sup>z</sup>*3)] − − 1 2 *T*∗(Λ) *dz*3 *∂ f*3(*<sup>z</sup>*1, *z*2, *<sup>z</sup>*3) *∂<sup>x</sup>*1 | *∂λ*(*<sup>z</sup>*1, *<sup>z</sup>*2) *∂p*1 − 12 *T*∗(Λ) *dz*3 *∂ f*3(*<sup>z</sup>*1, *z*2, *<sup>z</sup>*3) *∂<sup>x</sup>*2 | *∂λ*(*<sup>z</sup>*1, *<sup>z</sup>*2) *∂p*2 

Having now substituted temporal derivatives (374) and (375) into the equality (371) in their explicit form, one obtains the following functional relationship:

$$\begin{array}{ll} \frac{1}{2} \left( f\_1(z\_2) \frac{\partial f\_1(z\_1)}{\partial p\_1} \Big| \frac{\partial}{\partial x\_1} (V(\mathbf{x}\_1 - \mathbf{x}\_2) + \lambda(z\_1, z\_2) - \\ - \int\_{T^\*(\Lambda)} dz\_3 f\_1(z\_3) \left[ V(\mathbf{x}\_1 - \mathbf{x}\_3) + \lambda(z\_1, z\_3) \right] \right) \Big| + \\ + \frac{1}{2} \langle f\_1(z\_1) \frac{\partial f\_1(z\_2)}{\partial p\_2} \Big| \frac{\partial}{\partial x\_2} (V(\mathbf{x}\_2 - \mathbf{x}\_1) + \lambda(z\_2, z\_1) - \\ - \int\_{T^\*(\Lambda)} dz\_3 f\_1(z\_3) [V(\mathbf{x}\_2 - \mathbf{x}\_3) + \lambda(z\_2, z\_3)] \Big| \right) = 0, \end{array} \tag{376}$$

which is satisfied if

$$
\lambda(z\_1, z\_2) = -V(x\_1 - x\_2) \tag{377}
$$

for all *z*1, *z*2 ∈ *<sup>T</sup>*<sup>∗</sup>(Λ). Taking into account the result (377), one easily obtains from the Equation (374) the invariantly reduced on the submanifold *M*(1) F ⊂ *M*F kinetic equation on the one-particle distribution function:

$$\frac{\partial f\_1(z\_1)}{\partial t} + \langle p\_1/m \vert \frac{\partial f\_1(z\_1)}{\partial x\_1} \rangle = \langle \frac{\partial f\_1(z\_1)}{\partial p\_1} \vert \frac{\partial}{\partial x\_1} \int\_{T^\*(\Lambda)} dz\_2 f\_1(z\_2) V(x\_1 - x\_2) \rangle\_{\prime} \tag{378}$$

which can be rewritten in the following compact form:

$$\frac{\partial f\_1(z\_1)}{\partial t} = \left\{ f\_1(z\_1), \frac{\delta \mathcal{H}(\mathcal{F})}{\delta f\_1(z\_1)} \right\}^{(1)}\text{.}\tag{379}$$

where we put, by definition,

$$\tilde{\mathcal{H}}(\mathcal{F}) := \int\_{T^\*(\Lambda)} dz\_1 \frac{p\_1^2}{2m} f\_1(z\_1) + \frac{1}{2} \int\_{T^\*(\Lambda)^2} dz\_1 dz\_2 V(\mathbf{x}\_1 - \mathbf{x}\_2) f\_1(z\_1) f\_1(z\_2). \tag{380}$$

The kinetic Equation (378) naturally coincides exactly with that obtained previously from the naively reduced evolution equation

$$\frac{\partial f\_1(z\_1)}{\partial t} = \left\{ \left\{ \mathcal{H}(\mathcal{F}), f\_1(z\_1) \right\} \right\} \big|\_{M\_{\mathcal{F}}^{(1)}} \tag{381}$$

on the submanifold *M*(1) F ⊂ *M*F , as it is globally invariant [20,172] with respect to the classical Lie–Poisson–Bogolubov structure on *M*F .

The obtained result can be formulated as the following proposition.

**Proposition 13.** *The first correlation function Dirac type reduction on the functional submanifold M*(1) F ⊂ *M*F , *formed by relationships (369), reduces the corresponding Bogolubov chain of manyparticle kinetic equations to the well known classical Vlasov kinetic equation.*

**Remark 9.** *It is worth mentioning here that the well known classical Bogolubov approximation of the many-particle distribution functions as fn*(*<sup>z</sup>*1, *z*2, ... , *zn*) := *ϕn*(*<sup>z</sup>*1, *z*2, ... , *z*; *f*1), *zj* ∈ *<sup>T</sup>*(Λ), *j* = 2, *n*, *with mapping ϕn* : (...) × *Mf*1 → R, *n* ∈ N\{1}, *presenting smooth nonlinear functionals, independent of the temporal parameter t* ∈ R+, *define a suitably different functional submanifold M* ˜ (1) F⊂ *M*F , *upon which the reduced evolution flow*

$$\frac{\partial f\_1(z\_1)}{\partial t} = \left\{ \left\{ \mathcal{H}(\mathcal{F})\_\prime f\_1(z\_1) \right\} \right\} \vert\_{M\_{\mathcal{F}}^{(1)}} \tag{382}$$

*gives rise to a new Boltzmann type kinetic equation, being compatible with evolution equations for higher distribution functions, free of evolution inconsistencies and completely different from that derived previously by Bogolubov [86].*

In the same way as above, one can explicitly construct the system of invariantly reduced kinetic equations

$$\frac{\partial f\_1(z\_1)}{\partial t} = \left\{ \left\{ \mathcal{H}(\mathcal{F})\_\prime f\_1(z\_1) \right\} \right\}|\_{M\_{\tilde{\mathcal{F}}}^{(2)}} \frac{\partial f\_2(z\_1, z\_2)}{\partial t} = \left\{ \left\{ \mathcal{H}(\mathcal{F})\_\prime f\_2(z\_1, z\_2) \right\} \right\}|\_{M\_{\tilde{\mathcal{F}}}^{(2)}} \tag{383}$$

on the submanifold *M*(2) F ⊂ *M*F , which already is not *a priori* globally invariant with respect to the Hamiltonian evolution flows on *M*F and whose detail structure and analysis are postponed to another place.

#### *9.8. Conclusions and Perspectives*

We presented a review of the Boltzmann type kinetic equations in statistical physics as analytical objects based on the non-relativistic current algebra symmetry approach to constructing the Bogolubov generating functional of many-particle distribution functions. We then applied it to an important classical problem of describing Boltzmann–Bogolubov and Boltzmann–Vlasov type kinetic equations, naturally related with an invariantly reduced canonical Hamiltonian system on the infinite-dimensional space of distribution functions subject to the constraints imposed on suitably chosen many-particle correlation functions. As an interesting introductory example of deriving Boltzman–Vlasov type kinetic equations, we considered a quantum-mechanical model of spinless particles with delta-type interaction, having applications [159,185,186] for describing so called Benney type hydrodynamic particle flows. We also reviewed new results on a special class of dynamical systems of Boltzmann–Bogolubov and Boltzmann–Vlasov type on infinite dimensional functional manifolds modeling kinetic processes in many-particle media. There was demonstrated construction of the classical Bogolubov generating functional method in non-equilibrium statistical mechanics within the classical Wigner quasi-classical representation. We also analyzed and presented the kinetic Boltzmann type equation in non-equilibrium statistical mechanics in the frame of the Bogolubov functional hypothesis. Moreover, the Hamiltonian analysis of the infinite hierarchy of many-particle distribution functions was reviewed, and the algebraic structure of the Boltzmann–Bogolubov kinetic equations and their invariant Poissonian reductions were analyzed in detail, including the derivation of the related Boltzmann–Vlasov kinetic equations. Based on the methods and devised techniques, an approach was proposed to invariant reduction of the chain of Bogolubov distribution functions on suitably chosen correlation function constraints, which allowed the derivation of the related modified Boltzmann–Bogolubov kinetic equations for a finite set of multi-particle distribution functions.

We also elaborated in detail effective enough invariant analytical tools reducing the infinite Boltzmann–Bogolubov hierarchy of kinetic equations upon the two-particle correlation function constraint. Within this aspect of invariant reduction of the infinite Boltzmann– Bogolubov hierarchy of kinetic equations that has very important applications, there stays an interesting problem of analytically presenting this reduction upon the three-particle correlation function constraint and deriving a closed system of the Boltzmann type kinetic equations on the corresponding one- and two-particle distribution functions. The similar, ye<sup>t</sup> much more complicated, analytical problem for the future analysis consists of deriving invariantly reduced kinetic equations under the Bogolubov functional hypothesis and its modified versions.

#### **10. The Current Algebra Functional Representations and Geometric Structure of Quasi-Stationary Hydrodynamic Flows**

#### *10.1. Introductory Notes*

This section is devoted to compressible liquid or gas motions in which entropy remains locally constant throughout the flow field, i.e., the flow for which the entropy of a moving element along a streamline remains constant, is called isentropic. This means that along different streamlines, the entropy changes normal to the streamlines. As a typical example, one can mention the flow field behind a curved shock wave, where streamlines, passing through different locations along the curved shock wave, experience different increases in entropy. Hence, downstream from this shock, the entropy can be constant along a given streamline but differs from one streamline to another. This type of flow, with entropy constant along streamlines, is defined as isentropic. Flow with entropy constant everywhere is then called homentropic. Here we need to remark that owing to the second law of thermodynamics, an isentropic flow does not strictly exist. We know from thermodynamics that an isentropic flow is defined to be along streamlines both adiabatic and reversible. Yet, all real flows always experience to some extent the irreversible phenomena of friction, thermal conduction, and diffusion. For instance, any non-equilibrium, chemically reacting

flow is always irreversible, when considered to be a closed system. Nonetheless, there are a large number of liquid and gas dynamic problems with entropy increase negligibly slight, which for the purpose of analysis are assumed to be isentropic. Examples are flows through subsonic and supersonic nozzles, as in wind tunnels and rocket engines, or shock-free flows over a wing, fuselage, or other aerodynamic shapes. For all of them, except for a flow near the thin boundary-layer region, adjacent to the surface where friction and thermal conduction effects can be strong, the outer inviscid flow can be considered isentropic. In contrast, if shock waves exist in the flow, the entropy increase across these shocks destroys the assumption of isentropic flow, although the flow along streamlines between shocks may persist to be isentropic.

As an isentropic flow is governed by thermodynamically reversible processes, being adiabatic along a streamline, it needs to be specified with locally defined [187] thermodynamical parameters, such as the medium density *ρ*, the specific entropy *σ*, the local medium absolute temperature *T*, the pressure *p* and the specific energy *e*. All these quantities are related to each other in some way, which can be retrieved following the classical Gibbs reasonings. We assume from the very beginning that the reversible thermodynamical state of the medium under regard is completely locally described by means of the following first pair: (*p*-local *pressure*, *ρ-specific density*) of thermodynamical parameters. Assume now that the same thermodynamical state of this medium can also be simultaneously described by means of the following second pair: (*T*-local absolute temperature, *σ*-specific entropy). The latter, in particular, means that a suitable functional transformation from one parameter to another, if smooth, is diffeomorphic, which is the Jacobian *J*(*<sup>σ</sup>*,*<sup>T</sup>*)(*<sup>p</sup>*, *ρ*) of this transformation R<sup>2</sup> + (*<sup>σ</sup>*, *T*) → (*p*, *ρ*) ∈ R<sup>2</sup> + is not degenerate everywhere, i.e.,

$$J\_{\left(\sigma,T\right)}\left(p\_\prime\rho\right) = \frac{\partial\left(p\_\prime\rho\right)}{\partial\left(\sigma,T\right)} := \det\begin{pmatrix} \frac{\partial p}{\partial\sigma} & \frac{\partial p}{\partial T} \\ \frac{\partial\rho}{\partial\sigma} & \frac{\partial\rho}{\partial\sigma} \end{pmatrix} \neq 0 \tag{384}$$

at all points (*<sup>σ</sup>*, *T*) ∈ R<sup>2</sup> +. Taking into account that the local absolute temperature *T* and the adiabatic *σ* parameters are, in general, defined with some scaling ambiguity, we can always put, by definition, that *J*(*<sup>σ</sup>*,*<sup>T</sup>*)(*<sup>p</sup>*, *ρ*) = *ρ*2 = 0 everywhere. As a simple consequence of multiplying this expression by the unity Jacobian *J*(*<sup>σ</sup>*,*<sup>ρ</sup>*)(*<sup>σ</sup>*, *ρ*) = 1 one easily derives that

$$\begin{array}{lcl}\mathcal{I}\_{\left(\sigma,T\right)}\left(p,\rho\right)\times\mathcal{I}\_{\left(\sigma,\rho\right)}\left(\sigma,\rho\right)=\\\mathcal{\frac{\partial\left(p,\rho\right)}{\partial\left(\sigma,T\right)}\frac{\partial\left(\sigma,\rho\right)}{\partial\left(\sigma,\rho\right)}=\frac{\partial\left(\rho,p\right)}{\partial\left(\sigma,\rho\right)}\frac{\partial\left(\sigma,\rho\right)}{\partial\left(\sigma,T\right)}=\rho^{2},\end{array} \tag{385}$$

or, equivalently,

$$\frac{\partial(p,\rho)}{\partial(\sigma,\rho)} = \rho^2 \frac{\partial(\sigma,T)}{\partial(\sigma,\rho)} \Longleftrightarrow \left. \frac{\partial(p/\rho^2)}{\partial \sigma} \right|\_{\rho} = \left. \frac{\partial T}{\partial \rho} \right|\_{\sigma} \tag{386}$$

at all points (*<sup>σ</sup>*, *ρ*) ∈ R<sup>2</sup> +. The equality of partial derivatives above simply means, owing to the well known Montel–Menchoff–Young theorem [188–190], the existence of such a differentiable thermodynamic state function R<sup>2</sup> + (*ρ*, *σ*) → *e* ∈ R, that its differential satisfies the following equality:

$$
\delta\mathfrak{e}(\rho,\sigma) = T\delta\sigma + p\delta\rho/\rho^2. \tag{387}
$$

The latter expression presents exactly the written down second thermodynamic law with respect to the locally defined variables, if the smooth function R<sup>2</sup> + (*ρ*, *σ*) → *e* ∈ R is interpreted as the specific medium energy of the system at the internal absolute temperature *T* = *<sup>T</sup>*(*ρ*, *σ*) and pressure *p*(*ρ*, *σ*) at suitably fixed state parameters (*ρ*, *σ*) ∈ R<sup>2</sup> +. Taking into account that our medium is embedded into some domain *M* ⊂ R3, moving in space-time, our next task is to adequately describe the related motion spatial phase space variables, compatible with the corresponding Euler evolution equations.

#### *10.2. Diffeomorphism Group Structure and Functional Phase Space Description*

It is well known that the same physical system is often described using different sets of variables, related with their different physical interpretation. Simultaneously, this same system is endowed with different mathematical structures deeply depending on the geometric scenario used for its description. In general, these structures prove to be not equivalent but some special way connected to each other. In particular, such double descriptions commonly occur in systems with distributed parameters such as hydrodynamics, magnetohydrodynamics and diverse gauge systems, which are effectively described by means of both symplectic and Poisson structures on suitable phase spaces. In particular, it was observed [25–33] that these structures are canonically related to each other. Mathematical properties, lying in a background of their analytical description, make it possible to study additional important parameters [34–50] of different hydrodynamic and magnetohydrodynamic systems, amongs<sup>t</sup> which we will mention integral invariants, describing such internal fluid motion peculiarities as vortices, topological singularities [51] and other different instability states, strongly depending [52,53] on imposed isentropic fluid motion constraints. Being interested in their general properties and mathematical structures, responsible for their existence and behavior, we present a detailed enough differential geometrical approach to investigating thermodynamically quasi-stationary isentropic fluid motions, paying more attention to analytical argumentation of tricks and techniques used during the presentation.

In particular, we consider a compressible liquid filling a compact linearly-connected domain *M* ⊂ R<sup>3</sup> with smooth boundary *∂M*, and moving free of external forces. A configuration of this fluid is called the reference or Lagrangian configuration, its points are called material or Lagrangian points and are denoted by *X* ∈ *M* and are referred to as material, or Lagrangian coordinates. We shall not for now be specific about the correct choices of the related functional spaces to be used and refer to works [191,192], where this is discussed in grea<sup>t</sup> detail. The manifold *M* ⊂ R3, thought of as the target space of a configuration *η* ∈ Diff(*M*) of the fluid at a different time, is called the spatial or Eulerian configuration, whose points, called spatial or Eulerian points, will be denoted by small letters *x* ∈ *M*. Then a motion of the fluid is a time dependent family [26,29,41,48,122,192–195] of diffeomorphisms written as

$$M \ni \mathfrak{x}\_t = \eta(X, t) := \eta\_t(X) \in M \tag{388}$$

for any initial configuration *X* ∈ *M* and some mapping *ηt* ∈ Diff(*M*), *t* ∈ R. We also are given the mass density *ρ*0 ∈ R(*M*) ⊂ *C* <sup>∞</sup>(*<sup>M</sup>*; R+) and the specific entropy *σ*0 ∈ Σ(*M*) ⊂ *C* <sup>∞</sup>(*<sup>M</sup>*; R+) of the fluid in the reference configuration, changing in time in such a way that

$$
\rho\_0(X) = \rho\_l(\mathbf{x}\_l) I\_{\overline{\eta}\_l}(\mathbf{x}\_l), \sigma\_0(X) = \sigma\_l(\mathbf{x}\_l), \tag{389}
$$

where *J<sup>η</sup>t*(*xt*) denotes the standard Jacobian determinant of the motion *ηt* ∈ Diff(*M*) at *xt* ∈ *M* and *<sup>σ</sup>t*(*xt*) denotes the specific entropy for any *xt* = *ηt*(*X*) ∈ *M* and *t* ∈ R. For a motion *xt* = *ηt*(*X*) ∈ *M* and arbitrary *X* ∈ *M*, *t* ∈ R, one usually defines three velocities: thematerialvelocity

 or Lagrangian

$$V(X,t) = V\_t(X) := \partial \eta\_t(X) / \partial t,\tag{390}$$

the spatial or Eulerian velocity

$$
\upsilon(\mathbf{x}\_{\mathfrak{t}}, t) = \upsilon\_{\mathfrak{t}}(\mathbf{x}\_{\mathfrak{t}}) := \upsilon\_{\mathfrak{t}} \circ \eta\_{\mathfrak{t}}(\mathbf{X}) \tag{391}
$$

and convective or body velocity

$$\mathcal{V}(X,t) = \mathcal{V}\_t(X) := -\partial X(\mathbf{x}\_t, \mathbf{t})/\partial \mathbf{t} = -\partial \eta\_t^{-1}(\mathbf{x}\_t)/\partial \mathbf{t},\tag{392}$$

being equivalent to the expression V*t* = *η*<sup>−</sup><sup>1</sup> *t*,<sup>∗</sup> *vt* for all *t* ∈ R. Since the velocity *vt* : *M* ∈ *T*(*M*) is tangent to *M* for all *t* ∈ R at *xt* = *ηt*(*X*) ∈ *M*, it determines a time dependent vector field on *M*. On the other hand, tangency of *Vt*(*X*) and *ηt*(*X*), *X* ∈ *M*, means that the velocity *Vt* is a vector field over a configuration *ηt* ∈ Diff(*M*) on *M*, that is *Vt* : *M* → *T*(*M*) is such a map that *Vt*(*X*) is tangent to *M* not at *X* ∈ *M*, but at point *xt* = *ηt*(*X*) ∈ *M*. Simultaneously, the velocity V*t*(*X*) is a tangent vector to *M* at *X* ∈ *M*, that is V*t* is also a time dependent vector field on *M*. In what will follow, we will think of the fluid as moving smoothly in the domain *M* ⊂ R3, at any time filling it and producing no shocks and cavitation.

We present in the introductory section a modern differential geometric description of the isentropic fluid motion phase space and featuring diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. The next section is devoted to the Hamiltonian analysis of the adiabatic liquid dynamics, within which, following the general approach of [28,41,194], we explain the nature of the related Poissonian structure on the fluid motion phase space, as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie–Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product. A modification of the Hamiltonian analysis in case of the isothermal liquid dynamics is presented in the next section. We proceed further to studying the differential-geometric structure of the adiabatic magnetohydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and invariant theory. We construct there an infinite hierarchy of different kinds of integral magneto-hydrodynamic invariants, generalizing those previously constructed in [194,196], and analyzing their differential-geometric origins. The last section presents charged fluid dynamics on the phase space invariant with respect to an Abelian gauge group transformation.

#### *10.3. A Modified Current Algebra, Its Functional Representation And Geometric Description of the Ideal Liquid Dynamics*

It is well known that the motion of an ideal compressible and isentropic fluid is governed by the Euler equations

$$\begin{aligned} \partial v / \partial t + \langle v | \nabla \rangle v + \rho^{-1} \nabla p^{(0)} &= 0, \\ \partial \rho / \partial t + \langle \nabla | \rho v \rangle &= 0, \partial v / \partial t + \langle v | \nabla \rangle \sigma = 0, \end{aligned} \tag{393}$$

where *p*0 : *M* → R is the internal fluid pressure, *σ* = *<sup>σ</sup>*(*xt*, *t*) = *<sup>σ</sup>t*(*xt*) is the specific entropy at a spatial point *xt* = *ηt*(*X*) ∈ *M* for any *t* ∈ R, which is fixed owing to the Euler Equation (393), ∇ := *∂*/*∂x* is the usual gradient on the space of smooth functions *C* <sup>∞</sup>(*<sup>M</sup>*; R) and ·|· denotes the usual convolution on *T*(*M*) × *T*(*M*) subject to the usual metric in R3, reduced on the submanifold *M*. The evolution (393) is considered to be *a priori* thermodynamic equilibrium and quasi-stationary, meaning that the following *infinitesimal heat convective* and strictly mathematical relationship (387), derived above in the Introduction,

$$
\delta\sigma\_t(\rho\_t(\mathbf{x}\_t), \sigma\_t(\mathbf{x}\_t)) = T\_t(\mathbf{x}\_t)\delta\sigma\_t(\mathbf{x}\_t) + p\_t^{(0)}(\mathbf{x}\_t)\rho\_t^{-2}(\mathbf{x}\_t)\delta\rho\_t(\mathbf{x}\_t) \tag{394}
$$

holds for all *xt* ∈ *M* and *t* ∈ R, where *et* : R(*M*) × Σ(*M*) → *C* ∞(*M* × R; R) denotes the internal specific fluid energy, *Tt* : *M* → R+ denotes the internal fluid absolute temperature, *p* (0) *t* : *M* → R is the internal liquid pressure and the variation sign "*δ* means the change subject to both the temporal variable *t* ∈ R and the spatial variable *xt* ∈ *M*.

Let us now analyze the internal mathematical structure of quantities (*ρ<sup>t</sup>*, *<sup>σ</sup>t*) ∈ R(*M*) × Σ(*M*) as the *physical observables* subject to their evolution (393) with respect to the group diffeomorphisms *ηt* ∈ Diff(*M*), *t* ∈ R, generated by the liquid motion vector field *dxt*/*dt* = *vt*(*xt*), *xt* := *ηt*(*X*), *t* ∈ R, *X* ∈ *M*:

$$\begin{array}{ll} \mathcal{L}\_{d/dt}(\rho\_l d^3 \mathbf{x}\_l \langle \upsilon\_l | d\mathbf{x}\_l \rangle) = \rho\_l d^3 \mathbf{x}\_l (-\rho\_t^{-1} d\rho\_t^{(0)} + d|\upsilon\_l|^2/2), \\ \mathcal{L}\_{d/dt}(\rho\_l d^3 \mathbf{x}\_l) = 0, \ \mathcal{L}\_{d/dt} \mathbf{v}\_t = 0, \end{array} \tag{395}$$

where L*d*/*dt* : Λ(*M*) → Λ(*M*) denotes the corresponding Lie derivative with respect to the vector field *d*/*dt* := *∂*/*∂t* + *vt*|∇ ∈ Γ(*M* × R; *<sup>T</sup>*(*M*)), *t* ∈ R. The relationships (395) here simply mean that at every fixed *t* ∈ R the space of physical observables, being by definition, the adjoint space G∗ := (Λ<sup>1</sup>(*M*) ⊗ Λ<sup>3</sup>(*M*)) ⊕ (Λ<sup>3</sup>(*M*) ⊕Λ<sup>0</sup>(*M*)) to the vector space G := <sup>Γ</sup>(*<sup>M</sup>*; *T*(*M*)) × (Λ<sup>0</sup>(*M*) ⊕ Λ<sup>3</sup>(*M*)) - *TId*(*G*), the tangent space at the identity *Id* to the extended differential-functional current group manifold *G* := Diff(*M*)×(Λ<sup>0</sup>(*M*) ×Λ<sup>3</sup>(*M*)) - Diff(*M*)×(R(*M*) <sup>×</sup><sup>Σ</sup>(*M*)), where we have naturally identified the Abelian group product Λ<sup>0</sup>(*M*) ×Λ<sup>3</sup>(*M*) with its direct tangent space sum *T*(Λ<sup>0</sup>(*M*)) <sup>⊕</sup>*<sup>T</sup>*(Λ<sup>3</sup>(*M*)).

Consider now the natural action Diff(*M*) × *G* → *G* of the Diff(*M*)-group on the constructed differential-functional manifold *G*:

$$\begin{array}{rcl}(\eta \circ \varphi)(X) := \varphi(\eta(X)), (\eta \circ r)(X) := r(\eta(X)),\\ \eta \circ (s(X)d^3X) := \eta^\*(s(X)d^3X)\end{array} \tag{396}$$

for *η* ∈ Diff(*M*), *X* ∈ *M* and any (*ϕ*;*r*,*<sup>s</sup>*) ∈ Diff(*M*)×(R(*M*) × <sup>Σ</sup>(*M*)). Then, taking into account the suitably extended action (396) on the differential-functional manifold *G*, one can formulate the following easily checkable and crucial for what will follow further proposition.

**Proposition 14.** *The functional manifold G* := Diff(*M*) × (R(*M*) × Σ(*M*)) *in Eulerian coordinates is a smooth symmetry Banach group G* := Diff(*M*) - (R(*M*) × <sup>Σ</sup>(*M*)), *equal to the semidirect product of the diffeomorphism group* Diff(*M*) *and the direct product* R(*M*)× Σ(*M*) *of the Abelian functional* R(*M*) - Λ <sup>0</sup>(*M*), *and density* Σ(*M*) - Λ<sup>3</sup>(*M*) *group, endowed in Eulerian variables with the following right group multiplication law:*

$$\begin{aligned} & \left( (\boldsymbol{\varrho}\_{1}; r\_{1}, s\_{1} d^{3} \mathbf{x}) \circ (\boldsymbol{\varrho}\_{2}; r\_{2}, s\_{2} d^{3} \mathbf{x}) = \\ &= (\boldsymbol{\varrho}\_{2} \cdot \boldsymbol{\varrho}\_{1}; r\_{1} + r\_{2} \cdot \boldsymbol{\varrho}\_{1}, s\_{1} d^{3} \mathbf{x} + (s\_{2} d^{3} \mathbf{x}) \cdot \boldsymbol{\varrho}\_{1}) \end{aligned} \tag{397}$$

*for arbitrary elements ϕ*1, *ϕ*2 ∈ Diff(*M*),*r*1,*r*<sup>2</sup> ∈ Λ<sup>0</sup>(*M*) *and <sup>s</sup>*1*d*3*x*,*s*2*d*3*<sup>x</sup>* ∈ <sup>Λ</sup><sup>3</sup>(*M*).

This proposition allows a simple enough interpretation, namely, it means that the adiabatic mixing of the *G* (*ϕ*2;*r*2,*s*2*d*3*x*)-liquid configuration with the *G* (*ϕ*1;*r*1,*s*1*d*3*x*)- liquid configuration amounts to summation of their densities and entropies, simultaneously changing the common specific density owing to the fact that some space of the domain *M* is already occupied by the first liquid configuration and the second one should be diffeomorphically shifted from this configuration to another free part of the spatial domain *M*, whose volume is assumed to be fixed and bounded.

The second important observation concerns the variational one-form (394), which can be naturally interpreted as some constraint on the group manifold *G* for any fixed initial extended Lagrangian configuration (*η*; *ρ*0, *<sup>σ</sup>*0*d*3*X*) ∈ *G*, as it follows from the conditions (389):

$$J\_{\eta\_t}(X)\rho\_t \circ \eta\_t(X) := \rho\_0(X), \; \sigma\_t \circ \eta\_t(X) := \sigma\_0(X) \tag{398}$$

for all *X* ∈ *M*, *ηt* ∈ Diff(*M*) and *t* ∈ R. In addition, if to determine, owing to (394) and the streamline adiabatic constraint *δσt*(*xt*) = 0 for all *t* ∈ R, the specific energy density

$$w\_t(\rho\_t, \sigma\_t) := w\_t^{(0)}(\rho\_t, \sigma\_t) + c\_t(\sigma\_t) \tag{399}$$

for some still unknown mapping *ct* : Σ(*M*) → *C*∞(*M* × R; R) and the internal potential energy function *w*(0) *t* : R(*M*) × Σ(*M*) → *<sup>C</sup>*<sup>∞</sup>(*<sup>M</sup>*; R) of the liquid under regard, the local energy conservation property

$$\frac{d}{dt} \int\_{D\_t} \mathbf{e}\_t(\rho\_t, \sigma\_t) \rho\_t(\mathbf{x}\_t) d^3 \mathbf{x}\_t = -\int\_{D\_t} \langle \nabla | p\_t^{(0)}(\mathbf{x}\_t) v\_t(\mathbf{x}\_t) \rangle d^3 \mathbf{x}\_t \tag{400}$$

holds for all *t* ∈ R and the domain *Dt* := *ηt*(*<sup>D</sup>* ) ⊂ *M*, where a smooth submanifold *D* ⊂ *M* is chosen arbitrarily and *ηt* : *M* → *M* denotes the corresponding evolution subgroup of the diffeomorphism group Diff0(*M*), generated by the Euler evolution Equation (393), becomes compatible with constraint (394) if there holds the following equality:

$$p\_t^{(0)}(\\\mathbf{x}\_t) = \rho\_l(\mathbf{x}\_t)^2 \delta w\_t^{(0)}(\rho\_t, \sigma\_t) / \partial \rho\_t \tag{401}$$

for all *xt* ∈ *M* and *t* ∈ R. In particular, from (400) and (401) the following global internal energy functional

$$H := \int\_M \left[ w\_t^{(0)}(\rho\_{t\prime}\sigma\_t) + c\_t(\sigma\_t) \right] \rho\_t(\mathbf{x}\_t) d^3 \mathbf{x}\_t \tag{402}$$

is conserved that is *dH*/*dt* = 0 for all *t* ∈ R.

As the extended Lagrangian configuration (*η*; *ρ*0, *<sup>σ</sup>*0*d*3*X*) ∈ *G* is fixed for all whiles of time *t* ∈ R and the dynamical variables *ρt* ∈ R(*M*) and *σt* ∈ Σ(*M*) depend only on the evolution diffeomorphisms *ηt* ∈ Diff(*M*), *t* ∈ R, it is reasonable to consider the constraint (394) as an element of the cotangent space *<sup>T</sup>*<sup>∗</sup>*ηt*(Diff(*M*)) to the diffeomorphism group Diff(*M*) at the point *ηt* ∈ Diff(*M*) for any *t* ∈ R.

Determine first the tangent space *<sup>T</sup>η*(*G*) to the group manifold *G* at point (*η*; *ρ*0, *<sup>σ</sup>*0*d*3*X*) ∈ *G*, which will be the direct product of the tangent spaces *<sup>T</sup>η*(Diff(*M*)), *<sup>T</sup>ρ*0 (Λ<sup>0</sup>(*M*)) and *<sup>T</sup><sup>σ</sup>*0*d*3*X*(Λ<sup>3</sup>(*M*)). The last two tangent spaces are isomorphic, respectively, to themselves, that is *<sup>T</sup>ρ*0 (Λ<sup>0</sup>(*M*)) - Λ<sup>0</sup>(*M*) and *<sup>T</sup><sup>σ</sup>*0*d*3*X*(Λ<sup>3</sup>(*M*)) - Λ<sup>3</sup>(*M*) at any *X* ∈ *M*. Their adjoint spaces are naturally determined as suitably constructed density and functional spaces on the manifold *M* : *T*∗*ρ*0 (Λ<sup>0</sup>(*M*)) - Λ<sup>3</sup>(*M*) and *T*∗ *<sup>σ</sup>*0*d*3*X*(Λ<sup>3</sup>(*M*)) - <sup>Λ</sup><sup>0</sup>(*M*). Concerning the tangent space *<sup>T</sup>η*(Diff(*M*)) at a configuration *η* ∈ Diff(*M*) we will make use of the construction, devised before in [122,181,194]. Namely, let *η* ∈ Diff(*M*) be a Lagrangian configuration and determine the tangent space *<sup>T</sup>η*(Diff(*M*)) at *η* ∈ Diff(*M*) as the collection of left invariant vectors *ξη* := *<sup>L</sup>η*,<sup>∗</sup>*ξ* at *η* ∈ Diff(*M*), where *Lη* : Diff(*M*) → Diff(*M*) is, by definition, the left shift on the diffeomorphism group Diff(*M*) and *ξ* ∈ *TId*(Diff(*M*)) is a tangent vector at the unity *Id* ∈ Diff(*M*). It is obvious that for all reference points *X* ∈ *M* and any smooth curve R *τ* → *ητ* ∈ Diff(*M*) of diffeomorphisms of *M*, the set of right invariant vectors *ξ*(*X*)= (*η*<sup>−</sup><sup>1</sup> ◦ *dηt*/*dτ*)(*X*))|*<sup>τ</sup>*=<sup>0</sup> ∈ *TX*(*M*) at point *X* ∈ *M* defines a smooth vector field *ξ* : *M* → *T*(*M*) on the manifold *M*. Since, by definition, the tangent space *TId*(Diff(*M*)) coincides with the Lie algebra Diff(*M*) of the diffeomorphism group Diff(*M*), strictly isomorphic to the Lie algebra Γ(*T*(*M*)) of right invariant vector fields on *M*, the dual space *<sup>T</sup>*<sup>∗</sup>*Id*(Diff(*M*)) can be naturally determined from the geometric point of view as the space *di f f* <sup>∗</sup>(*M*), consisting of analytic functions on *di f f*(*M*) and coinciding with the set of one-form densities on *M*:

$$
\hat{diff}^\*(M) \simeq \Lambda^1(M) \otimes |\Lambda^3(M)|.\tag{403}$$

Similarly, the cotangent space *T*∗*η* (Diff(*M*)) consists of all one-form densities on *M* over *η* ∈ Diff(*M*):

$$T^\*\_{\eta}(\text{Diff}(M)) = \{ a\_{\eta} : M \to T^\*(M) \otimes |\Lambda^3(M)| : a\_{\eta}(X) \in T^\*\_{\eta(X)}(M) \otimes |\Lambda^3(M)| \}\tag{404}$$

subject to the canonical nondegenerate convolution (·|·)*c* on *T*∗*η* (Diff(*M*)) × *<sup>T</sup>η*(Diff(*M*)) : if *α η* ∈ *T*∗*η* (Diff(*M*)), *ξη* ∈ *<sup>T</sup>η*(Diff(*M*)), where *αη*|*X* = *αη*(*X*)|*dx* ⊗ *d*3*X*, *ξη*|*X* = *ξη*(*X*)|*∂*/*∂x* , then

$$(\mathfrak{a}\_{\eta}|\xi\_{\eta})\_{\varepsilon} := \int\_{M} \langle \mathfrak{a}\_{\eta}(X)|\xi\_{\eta}(X)\rangle d^{3}X. \tag{405}$$

The construction above makes it possible to identify the cotangent bundle *T*∗*η* (Diff(*M*)) at the fixed Lagrangian configuration *η* ∈ Diff(*M*) to the tangent space *<sup>T</sup>η*(Diff(*M*)), insomuch as the tangent space *T*(*M*) is endowed with the natural internal tangent bundle metric ·| · *g* at any point *η*(*X*) ∈ *M*, identifying *T*(*M*) with *T*∗(*M*) via the metric isomorphism : *T*∗(*M*) → *<sup>T</sup>*(*M*). The latter can also be naturally lifted to *T*∗*η* (Diff(*M*)) at *η* ∈ Diff(*M*), namely: for any elements *αη*, *βη* ∈ *T*∗*η* (Diff(*M*)), *αη*|*X* = *αη*(*X*)|*dx* ⊗ *d*3*X* and *βη*|*X* = *βη*(*X*)|*dx* ⊗ *d*3*X* ∈ *T*∗*η* (Diff(*M*)) we can define the metric

$$(a\_{\eta}|\beta\_{\eta})\_{\mathcal{S}} := \int\_{M} \rho\_{0}(X) \langle a\_{\eta}^{\sharp}(X)|\beta\_{\eta}^{\sharp}(X)\rangle\_{\mathcal{S}} d^{3}X,\tag{406}$$

where, by definition, *<sup>α</sup>η*(*X*) := (*ρ*0(*X*)−<sup>1</sup> *αη*(*X*)|*dx* ), *<sup>β</sup>η*(*X*) := (*ρ*0(*X*)−<sup>1</sup> *βη*(*X*)|*dx* ) ∈ *<sup>T</sup>η*(*X*)(*M*) for any *X* ∈ *M*.

The diffeomorphism group Diff(*M*) can be naturally restricted to the factor-group Diff0(*M*) := Diff(*M*)/*Dif f<sup>ρ</sup>*0,*σ*0 (*M*) subject to the stationary normal symmetry subgroup Diff*ρ*0,*σ*0 (*M*) ⊂ Diff(*M*), where

$$Diff\_{\rho\_0, \sigma\_0}(M) := \{ \varphi \in \text{Diff}(M) : \rho\_0(X) = I\_{\varphi(X)} \rho\_0(\varphi(X)), \sigma\_0(X) = \sigma\_0(\varphi(X)) \}\tag{407}$$

for any *X* ∈ *M*. Based on the construction above, one can proceed to constructing smooth flows and functionals on the specially extended group manifold *G*0 := Diff0(*M*)- (Λ<sup>0</sup>(*M*)× Λ<sup>3</sup>(*M*)) and consider their coadjoint action on the cotangent bundle *T*∗*gη* (*<sup>G</sup>*0), *gη* := (*η*; *ρ*0, *<sup>σ</sup>*0) ∈ *G*0, and relate them in some way to the evolution with respect to the Euler Equation (393). Moreover, as the cotangent bundle *T*∗*gη* (*<sup>G</sup>*0), *gη* ∈ *G*0, is a priori endowed with the canonical Poisson structure, one can study both the Hamiltonian flows on it, related with the Euler Equation (393), and a hidden geometrical meaning of the differential constraints like (394).

#### *10.4. The Hamiltonian Analysis and Related Adiabatic Liquid Dynamics*

We observed above that the liquid motion is adequately described by means of the symmetry diffeomorphism group Diff0(*M*), acting on the target manifold *M* ⊂ R3, and in this way the modeling liquid motion, generated by suitable vector fields on Diff0(*M*). This also means that the fluid motion strongly depends on the constraint (394) on the cotangent bundle *T*∗*gη* (*<sup>G</sup>*0), *gη* ∈ *G*0, and *a priori* possesses the canonical Poisson structure on it. Taking into account that the diffeomorphism group Diff0(*M*) acts on the extended group density manifold *G*0 := Diff0(*M*) - (Λ<sup>0</sup>(*M*) × <sup>Λ</sup><sup>3</sup>(*M*)), fixed by the element (*η*; *ρ*0, *<sup>σ</sup>*0*d*3*X*) ∈ *G*, one can suitably construct the canonical Poisson bracket on the cotangent bundle *T*∗*gη* (*<sup>G</sup>*0), *gη* ∈ *G*0, using the canonical coordinate variables on it. Namely, let (*μη*; *<sup>ρ</sup>*0*d*3*X*, *<sup>σ</sup>*0) ∈ *T*∗*gη* (*<sup>G</sup>*0), *gη* ∈ *G*0, be coordinates on *T*∗*gη* (*<sup>G</sup>*0), where

$$\begin{split} \mu\_{\eta}(X) &= \rho\_{0}(X)[V\_{\eta}^{b}(X)]d^{3}X|\_{x=\eta(X)} = \\ &= \rho\_{0}(X)\upsilon^{b}(\eta(X))f\_{\eta^{-1}}(\mathbf{x})d^{3}\mathbf{x} := \rho(\mathbf{x})\upsilon(\mathbf{x})d^{3}\mathbf{x}, \\ r\_{\eta}(X) &= \rho\_{0}(X)d^{3}X = \rho\_{0}(X)d^{3}X|\_{x=\eta(X)} := \rho(\mathbf{x})d^{3}\mathbf{x}, \\ s\_{\eta}(X) &= \sigma\_{0}(X) = \sigma(\eta(X))|\_{x=\eta(X)} := \sigma(\mathbf{x}) \end{split} \tag{400}$$

and := −1, being suitably represented into the Eulerian spatial variables on *T*∗*gη* (*<sup>G</sup>*0) at point (*η*; *ρ*, *σd*3*x*) ∈ *G*0. In particular, the quantities *μ*(*x*) := *ρ*(*x*)*v*(*x*)*d*3*x* = (*η*<sup>∗</sup>*μη*)(*X*), *r*(*x*) := *ρ*(*x*)*d*3*x* = (*η*<sup>∗</sup>*rη*)(*X*) and *s*(*x*) := *<sup>σ</sup>*(*x*)=(*η*<sup>∗</sup>*sη*)(*X*) are called, respectively, the Eulerian momentum density, the Eulerian fluid density and entropy variables at point *x* = *η*(*X*) ∈ *M*. The corresponding metric on *T*∗*gη* (*<sup>G</sup>*0) is given by the expression

$$\begin{split} \left( (\boldsymbol{\alpha}\_{\eta,1}; \boldsymbol{r}\_{\eta,1} \mathbf{s}\_{\eta,1}) \, | \, (\boldsymbol{\alpha}\_{\eta,2}; \boldsymbol{r}\_{\eta,2} \mathbf{s}\_{\eta,2}) \right) &:= (\boldsymbol{\alpha}\_{\eta,1} | \boldsymbol{\alpha}\_{\eta,2}) + \\ &+ (\boldsymbol{r}\_{\eta,1} | \boldsymbol{r}\_{\eta,2}) + (\boldsymbol{s}\_{\eta,1} | \boldsymbol{s}\_{\eta,2}), \end{split} \tag{409}$$

where (*αη*,<sup>1</sup>|*αη*,<sup>2</sup>) for *αη*,1, *αη*,<sup>2</sup> ∈ *T*∗*η* (Diff0(*M*)) is determined by (406) and for any *<sup>r</sup>η*,1,*rη*,<sup>2</sup> ∈ *T*∗*η* (Λ<sup>3</sup>(*M*)) and *<sup>s</sup>η*,1,*sη*,<sup>2</sup> ∈ *T*∗*η* (Λ<sup>0</sup>(*M*)) one determines, respectively, as

$$\rho\_1(r\_{\eta,1}|r\_{\eta,2}) := \int\_M (\rho\_1(\mathbf{x})\rho\_2(\mathbf{x}))d^3\mathbf{x},\ (s\_{\eta,1}|s\_{\eta,2}) := \int\_M (\sigma\_1(\mathbf{x})\sigma\_2(\mathbf{x}))d^3\mathbf{x}.\tag{410}$$

Consider now the cotangent bundle *T*∗*gη* (*<sup>G</sup>*0) at point *gη* = (*η*; *ρ*, *σd*3*x*) ∈ *G*0 as a smooth manifold endowed with the canonical symplectic structure on it, equivalent to the corresponding canonical Poisson bracket on *T*∗*gη* (*<sup>G</sup>*0). Taking into account that the manifold *T*∗*gη* (*<sup>G</sup>*0), shifted by the right *<sup>R</sup>η*−<sup>1</sup> -action to the manifold *<sup>T</sup>*<sup>∗</sup>*Id*(*<sup>G</sup>*0), *Id* ∈ *G*0, becomes diffeomorphic to the adjoint space G∗ to the Lie algebra G of the group *G*0, as was stated [30–33,41] by S. Lie in 1887, this canonical Poisson bracket on *T*∗*η* (*<sup>G</sup>*0) transforms [26,31,32,41,181,195] into the classical Lie–Poisson bracket on the adjoint space G∗. Moreover, the orbits of the group *G*0 on *T*∗*gη* (*<sup>G</sup>*0), *gη* = (*η*; *ρ*, *σd*3*x*) ∈ *G*0, transform into the corresponding coadjoint orbits on the adjoint space G∗, generated by elements of the Lie algebra G. To construct this Lie–Poisson bracket, we formulate the following preliminary proposition.

**Proposition 15.** *The Lie algebra* G - <sup>Γ</sup>(*<sup>M</sup>*; *T*(*M*)) - (Λ<sup>0</sup>(*M*)) ⊕ Λ<sup>3</sup>(*M*)) *is determined by the following Lie commutator relationships:*

$$\begin{aligned} \left[ \left( a\_1; r\_1, s\_1 \right), \left( a\_2; r\_2, s\_2 \right) \right] &= \left( \left[ a\_1, a\_2 \right]; \\\\ \left\langle a\_1 \middle| \nabla r\_2 \right\rangle - \left\langle a\_2 \middle| \nabla r\_1 \right\rangle, \left\langle \nabla \middle| a\_1 s\_2 \right\rangle - \left\langle \nabla \middle| a\_2 s\_1 \right\rangle \right) \end{aligned} \tag{411}$$

*for any vector fields a*1, *a*2 ∈ *di f f*0(*M*) - <sup>Γ</sup>(*<sup>M</sup>*; *T*(*M*)) *and scalar quantities r*1,*r*2 ∈ Λ<sup>0</sup>(*M*) *and s*1,*s*2 ∈ Λ<sup>3</sup>(*M*) *on the manifold M*.

**Proof.** Proof of the commutation relationships (411) easily follows from the group multiplication (397), if to take into account that tangent spaces *T*(Λ<sup>0</sup>(*M*)) - Λ<sup>0</sup>(*M*) and *T*(Λ<sup>3</sup>(*M*)) - (Λ<sup>3</sup>(*M*)).

As an example, we calculate, for brevity, the Poisson bracket on the cotangent space *T*∗*η* (Diff(T*n*)) at any *η* ∈ Diff(T*<sup>n</sup>*). Consider the cotangent space *T*∗*η* (Diff(T*n*)) - *di f f* <sup>∗</sup>(T*<sup>n</sup>*), the adjoint space to the tangent space *<sup>T</sup>η*(Diff(T*<sup>n</sup>*)) of left invariant vector fields on Diff(T*n*) at any *η* ∈ Diff(T*<sup>n</sup>*), and take the canonical symplectic structure on *T*∗*η* (Diff(T*n*)) in the form *<sup>ω</sup>*(2)(*<sup>μ</sup>*, *η*) := *δα*(*μ*, *η*), where the canonical Liouville form *<sup>α</sup>*(*μ*, *η*) := (*μ*|*δη*)*c* ∈ <sup>Λ</sup><sup>1</sup>(*μ*,*η*)(*T*<sup>∗</sup>*<sup>η</sup>* (Diff(T*n*))) at a point (*μ*, *η*) ∈ *T*∗*η* (Diff(T*n*)) is defined *a priori* on the tangent space *<sup>T</sup>η*(Diff(T*<sup>n</sup>*)) - Γ(*T*(*M*)) of right-invariant vector fields on the torus manifold T*<sup>n</sup>*. Having calculated the corresponding Poisson bracket of smooth functions (*μ*|*a*)*<sup>c</sup>*,(*μ*|*b*)*<sup>c</sup>* ∈ *<sup>C</sup>*<sup>∞</sup>(*T*<sup>∗</sup>*η* (Diff(T*<sup>n</sup>*)); R) on *T*∗*η* (Diff(T*n*)) - *di f f* <sup>∗</sup>(T*<sup>n</sup>*), *η* ∈ Diff(T*<sup>n</sup>*), one can formulate the following proposition.

**Proposition 16.** *The Lie–Poisson bracket on the coadjoint space T*∗*η* (Diff(T*n*)) - *di f f* ∗(T*n*) *is equal to the expression*

$$(f, \emptyset)(\mu) = (\mu | [\delta f(\mu) / \delta \mu, \delta \lg(\mu) / \delta \mu])\_{\mathbb{C}} \tag{412}$$

*for any smooth functionals f* , *g* ∈ *<sup>C</sup>*<sup>∞</sup>(G∗; <sup>R</sup>).

**Proof.** By definition [26,122] of the Poisson bracket of smooth functions (*μ*|*a*)*<sup>c</sup>*,(*μ*|*b*)*<sup>c</sup>* ∈ *<sup>C</sup>*<sup>∞</sup>(*T*<sup>∗</sup>*η* (Diff(T*<sup>n</sup>*)); R) on the symplectic space *T*∗*η* (Diff(T*<sup>n</sup>*)), it is easy to calculate that

$$\begin{array}{lcl}\{\mu(a),\mu(b)\} := -\delta a(X\_{a},X\_{b}) =\\ = -X\_{a}(a|X\_{b})\_{c} + X\_{b}(a|X\_{a})\_{c} + (a|[X\_{a},X\_{b}])\_{c},\end{array} \tag{413}$$

where *Xa* := *<sup>δ</sup>*(*μ*|*a*)*c*/*δμ* = *a* ∈ *di f f*(T*<sup>n</sup>*), *Xb* := *<sup>δ</sup>*(*μ*|*b*)*c*/*δμ* = *b* ∈ *di f f*(T*<sup>n</sup>*). Since the expressions *Xa*(*α*|*Xb*)*c* = 0 and *Xb*(*α*|*Xa*)*c* = 0 owing the right-invariance of the vector fields *Xa*, *Xb* ∈ *<sup>T</sup>η*(Diff(T*<sup>n</sup>*)), the Poisson bracket (412) transforms into

$$\begin{array}{lcl} \{ (\mu|a)\_{\mathfrak{c}}, (\mu|b)\_{\mathfrak{c}} \} = (a|[X\_{\mathfrak{a}}, X\_{\mathfrak{b}}])\_{\mathfrak{c}} = \\ = (\mu|[a, b])\_{\mathfrak{c}} = (\mu|[\delta(\mu|a)\_{\mathfrak{c}}/\delta\mu, \delta(\mu|b)\_{\mathfrak{c}}/\delta\mu])\_{\mathfrak{c}} \end{array} \tag{414}$$

for all (*μ*, *η*) ∈ *T*∗*η* (Diff(T*n*)) - *di f f* <sup>∗</sup>(T*<sup>n</sup>*), *η* ∈ Diff(T*n*) and any *a*, *b* ∈ *di f f*(T*<sup>n</sup>*). The Poisson bracket (412) is easily generalized to

$$(f, \emptyset)(\mu) = (\mu | [\delta f(\mu) / \delta \mu, \delta \lg \mu) / \delta \mu |)\_{\varepsilon} \tag{415}$$

for any smooth functionals *f* , *g* ∈ *<sup>C</sup>*<sup>∞</sup>(G∗; <sup>R</sup>), finishing the proof.

Proceed now to the Grassmann algebra Λ(*M*) endowed with Hodge [197] starisomorphism ∗ : Λ(*M*) → Λ(*M*) subject to the usual metric on the tangent space *T*(*M*) and determine the adjoint space to the Abelian subalgebra R(*M*) ⊕ Σ(*M*) - Λ<sup>0</sup>(*M*) ⊕ Λ<sup>3</sup>(*M*) as the space ∗Λ<sup>3</sup>(*M*) ⊕ ∗Λ<sup>0</sup>(*M*) with respect to the following scalar product on Λ(*M*) :

$$(\alpha^{(n)}|\beta^{(m)}) := \delta\_{mn} \int\_M (\alpha^{(n)} \wedge \ast \beta^{(m)}) \tag{416}$$

for any *<sup>α</sup>*(*n*), *β*(*m*) ∈ <sup>Λ</sup>(*M*), *m*, *n* = 0, 3. Then the adjoint space G∗, owing to the expressions (409) and (389), is described by means of the Eulerian variables (*μ*; *<sup>ρ</sup>d*3*x*, *σ*) ∈ G∗ - (Λ<sup>1</sup>(*M*) ⊗ |Λ<sup>3</sup>(*M*)|) - (Λ<sup>3</sup>(*M*) ⊕ <sup>Λ</sup><sup>0</sup>(*M*)). The latter makes it possible to calculate the corresponding Lie–Poisson bracket on the adjoint space G∗ at a point *l* := (*μ*; *<sup>ρ</sup>d*3*x*, *σ*) ∈ G∗, generalizing the Poisson bracket (414):

$$\begin{aligned} \{f, \emptyset\}(l) &= \langle l | [\delta f/\delta l, \delta \emptyset/\delta l] \rangle\_{\mathfrak{c}} = \\\\ = \int\_{M} d^3 \mathbf{x} \Big\langle m \Big| \left[ \left\langle \frac{\delta f}{\delta m} \big| \nabla \right\rangle \frac{\delta \mathcal{X}}{\delta m} - \left\langle \frac{\delta \mathcal{X}}{\delta m} \big| \nabla \right\rangle \frac{\delta f}{\delta m} \right] \Big\rangle + \\ &+ \int\_{M} \rho d^3 \mathbf{x} \Big[ \left\langle \frac{\delta f}{\delta m} \bigg| \nabla \frac{\delta \mathcal{X}}{\delta \rho} \right\rangle - \left\langle \frac{\delta \mathcal{X}}{\delta m} \bigg| \nabla \frac{\delta f}{\delta \rho} \right\rangle \Big] + \\ &+ \int\_{M} \sigma \Big[ \left\langle \nabla \Big| \frac{\delta f}{\delta m} \frac{\delta \mathcal{X}}{\delta \sigma} \right\rangle - \left\langle \nabla \Big| \frac{\delta \mathcal{X}}{\delta m} \frac{\delta f}{\delta \sigma} \right\rangle \Big] d^3 \mathbf{x} \end{aligned} \tag{417}$$

for any smooth functionals *f* , *g* ∈ *<sup>C</sup>*<sup>∞</sup>(G∗; <sup>R</sup>), where we put, by definition, *μ*(*x*) := *m*(*x*)|*dx* ⊗ *d*3*x*, *m*(*x*) = *ρ*(*x*)*v*(*x*) ∈ *T*∗(*M*) for all *x* ∈ *M* and any *t* ∈ R.

Return now to the constraint (394) in the following variational form:

$$
\delta\varepsilon\_t(\rho\_t, \sigma\_t) / \delta t = T\_t(\mathbf{x}\_t) \delta\sigma\_t(\mathbf{x}\_t) / \delta t + p\_t^{(0)}(\mathbf{x}\_t)\rho\_t^{-2}(\mathbf{x}\_t)\delta\rho\_t(\mathbf{x}\_t) / \delta t,\tag{418}
$$

which should hold at any *xt* ∈ *M* for all *t* ∈ R. Insomuch as, owing to the Euler Equation (393), the full (convective) derivative *δσt*(*xt*)/*δ<sup>t</sup>* = 0 at any *xt* ∈ *M* for all *t* ∈ R, one checks once more that the expression (399) holds at any *xt* ∈ *M* for all *t* ∈ R. To determine the energy density function (399), we consider the Euler Equation (393) and

check their Hamiltonian structure subject to the Poisson bracket (417), i.e., the existence of a Hamiltonian functional *H* : G∗ → R, for which

$$\frac{\partial}{\partial t}(m;\rho,\sigma)^{\mathsf{T}} = \{H,(m,\rho,\sigma)^{\mathsf{T}}\}\tag{419}$$

at any element *l* = (*m* := *ρv*; *ρ*, *σ*)- ∈ G∗. By means of easy calculations, one obtains from the system (419) the variational gradient vector

$$\delta H(\mathbf{l})/\delta \mathbf{l} = (m\rho^{-1}; -|m|^2/(2\rho^2) + w^{(0)}\left(\rho, \sigma\right) + \rho \partial w^{(0)}(\rho, \sigma)/\partial \rho, \rho \partial w^{(0)}(\rho, \sigma)/\partial \sigma), \tag{420}$$

from which one derives [11,59,120] via the Volterra homotopy mapping

$$H = \int\_0^1 (\delta H(\lambda l) / \delta l | l)\_c d\lambda \tag{421}$$

the exact Hamiltonian expression

$$H = \int\_{M} \left( |m|^2 / (2\rho) + \rho w^{(0)}(\rho, \sigma) \right) d^3 \mathbf{x},\tag{422}$$

coinciding with the expression (402) at *c*(*σ*) := |*m*|2/(<sup>2</sup>*ρ*<sup>2</sup>) = |*v*|2/2, as *m* := *ρv* for *v* ∈ *<sup>T</sup>*(*M*). Thus, we obtain the internal energy density functional (399) as

$$\left|\upsilon\_{t}(\rho\_{t},\sigma\_{t}) = |\upsilon\_{t}|^{2}/2 + w\_{t}^{(0)}(\rho\_{t},\sigma\_{t}),\tag{423}$$

for all *ρ* := *ρt* ∈ R(*M*), *σ* := *σt* ∈ Σ(*M*) and *vt* ∈ *<sup>T</sup>*(*M*), satisfying simultaneously both the constraint (394) and the Euler evolution Equation (393) for all *t* ∈ R. Moreover, from the condition (400) one easily finds [194] the following important local differential relationship:

$$\begin{split} \left\| \left[ \rho\_{l}(\mathbf{x}\_{l}) v\_{l}(\rho\_{l}, \sigma\_{l}) \right] / \right\| &+ \left< \nabla \left| \rho\_{l}(\mathbf{x}\_{l}) v\_{l}(\mathbf{x}\_{l}) \right| e\_{l}(\rho\_{l}, \sigma\_{l}) + \\ &+ \rho\_{l}(\mathbf{x}\_{l}) \left\| w\_{l}^{(0)}(\rho\_{l}, \sigma\_{l}) / \partial \rho\_{l} \right\| \right> = 0, \end{split} \tag{424}$$

 satisfied for all *xt* ∈ *M* and *t* ∈ R, also confirming the energy conservation (422).

#### *10.5.TheHamiltonianAnalysisand RelatedIsothermalLiquidDynamics*

ConsideraliquidmotiongovernedbytheEulerequations

$$\begin{aligned} \partial \boldsymbol{\upsilon}/\partial \boldsymbol{t} + \langle \boldsymbol{\upsilon} | \boldsymbol{\nabla} \rangle \boldsymbol{\upsilon} + \boldsymbol{\rho}^{-1} \boldsymbol{\nabla} \boldsymbol{p}^{(0)} &= \boldsymbol{0}, \\ \partial \boldsymbol{\rho}/\partial \boldsymbol{t} + \langle \boldsymbol{\nabla} | \boldsymbol{\rho} \boldsymbol{\upsilon} \rangle &= \boldsymbol{0}, \partial \boldsymbol{T}/\partial \boldsymbol{t} + \langle \boldsymbol{\upsilon} | \boldsymbol{\nabla} \boldsymbol{T} \rangle = \boldsymbol{0}, \end{aligned} \tag{425}$$

and describing the ideal compressible and isothermal motion of an ideal compressible and adiabatic fluid in a spatial domain *M* ⊂ R3, as the temperature *Tt*(*xt*) = *<sup>T</sup>*0(*xt*) at any evolution point *xt* := *ηt*(*X*) ∈ *M* for all *X* ∈ *M* and *t* ∈ R. The evolution (425) is considered to be *a priori* thermodynamically quasi-stationary, i.e., the following, *infinitesimal convective* energy relationship

$$
\delta \tilde{h}\_t(\rho\_t, T\_t) = -\sigma\_t(\mathbf{x}\_t) \, \delta T\_t + p\_t^{(0)}(\mathbf{x}\_t) \, \rho\_t^{-2} \delta \rho\_t \tag{426}
$$

holds for all densities *ρt* ∈ R(*M*), temperature *Tt* ∈ T (*M*) and specific entropy *σt* ∈ <sup>Σ</sup>(*M*), where ˜ *h* : R(*M*) × T (*M*)→ R denotes the corresponding internal specific fluid "energy"and the variation sign "*δ* means the change subject to both the temporal variable *t* ∈ R and the spatial variable *xt* ∈ *M*. Under the imposed *isothermal condition δTt* = 0 the expression (426) transforms into

$$
\tilde{h}\_t(\rho\_{t\prime}T\_t) = |\upsilon\_t|^2/2 + \tilde{w}\_t^{(0)}(\rho\_{t\prime}T\_t),
\tag{427}
$$

where *w*˜(0) *t* (*ρ<sup>t</sup>*, *Tt*) := *w*(0) *t* (*ρ<sup>t</sup>*, *<sup>σ</sup>t*)|*<sup>σ</sup>t*:=*σ*˜(*ρ<sup>t</sup>*,*Tt*) − *Tt<sup>σ</sup>t*(*ρ<sup>t</sup>*, *Tt*), is the specific potential liquid energy for *the isothermal flow*, determined at *σt* := *<sup>σ</sup>t*(*ρ<sup>t</sup>*, *Tt*), solving the functional relation *Tt* = *<sup>∂</sup>w*(0) *t* (*ρ<sup>t</sup>*, *<sup>σ</sup>t*)/*∂σt* ∈ T (*M*) subject to the entropy argumen<sup>t</sup> *σt* ∈ <sup>Σ</sup>(*M*), if the condition *<sup>∂</sup>*2*w*(0) *t*(*ρ<sup>t</sup>*, *<sup>σ</sup>t*)/*∂σ*2*t*= 0 holds for all densities *ρt* ∈ R(*M*) and *t* ∈ R.

Observe now that the third equation of (425) is exactly equivalent to the internal average fluid kinetic energy conservation integral relationship

$$\frac{d}{dt} \int\_{D\_t} \rho\_t(\mathbf{x}\_t) T\_t(\mathbf{x}\_t) d^3 \mathbf{x}\_t = 0 \tag{428}$$

over the domain *Dt* := *ηt*(*D*) ⊂ *M*, where a smooth submanifold *D* = *Dt*|*<sup>t</sup>*=<sup>0</sup> ⊂ *M* is chosen arbitrarily and *ηt* : *M* → *M*, *t* ∈ R, denotes the corresponding evolution subgroup of the diffeomorphism group Diff0(*M*), generated by the Euler evolution Equation (425). The relationship (428) simply means that if the density function *ρt* ∈ R(*M*) transforms under diffeomorphism group Diff0(*M*) action as the Abelian functional group R(*M*) - <sup>Λ</sup><sup>0</sup>(*M*), the corresponding transformation of the temperature *Tt* ∈ T (*M*) is induced by the diffeomorphism group Diff0(*M*) action on the related Abelian group T (*M*) - <sup>Λ</sup><sup>3</sup>(*M*). Concerning the energy density (427) one easily obtains the following differential relationship:

$$\left[\partial \left[\rho\_t(\mathbf{x}\_t)\tilde{h}\_t(\rho\_{t\star}T\_t)\right]/\partial t + \left<\nabla \left|\rho\_t(\mathbf{x}\_t)v\_t(\mathbf{x}\_t)\right|\right>\left[\tilde{h}\_t(\rho\_{t\star}T\_t)\right> + \rho\_t\partial\tilde{w}\_t^{(0)}(\rho\_{t\star}T\_t)/\partial\rho\_t\right]/\left> = 0,\tag{429}$$

satisfied for all *t* ∈ R. As a simple consequence of the relationship (429), one obtains that the following functional

$$
\tilde{H} = \int\_{D\_t} \rho\_t(\mathbf{x}\_t) \tilde{h}\_t(\rho\_t, T\_t) d^3 \mathbf{x}\_t \tag{430}
$$

is conserved over the domain *Dt* := *ηt*(*D*) ⊂ *M*, *t* ∈ R, where a smooth submanifold *D* = *Dt*|*<sup>t</sup>*=<sup>0</sup> ⊂ *M* is chosen arbitrarily.

Similarly to the reasoning above, one can now construct the differential-functional group space Diff(*M*) × (R(*M*) × T (*M*)) and formulate the following easily checkable proposition. The differential-functional group functional manifold Diff(*M*) × (R(*M*) × T (*M*)) in Eulerian coordinates is a smooth Banach group *G* := Diff(*M*)-(R(*M*) × T (*M*)), equal to the semidirect product of the diffeomorphism group Diff(*M*) and the direct product R(*M*) × T (*M*) of Abelian functional R(*M*) - Λ<sup>0</sup>(*M*) and density T (*M*) - Λ<sup>3</sup>(*M*) groups, endowed with the following group multiplication law:

$$\begin{array}{ll}(\;\!\!\begin{array}{c}(\;\!\!p\_{1};r\_{1},\mathsf{r}\_{1}d^{3}\mathsf{x} & \;\!\right)\circ\left(\;\!\!p\_{2};r\_{2},\mathsf{r}\_{2}d^{3}\mathsf{x}\right)=\\+\left(\;\!\!p\_{2}\cdot\mathsf{q}\_{1};r\_{1}+r\_{2}\cdot\mathsf{q}\_{1},\mathsf{r}\_{1}d^{3}\mathsf{x}+\left(\mathsf{r}\_{2}d^{3}\mathsf{x}\right)\cdot\mathsf{q}\_{1}\right)\end{array} \tag{431}$$

for arbitrary elements *ϕ*1, *ϕ*2 ∈ Diff(*M*),*r*1,*r*<sup>2</sup> ∈ Λ<sup>0</sup>(*M*) and *<sup>τ</sup>*1*d*3*x*, *<sup>τ</sup>*2*d*3*<sup>x</sup>* ∈ <sup>Λ</sup><sup>3</sup>(*M*).

This proposition allows a simple enough interpretation, namely, it means that the adiabatic mixing of the *G* (*ϕ*2;*r*2, *<sup>τ</sup>*2*d*3*x*) liquid configuration with the *G* (*ϕ*1;*r*1, *<sup>τ</sup>*1*d*3*x*) liquid configuration amounts to summation of their spatially shifted densities, simultaneously changing the common specific kinetic energy, proportional [55,198,199] to the liquid temperature, owing to the fact that some space in the domain *M* is already occupied by the first liquid configuration and the second one should be diffeomorphically shifted from this configuration to another free part of the spatial domain *M* with fixed and bounded volume. The diffeomorphism group Diff(*M*) can be naturally restricted to the factor-group Diff0(*M*) := Diff(*M*)/Diff*<sup>ρ</sup>*0,*T*0 (*M*) subject to the stationary normal symmetry subgroup Diff0(*M*) := Diff*ρ*0,*T*0 (*M*) ⊂ Diff(*M*), where

$$\text{Diff}\_{\rho\_0, T\_0}(M) := \{ \varphi \in \text{Diff}(M) : \rho\_0(X) = I\_{\varphi(X)} \rho\_0(\varphi(X)), T\_0(X) = T\_0(\varphi(X)) \}\tag{432}$$

for any *X* ∈ *M*. Based on the construction above one can proceed to studying the extended Banach group *G* := Diff0(*M*) - (Λ<sup>0</sup>(*M*) × Λ<sup>3</sup>(*M*)) action on the cotangent bundle *T*∗*gη* (*G*) at *gη* := (*η*; *ρ*0, *<sup>T</sup>*0) ∈ *G*0, generated by the fluid evolution with respect to the Euler Equation (425). The related fluid motion is naturally modelled by means of the coadjoint action of the corresponding Lie algebra G - *Tgη* (*<sup>G</sup>*0) - <sup>Γ</sup>(*<sup>M</sup>*; *T*(*M*)) - (Λ<sup>0</sup>(*M*) ⊕ Λ<sup>3</sup>(*M*)) of the group *G*0, *gη* = *Id* ∈ *G*0, on its adjoint space G∗ - (Λ<sup>1</sup>(*M*) ⊗ Λ<sup>3</sup>(*M*)) - (∗Λ<sup>0</sup>(*M*) ⊕ ∗Λ<sup>3</sup>(*M*)) = (Λ<sup>1</sup>(*M*) ⊗ Λ<sup>3</sup>(*M*)) -(Λ<sup>3</sup>(*M*) ⊕ <sup>Λ</sup><sup>0</sup>(*M*)).

The related Lie structure on G easily ensues from the action (431):

$$\begin{array}{l} \left[ \left( a\_1; r\_1, r\_1 \right), \left( a\_2; r\_2, r\_2 \right) \right] = \left( \left[ a\_1, a\_2 \right]; \\\left\langle a\_1 \middle| \nabla r\_2 \right\rangle - \left\langle a\_2 \middle| \nabla r\_1 \right\rangle, \left\langle \nabla \middle| a\_1 r\_2 \right\rangle - \left\langle \nabla \middle| a\_2 r\_1 \right\rangle \right) \end{array} \tag{433}$$

for any representative elements (*<sup>a</sup>*1;*r*1, *<sup>τ</sup>*1) and (*<sup>a</sup>*2;*r*2, *<sup>τ</sup>*2) ∈ G. Moreover, as the cotangent bundle *T*∗*gη* (*<sup>G</sup>*0) at *gη* = *Id* ∈ *G*0 is diffeomorphic to the adjoint space G∗ to the Lie algebra G of the Banach group *G*0, it is *a priori* endowed with the canonical Lie–Poisson structure

$$\begin{array}{l} \{f, \emptyset\}(l) = (l | [\delta \mathfrak{g}/\delta l, \delta f/\delta l])\_c = \\ = \int\_M d^3 x \left< m \left[ \left< \frac{\delta f}{\delta m} | \nabla \right> \frac{\delta \mathfrak{g}}{\delta m} - \left< \frac{\delta \mathfrak{g}}{\delta m} | \nabla \right> \frac{\delta f}{\delta m} \right] \right> + \\ + \int\_M \rho d^3 x \left[ \left< \frac{\delta f}{\delta m} | \nabla \frac{\delta \mathfrak{g}}{\delta \rho} \right> - \left< \frac{\delta \mathfrak{g}}{\delta m} | \nabla \frac{\delta f}{\delta \rho} \right> \right] + \\ + \int\_M T \left[ \left< \nabla \left| \frac{\delta f}{\delta m} \frac{\delta \mathfrak{g}}{\delta T} \right> - \left< \nabla \left| \frac{\delta \mathfrak{g}}{\delta m} \frac{\delta f}{\delta T} \right> \right] \right] d^3 x \end{array} \tag{434}$$

for any smooth functional *f* , *g* ∈ *<sup>C</sup>*<sup>∞</sup>(G∗; <sup>R</sup>), where we put, by definition, an element *l* := (*m*; *ρ*, *T*) - (*μ*; *<sup>ρ</sup>d*3*x*, *T*) ∈ G∗, *μ*(*x*) := *m*(*x*)|*dx* ⊗ *d*3*x*, *m*(*x*) = *ρ*(*x*)*v*(*x*) ∈ *T*∗(*M*) for all *x* ∈ *M* and *t* ∈ R, one can easily check that the flow (425) is Hamiltonian:

$$\text{dl}/\text{dt} = \{\text{H}, \text{l}\}\tag{435}$$

subject to the adjusted Hamiltonian functional (430):

$$\tilde{H} := \int\_M \rho\_l h\_l(\rho\_l, T\_l) d^3 \mathbf{x}\_l = \int\_M \rho\_l (|m\_l|^2 / 2\rho\_l^2 + \tilde{w}\_l^{(0)}(\rho\_l, T\_l)) d^3 \mathbf{x}. \tag{436}$$

satisfying the conservative condition *dH* ˜ /*dt* = 0 for all *t* ∈ R, following simultaneously both from (435) and from the differential relationship (429).

#### *10.6. The Hamiltonian Analysis and Adiabatic Magneto-Hydrodynamic Superfluid Motion*

We start with considering a quasi-neutral superfluid contained in a domain *M* ⊂ R<sup>3</sup> and interacting with a "frozen" sourceless magnetic field *B* ∈ B(*M*) ⊂ *<sup>C</sup>*<sup>∞</sup>(*<sup>M</sup>*;E<sup>3</sup>), satisfying the superconductivity conditions

$$E := E + \boldsymbol{\upsilon} \times \boldsymbol{B} = 0,\ \partial E/\partial \mathfrak{t} = \nabla \times \boldsymbol{B},\tag{437}$$

where *E* ˜ : *M* → E<sup>3</sup> is the internal net superfluid electric field, *E* = −*∂A*/*∂t* : *M* → E<sup>3</sup> and *B* = ∇ × *A* : *M* → E<sup>3</sup> are the internal electric and magnetic fields, respectively, generated by the corresponding magnetic vector field potential *A* : *M* → E3, *v* : *M* −→ *T*(*M*) is the superfluid velocity and "×" denotes the usual vector product in the Euclidean space E3. The following natural boundary conditions *n*|*v* |*∂M* = 0 and *n*|*<sup>B</sup>* |*∂M* = 0 are imposed on the superfluid flow, where *n* ∈ *T*∗(*M*) is the vector normal to the boundary *∂M*, which is considered to be smooth almost everywhere.

Then, in adiabatic magneto-hydrodynamics (MHD), quasi-neutral superconductive superfluid motion is described by the following system of evolution equations:

$$\begin{aligned} \partial v/\partial t + \langle v|\nabla\rangle v + \rho^{-1}\nabla p - \rho^{-1}(\nabla \times B) \times B &= 0, \\ \partial\rho/\partial t + \langle \nabla|\rho v\rangle = 0, \partial\sigma/\partial t + \langle u|\nabla\sigma\rangle &= 0, \partial B/\partial t = \nabla \times (v \times B), \end{aligned} \tag{438}$$

where, as before, *ρ* := *ρt* ∈ R(*M*) is the superfluid density, *B* := *Bt* : *M* −→ E<sup>3</sup> is the "frozen" into the superfluid magnetic field, *p* := *pt* : *M* −→ R is the internal liquid pressure and *σ* := *σt* : *M* −→ R is the specific superfluid entropy at time *t* ∈ R. The latter is related to the internal MHD superfluid specific energy function *e* = *et*(*ρ<sup>t</sup>*, *<sup>σ</sup>t*) owing to the first thermodynamic law:

$$T\_t(\rho\_{t\prime}, \sigma\_t) \; \delta \sigma\_t = \delta e\_t(\rho\_{t\prime}, \sigma\_t) - p\_t \rho\_t^{-2} \delta \rho\_{t\prime} \tag{439}$$

satisfied for any admissible variations of the phase space parameters *ρt* ∈ R(*M*), *σt* ∈ <sup>Σ</sup>(*M*), where *Tt* = *Tt*(*ρ<sup>t</sup>*, *<sup>σ</sup>t*) is the internal absolute temperature in the superfluid for *t* ∈ R. The isentropic condition *δσt*(*xt*) = 0, where *xt* := *ηt*(*X*) ∈ *M* for all *X* ∈ *M* and that related to (438) evolution diffeomorphism *ηt* ∈ Diff(*M*), *t* ∈ R, entails the following expression for the specific internal energy

$$w\_t(\rho\_t, \sigma\_t) = w\_t^{(0)}(\rho\_{t\prime}\sigma\_t) + c\_t(\rho\_{t\prime}B\_t),\tag{440}$$

where *w*(0) *t* : R(*M*) × Σ(*M*) → *<sup>C</sup>*<sup>∞</sup>(*<sup>M</sup>*; R) is the corresponding internal potential specific energy density and *ct* : R(*M*) × B(*M*) → *<sup>C</sup>*<sup>∞</sup>(*<sup>M</sup>*; R) is some still unknown function, depending in general on the imposed magnetic field *Bt* : *M* −→ E3, *t* ∈ R.

Let us now analyze, as before, the mathematical structure of quantities (*ρ<sup>t</sup>*, *σt*, *Bt*) ∈ R(*M*) × Σ(*M*) × B(*M*) as the *physical observables* subject to their evolution (438) with respect to the group diffeomorphisms *ηt* ∈ Diff(*M*), *t* ∈ R, generated by the liquid motion vector field *dxt*/*dt* = *vt*(*xt*), *xt* := *ηt*(*X*), *t* ∈ R, *X* ∈ *M*:

$$\begin{cases} \mathcal{L}\_{d/dt}(\langle \rho\_t \upsilon\_t | d\mathbf{x}\_t \rangle d^3 \mathbf{x}\_t) = [-dp\_t^{(0)} + \rho\_t^{-1} d |\upsilon\_t|^2 / 2 + \langle B\_t | \nabla \rangle \langle B\_t | d\mathbf{x}\_t \rangle)] \rho\_t d^3 \mathbf{x}\_t \\\ \mathcal{L}\_{d/dt}(\rho\_t d^3 \mathbf{x}\_t) = 0, \quad \mathcal{L}\_{d/dt} \sigma\_t = 0, \quad \mathcal{L}\_{d/dt}(\* \langle B\_t | d\mathbf{x}\_t \rangle) = 0, \end{cases} \tag{441}$$

where L*d*/*dt* : Λ(*M*) → Λ(*M*) denotes the corresponding Lie derivative with respect to the vector field *d*/*dt* := *∂*/*∂t* + *vt*|∇ ∈ Γ(*M* × R; *<sup>T</sup>*(*M*)), *t* ∈ R. The relationships (441) mean that the space of physical observables, being by definition, the adjoint space G∗*em* := <sup>Λ</sup><sup>1</sup>(*M*)*d*3*x* × (Λ<sup>3</sup>(*M*) ⊕ Λ<sup>0</sup>(*M*) ⊕ Λ<sup>2</sup>(*M*)) to the extended configuration space is equal to G*em* := *di f f*(*M*) × (Λ<sup>0</sup>(*M*) ⊕ Λ<sup>3</sup>(*M*) ⊕ Λ<sup>1</sup>(*M*)) - *TId*(*Gem*), the tangent space at the identity *Id* to the extended differential-functional group manifold *Gem* := Diff(*M*) × Λ<sup>0</sup>(*M*) ×Λ<sup>3</sup>(*M*) × Λ<sup>1</sup>(*M*) - Diff(*M*) × R(*M*) ×<sup>Σ</sup>(*M*) × B(*M*), where we have naturally identified the Abelian group product Λ<sup>0</sup>(*M*) ×Λ<sup>3</sup>(*M*) × Λ<sup>1</sup>(*M*) with its direct tangent space sum *T*(Λ<sup>0</sup>(*M*)) ⊕*<sup>T</sup>*(Λ<sup>3</sup>(*M*)) <sup>⊕</sup>*<sup>T</sup>*(Λ<sup>1</sup>(*M*)).

Consider now the constructed differential-functional current group manifold *Gem* in Eulerian variables, on which one naturally acts the Diff(*M*)-group Diff(*M*) × *Gem* → *Gem* the standard way:

$$\begin{array}{lcl}(\eta \circ \rho)(X) := \varphi(\eta(X)), (\eta \circ r)(X) := r(\eta(X)),\\ \eta \circ (s(X)d^3X) := \eta^\*(s(X)d^3X),\\ \eta \circ \langle b(X)|dX\rangle := \eta^\*\langle b(X)|d^3X\rangle\end{array} \tag{442}$$

for *η* ∈ Diff(*M*), *X* ∈ *M* and any (*ϕ*;*r*,*s*, *b*) ∈ Diff(*M*) × R(*M*) × Σ(*M*) × B(*M*). Then, taking into account the suitably extended action (442) on the differential-functional manifold *Gem*,, one can formulate the following easily checkable further proposition that is crucial for what will follow.

**Proposition 17.** *The differential-functional current group manifold Gem* := Diff(*M*) × R(*M*)× Σ(*M*) ×B(*M*) *in Eulerian coordinates is a smooth symmetry Banach group Gem* := Diff(*M*) - (R(*M*) × Σ(*M*) × B(*M*)), *equal to the semidirect product of the diffeomorphism group* Diff(*M*) *and the direct product* R(*M*)× <sup>Σ</sup>(*M*)× B(*M*) *of abelian functional* R(*M*) - <sup>Λ</sup><sup>0</sup>(*M*), *density* Σ(*M*) - Λ<sup>3</sup>(*M*) *and one-form* B(*M*) - Λ<sup>1</sup>(*M*) *groups, endowed with the following group multiplication law in Eulerian variables:*

$$\begin{aligned} \left( \begin{aligned} (\boldsymbol{\varrho}\_{1}; \boldsymbol{r}\_{1}, \boldsymbol{s}\_{1} d^{3} \mathbf{x}, \langle b\_{1} | d\boldsymbol{x} \rangle) \circ (\boldsymbol{\varrho}\_{2}; \boldsymbol{r}\_{2}, \boldsymbol{s}\_{2} d^{3} \mathbf{x}, \langle b\_{2} | d\boldsymbol{x} \rangle) = \\ = (\boldsymbol{\varrho}\_{2} \cdot \boldsymbol{\varrho}\_{1}; \boldsymbol{r}\_{1} + \boldsymbol{r}\_{2} \cdot \boldsymbol{\varrho}\_{1}, \boldsymbol{s}\_{1} d^{3} \mathbf{x} + (\boldsymbol{s}\_{2} d^{3} \mathbf{x}) \cdot \boldsymbol{q}\_{1}, \langle b\_{1} | d\boldsymbol{x} \rangle + \langle b\_{2} | d\boldsymbol{x} \rangle \diamond \boldsymbol{\varrho}\_{1}) \end{aligned} \end{aligned} \tag{443}$$

*for arbitrary elements ϕ*1, *ϕ*2 ∈ Diff(*M*),*r*1,*r*<sup>2</sup> ∈ <sup>Λ</sup><sup>0</sup>(*M*), *<sup>s</sup>*1*d*3*x*,*s*2*d*3*<sup>x</sup>* ∈ Λ<sup>3</sup>(*M*) *and b*1|*dx* , *b*2|*dx* ∈ <sup>Λ</sup><sup>1</sup>(*M*).

Thus, one can proceed to studying the corresponding coadjoint action of the Lie algebra G*em* - *TId*(*Gem*), *Id* ∈ *Gem*, on the adjoint space G∗*em*. As the Lagrangian configuration *η*0 ∈ Diff(*M*) and the entropy *σ*0 ∈ Σ(*M*) are assumed to be invariant under the Banach diffeomorphism group action Diff(*M*), the resulting group action can be reduced to the factor-group *Dif f*0(*M*) := Diff(*M*)/*Dif f<sup>η</sup>*0,*σ*0 (*M*) action on the semidirect group product *Gem*,<sup>0</sup> := *Dif f*0(*M*) - (R(*M*) × Σ(*M*) × B(*M*). Based on the multiplication law (443), one easily calculates the following Lie algebra commutation relationships:

$$\begin{array}{l} \left[ (a\_1; r\_1, s\_1, b\_1), (a\_2; r\_2, s\_2, b\_2) \right] = (\left[ a\_1, a\_2 \right]; \left< a\_1 \right| \nabla r\_2 \rangle - \\\ - \left< a\_2 \right| \nabla r \rangle, \left< \nabla \left| a\_1 b\_2 \right> - \left< \nabla \left| a\_2 s\_1 \right> , \left< a\_1 \right| \nabla \right> b\_2 - \\\ - \left< a\_2 \right| \nabla \right> b\_1 + \left< b\_2 \right| \diamond \nabla a\_1 \rangle - \left< b\_1 \right| \diamond \nabla a\_2 \rangle \end{array} \tag{444}$$

for any elements *a*1, *a*2 ∈ *di f f*(*M*) - *<sup>T</sup>*(*M*),*r*1,*r*<sup>2</sup> ∈ R(*M*) - <sup>Λ</sup><sup>0</sup>(*M*), *s*1,*s*2 ∈ Σ(*M*) - Λ<sup>3</sup>(*M*) and *b*1, *b*2 ∈ B(*M*) - <sup>Λ</sup><sup>1</sup>(*M*).

The adjoint space to the semidirect product Lie algebra G*em*,<sup>0</sup> = *di f f*(*M*) - (R(*M*) ⊕ Σ(*M*) ⊕B(*M*)) can be, naturally, written symbolically as the space G∗*em*,<sup>0</sup> = (Λ<sup>1</sup>(*M*) ⊗ Λ<sup>3</sup>(*M*)) × (∗Λ<sup>0</sup>(*M*) ⊕ ∗Λ<sup>3</sup>(*M*) ⊕ ∗ Λ<sup>1</sup>(*M*)) = *di f f* ∗(*M*) × (Λ<sup>3</sup>(*M*)⊕ Λ<sup>0</sup>(*M*) ⊕ <sup>Λ</sup><sup>2</sup>(*M*)), whereas before, the mapping ∗ : Λ(*M*) → Λ(*M*) denotes the Hodge isomorphism. Then, taking into account the adjoint space G∗*em*,<sup>0</sup> to the current Lie algebra G*em*,<sup>0</sup> is endowed with the following [27,28,41,177,194,200] canonical Lie–Poisson bracket

$$\begin{cases} \{f,g\} := \int\_M \langle m \vert \langle \frac{\delta f}{\delta m} \vert \nabla \rangle \frac{\delta g}{\delta m} - \langle \frac{\delta g}{\delta m} \vert \nabla \rangle \frac{\delta f}{\delta m} \rangle d^3 \mathbf{x} + \\ \quad + \int\_M \rho \left( \langle \frac{\delta f}{\delta m} \vert \nabla \frac{\delta g}{\delta \rho} \rangle - \langle \frac{\delta g}{\delta m} \vert \nabla \frac{\delta f}{\delta \rho} \rangle \right) d^3 \mathbf{x} + \int\_M \rho \langle \nabla \vert (\langle \frac{\delta f}{\delta m} \frac{\delta g}{\delta \sigma} - \frac{\delta g}{\delta m} \frac{\delta f}{\delta \sigma}) \rangle d^3 \mathbf{x} + \\ \quad + \int\_M \left( \langle B \vert \langle \frac{\delta f}{\delta m} \vert \nabla \rangle \frac{\delta g}{\delta \sigma} - \langle \frac{\delta g}{\delta m} \vert \nabla \rangle \frac{\delta f}{\delta \sigma} \rangle + \langle \frac{\delta f}{\delta B} \vert \langle B \vert \nabla \rangle \frac{\delta g}{\delta m} \rangle - \langle \frac{\delta g}{\delta B} \vert \langle B \vert \nabla \rangle \frac{\delta f}{\delta m} \rangle \right) d^3 \mathbf{x} \end{cases} \tag{445}$$

for any smooth functionals *f* , *g* ∈ <sup>D</sup>(G∗*em*,<sup>0</sup>) on the adjoint space G∗, where, as before, we denoted by *m* := *ρv* ∈ *T*∗(*M*) the specific momentum of the superfluid. The bracket (445) naturally ensues from the canonical symplectic structure on the cotangent phase space *<sup>T</sup>*<sup>∗</sup>(*Gem*,<sup>0</sup>), as it was previously demonstrated in the section above.

We now write down the first two equations of the Euler MHD system (438) as the local fluid mass and momentum conservation laws in the integral Ampere–Newton form

$$\begin{aligned} \frac{d}{dt} \int\_{D\_t} \rho\_t d^3 \mathbf{x}\_t &= 0, & \frac{d}{dt} \int\_{D\_t} \rho\_t \mathbf{v}\_t \, d^3 \mathbf{x}\_t + \\ + \int\_{\partial D\_t} p\_t^{(0)}(\mathbf{x}\_t) d^2 \mathbf{S}\_t - \int\_{D\_t} \langle B\_t(\mathbf{x}\_t) \rangle \nabla \rangle B\_t(\mathbf{x}\_t) d^3 \mathbf{x}\_t &= 0, \end{aligned} \tag{446}$$

which is completely equivalent to the relationships (441) and where *p*(0) *t* : *M* → R+ is the net internal superfluid pressure, (∇ × *Bt*(*xt*)) × *Bt*(*xt*) : *M* → *<sup>C</sup>*<sup>∞</sup>(*<sup>M</sup>*;E<sup>3</sup>) is the spatially distributed Lorentz force on the superfluid, *d*<sup>2</sup>*St* is the respectively oriented surface measure on the boundary *∂Dt* for the domain *Dt* := *ηt*(*<sup>D</sup>* ) ⊂ *M*, *t* ∈ R, and a smooth submanifold *D* ⊂ *M* is chosen arbitrarily. Taking into account that (∇ × *Bt*(*xt*)) × *Bt*(*xt*) = *Bt*|∇ *Bt* − ∇ *Bt*|*Bt* /2 for any *Bt* ∈ B(*M*), the second integral relationship (446) becomes equivalent to the following:

$$
\partial v\_t / \partial t + \langle v\_t | \nabla \rangle v\_t + \rho\_t^{-1} \nabla p\_t^{(0)} (\rho\_t, \sigma\_t) - \rho\_t^{-1} \langle B\_t | \nabla \rangle B\_t = 0,\tag{447}
$$

where we have represented the internal superfluid pressure quantity as

$$p\_l(\mathbf{x}\_l) := p\_t^{(0)}(\rho\_l, \sigma\_l) - \langle B\_l | B\_l \rangle / 2 \tag{448}$$

for some mapping *p*(0) *t* : R(*M*) × Σ(*M*) → *<sup>C</sup>*<sup>∞</sup>(*<sup>M</sup>*; <sup>R</sup>), strictly depending only on the internal liquid configuration *ηt* ∈ Diff(*M*) for all *t* ∈ R.

Based on the Poisson bracket expression (445), one can now easily determine the Hamiltonian function *H* : *M* → R, corresponding to the Euler evolution equation (438) on the adjoint space G∗:

$$\begin{array}{l} H = \int\_M \rho\_t(|m\_t|^2/(2\rho\_t^2) + w\_t^{(0)}(\rho\_{t\prime}\sigma\_t) + \\ + |B\_t|^2/(2\rho\_t))d\mathbf{x}\_t^3 := \int\_M \rho(\mathbf{x}\_t)\boldsymbol{\varepsilon}\_t(\rho\_{t\prime}\sigma\_t)d^3\mathbf{x}\_{t\prime} \end{array} \tag{449}$$

where the quantity

$$\begin{aligned} \varepsilon\_t(\rho\_t, \sigma\_t) &= |m\_t|^2 / (2\rho\_t^2) + |w\_t^{(0)}(\rho\_t, \sigma\_t) + \\ + |B\_t|^2 / (2\rho\_t) &:= |m\_t|^2 / (2\rho\_t^2) + |w\_t(\rho\_t, \sigma\_t)| \end{aligned} \tag{450}$$

denotes the specific internal superfluid energy, modified by means of the "frozen" magnetic field *Bt* ∈ B(*M*), *t* ∈ R, replacing the previously defined internal specified potential energy *w*(0) *t* (*ρ<sup>t</sup>*, *<sup>σ</sup>t*) by the shifted specified potential energy quantity *wt*(*ρ<sup>t</sup>*, *<sup>σ</sup>t*) := *w*(0) *t* (*ρ<sup>t</sup>*, *<sup>σ</sup>t*) <sup>+</sup>|*Bt*|2/(<sup>2</sup>*ρt*). In particular, the Equation (447) reduces to the equivalent Hamilton expression

$$
\partial \mathfrak{m}\_t / \partial \mathfrak{t} = \{H, \mathfrak{m}\_t\} \tag{451}
$$

for *mt* ∈ *T*∗(*M*) - *di f f* ∗(*M*) and all *t* ∈ R. It is also seen that if *Bt* → 0 uniformly with respect to time *t* ∈ R, the internal energy expression (450) brings about that (423). Recall now that the following quasi-stationary second thermodynamic energy conservation law

$$
\delta\boldsymbol{e}\_{t}(\rho\_{t}, \sigma\_{t}) = \rho\_{t}^{-2} p\_{t}(\mathbf{x}\_{t}) \delta\rho\_{t} + T\_{t}(\mathbf{x}\_{t}) \delta\sigma\_{t} \tag{452}
$$

holds for all admitted superfluid variations *δρt* ∈ R(*M*) and *δσt* ∈ <sup>Σ</sup>(*M*), *t* ∈ R. As, by isentropic assumption, *δσt* = 0 for all *t* ∈ R along fluid streamlines, for the internal pressure one easily obtains the expression *pt*(*xt*) = *ρ*2*t <sup>∂</sup>w*(0) *t* (*ρ<sup>t</sup>*, *<sup>σ</sup>t*)/*∂ρt* − *Bt*|*Bt* /2, exactly coinciding with that of (448).

The Hamiltonian function (449) evidently satisfies the conservation condition *dH*/*dt* = 0 for all *t* ∈ R. To check this directly, it is enough to observe [194] that the following differential relationship

$$\left(\partial c\_t(\rho\_t, \sigma\_t) / \partial t + \langle \nabla | \rho\_t v\_t \left[ c\_t(\rho\_t, \sigma\_t) + \rho\_t \partial w\_0(\rho\_t, \sigma\_t) / \partial \rho\_t - |B\_t|^2 / 2 \right] \right) = 0 \tag{453}$$

holds for all *t* ∈ R and whose integration over the domain *M* ⊂ R<sup>3</sup> easily gives rise to the conservation of the Hamiltonian function (449).

#### *10.7. A Modified Current Lie Algebra, Magneto-Hydrodynamic Invariants and Their Geometry*

The importance of spatial invariants describing the stability [194] of MHD superfluid motion was previously stated long ago [181,193,194,201]. Based on the modern symplectic theory of differential–geometric structures on manifolds, we devise a unified approach to study MHD invariants of compressible superfluid flow, related with specially constructed symmetry structures and commuting to each other vector fields on the phase space.

We start from a useful differential-geometric observation that the magnetohydrodynamic Euler equations <sup>Γ</sup>(*<sup>M</sup>*; *T*(*M*)) action on the adjoint space to the Lie algebra G of the modified Banach current group *G* = Diff(*M*) - (Λ<sup>0</sup>(*M*) ⊕ Λ<sup>3</sup>(*M*) ⊕ <sup>∗</sup><sup>1</sup>(*M*)), generated by the following vector field differential relationship:

$$d\mathbf{x}\_{l}/dt = \boldsymbol{\upsilon}\_{l}(\mathbf{x}\_{l}),\tag{454}$$

where *xt* = *ηt*(*X*) ∈ *M*, *X* ∈ *M*, and *vt* : *M* → *<sup>T</sup>*(*M*), *t* ∈ R, is an acceptable timedependent vector field on the domain *M*, describing the adiabatic superfluid and superconductive motion via the diffeomorphism subgroup mappings *ηt* ∈ Diff(*M*), *ηt*|*<sup>t</sup>*=<sup>0</sup> = *η*0 ∈ Diff0(*M*). Taking into account that the initial superfluid configuration *η*0 ∈ Diff(*M*) is fixed, one can define, following reasonings from [81], a new differential relationship

$$d\mathbf{x}\_{\tau}/dt = u\_t(\mathbf{x}\_{\tau})\tag{455}$$

on the domain *M* with respect to the evolution variable *τ* ∈ R, parameterized by the time parameter *t* ∈ R, where *ut* : *M* → *<sup>T</sup>*(*M*), is a *τ*-independent vector field on *M*, generating the diffeomorphism subgroup *ψt* ∈ Diff(*M*), *xτ* := *ψι*(*η*0(*X*)), *X* ∈ *M*, commuting to that generated by the vector field (454), i.e., *ηt* ◦ *ψι* = *ψt* ◦ *ηι* for all *t*, *τ* ∈ R. The action of the diffeomorphism subgroup *ψt* ∈ Diff(*M*) at any fixed time *t* ∈ R can be naturally interpreted as rearranging the particle configurations in the superfluid, not changing its other dynamic characteristics. If to denote the corresponding Lie derivatives with respect to the vector fields (454) and (455) by differential expressions L*d*/*dt* := *∂*/*∂t* + *vt*|∇ ◦ : *C* <sup>∞</sup>(*<sup>M</sup>*; R) → *C* <sup>∞</sup>(*<sup>M</sup>*; R) and L*ut* := *ut*|∇ ◦ : *C* <sup>∞</sup>(*<sup>M</sup>*; R) → *C* <sup>∞</sup>(*<sup>M</sup>*; <sup>R</sup>), the commutation condition *ηt* ◦ *ψι* = *ψt* ◦ *ηι* for all *t*, *τ* ∈ R is equivalently rewritten as the operator commutator

$$\left[\mathcal{L}\_{d/dt}, \mathcal{L}\_{u\_l}\right] = 0.\tag{456}$$

Consider now an arbitrary integral invariant of the MHD superfluid, governed by the Euler system (438):

$$I = \int\_{D\_t} \rho\_t(\mathbf{x}\_t) \gamma\_t(m\_t; \rho\_t, \sigma\_t, B\_t) d^3 \mathbf{x}\_{t\*} \tag{457}$$

generated by some specific density functional *γt* : G∗ → *C* ∞(*M* × R; R) and held over the domain *Dt* = *ηt*(*D*) for any domain *D* ⊂ *M*, corresponding to the diffeomorphism subgroup *ηt* ∈ Diff(*M*), *t* ∈ R, generated by flow (454). Taking into account that there holds the following density relationship

$$\mathcal{L}\_{d/dt}(\rho\_t(\mathbf{x}\_t)\dot{d}^\circ \mathbf{x}\_t) = 0 \tag{458}$$

for any *t* ∈ R, one easily derives from (457) and (458) that also

$$
\mathcal{L}\_{d/dt} \gamma\_t(m\_t; \rho\_t, \sigma\_t, B\_t) = 0 \tag{459}
$$

for any *t* ∈ R. Thus, based on the commutation relationship (456) one can formulate the following important lemma.

**Lemma 1.** *Let vector fields (454) and (455) commute to each other and a density functional γ*0 : G∗ × R → *C* ∞(*M* × R; R) *satisfies for all t* ∈ R *the condition*

$$\mathcal{L}\_{d/dt}\gamma\_0(m\_t;\rho\_t,\sigma\_t,B\_t) = 0,\tag{460}$$

*then the following expressions*

$$I\_{n,k} = \int\_{D\_l} \rho\_t(\mathcal{L}\_{u\_l}^n \gamma\_0(m\_t; \rho\_{t\prime}, \sigma\_{t\prime} B\_t)^k d^3x\_t \tag{461}$$

*over the domain Dt* = *ηt*(*D*), *generated by corresponding to the flow (454) diffeomorphism subgroup ηt* ∈ Diff(*M*), *t* ∈ R, *and arbitrary domain D* ⊂ *M*, *are for all integers n* ∈ Z+, *k* ∈ Z, *the MHD invariants of the superfluid flow (438).*

**Proof.** A proof easily follows from the commutation condition (456) and the superfluid density relationship (458).

As examples, let us take following [81,194], the vector field *ut* := *ρ*<sup>−</sup><sup>1</sup> *t Bt* ∈ <sup>Γ</sup>(*T*(*M*)), commuting to the vector field *vt* ∈ <sup>Γ</sup>(*T*(*M*)), and *γ*0 = *iut At* |*dxt* = *At* |*ρ*<sup>−</sup><sup>1</sup> *t Bt* ∈ *C* ∞(*M* × R; <sup>R</sup>), where the magnetic vector potential *At* ∈ *C* <sup>∞</sup>(*<sup>M</sup>*; <sup>R</sup>), *t* ∈ R, satisfies the

classical Maxwell relationships: the magnetic field *Bt* = ∇ × *At* and the electric field *Et* = −*∂At*/*∂t* = −*vt* × *Bt*, owing to the net electric field superconductivity (437) condition *E* ˜ *t* = *Et* + *vt* × *Bt* = 0. Really, the commutativity condition (456) means that

$$\mathcal{L}\_{d/dt}(\rho\_t^{-1}B\_t) - \langle \rho\_t^{-1}B\_t|\nabla \rangle \,\forall \,\, v\_t = 0,\tag{462}$$

which is satisfied, owing to the second and fourth equations of the Euler MHD system (438), as well as to the invariance

$$\begin{aligned} \mathcal{L}\_{d/dt}\gamma\_0 &= \mathcal{L}\_{d/dt}i\_{\mathrm{ul}\_l}\langle A\_{\mathrm{t}} \,|\,\mathrm{d}\mathbf{x}\_{\mathrm{t}}\rangle = \left[\mathcal{L}\_{d/dt}i\_{\mathrm{ul}\_l}\right]\langle A\_{\mathrm{t}} \,|\,\mathrm{d}\mathbf{x}\_{\mathrm{t}}\rangle + \\ + i\_{\mathrm{ul}\_l}\mathcal{L}\_{d/dt}\langle A\_{\mathrm{t}} \,|\,\mathrm{d}\mathbf{x}\_{\mathrm{t}}\rangle &= i\_{\mathrm{[d/dt,u\_t]}}\langle A\_{\mathrm{t}} \,|\,\mathrm{d}\mathbf{x}\_{\mathrm{t}}\rangle + i\_{\mathrm{ul\_l}}\mathcal{L}\_{\mathrm{d}/dt}\langle A\_{\mathrm{t}} \,|\mathrm{d}\mathbf{x}\_{\mathrm{t}}\rangle = 0, \end{aligned} \tag{463}$$

which holds owing to the algebraic relationship

$$|\mathcal{L}\_{d/dt}, i\_{u\_l}| = i\_{[\partial/\partial t + v\_l v\_l, u\_l]},\tag{464}$$

commutativity of vector fields *ut* and *vt* ∈ Γ(*M*) and the integral relationship

$$\begin{array}{lcl}\frac{d}{dt}\int\_{\partial\mathcal{S}\_{l}} \langle A\_{l} \mid d\mathbf{x}\_{l} \rangle = \int\_{\partial\mathcal{S}\_{l}} \mathcal{L}\_{d/dt} \langle A\_{l} \mid d\mathbf{x}\_{l} \rangle = \\ = \int\_{\partial\mathcal{S}\_{l}} \left[ \langle \mathcal{L}\_{d/dt} A\_{l} \mid d\mathbf{x}\_{l} \rangle + \langle A\_{l} \mid \mathcal{L}\_{d/dt} d\mathbf{x}\_{l} \rangle \right] = \\ = \int\_{\partial\mathcal{S}\_{l}} \left[ \langle \mathcal{L}\_{d/dt} A\_{l} \mid d\mathbf{x}\_{l} \rangle + \langle A\_{l} \mid d\mathbf{v}\_{l} \rangle \right] = \\ = \int\_{\partial\mathcal{S}\_{l}} \left[ \langle v\_{l} \times B + \langle v\_{l} \vert \nabla \rangle A\_{l} \mid d\mathbf{x}\_{l} \rangle + \langle A\_{l} \mid d\mathbf{v}\_{l} \rangle \right] = \\ = \int\_{\partial\mathcal{S}\_{l}} \left[ \langle v\_{l} \times (\nabla \times A) + \langle v\_{l} \vert \nabla \rangle A\_{l} \mid d\mathbf{x}\_{l} \rangle + \langle A\_{l} \mid d\mathbf{v}\_{l} \rangle \right] = \\ = \int\_{\partial\mathcal{S}\_{l}} \left[ \langle dA\_{l} \vert v\_{l} \rangle + \langle A\_{l} \mid d\mathbf{v}\_{l} \rangle \right] = \int\_{\partial\mathcal{S}\_{l}} \left[ d\langle A\_{l} \mid v\_{l} \rangle \right] = 0, \end{array} \tag{465}$$

equivalent to the condition <sup>L</sup>*d*/*dt At* |*dxt* = 0 for all *t* ∈ R. The same statement we obtain from the slightly simpler reasoning:

$$\begin{array}{ll} \frac{d}{dt} \int\_{\partial S\_t} \langle A\_t \mid d\mathbf{x}\_t \rangle = \frac{d}{dt} \int\_{S\_t} \langle \nabla \times A\_t \vert dS\_t^2 \rangle = \\ = \frac{d}{dt} \int\_{S\_t} \langle B\_t \vert dS\_t^2 \rangle := -\int\_{\partial S\_t} \langle \tilde{E}\_t \vert d\mathbf{x}\_t \rangle = 0, \end{array} \tag{466}$$

following from the net electric field *E* ˜ *t* = 0 superconductivity condition (437) along the boundary *∂St*, where *St* := *ηt*(*<sup>S</sup>*0) ⊂ *M* is the surface, generated by the diffeomorphism subgroup *ηt* ∈ Diff(*M*), *t* ∈ R, and an arbitrarily chosen surface *S*0 = *St*|*<sup>t</sup>*=<sup>0</sup> ⊂ *M*. The latter is, evidently, equivalent to the equality <sup>L</sup>*d*/*dt At* |*dxt* = 0 modulo the gauge transformation *At* → *At* + ∇*ξ<sup>t</sup>*, where L*d*/*dtξt* + *At*|*vt* = 0 for some function *ξt* ∈ *<sup>C</sup>*<sup>∞</sup>(*<sup>M</sup>*; R) and all *t* ∈ R. Thus, one can formulate [81,194] the following proposition.

**Proposition 18.** *The functionals*

$$I\_{n\,k}^{(B)} = \int\_{D\_t} \rho\_t \left( \mathcal{L}\_{\rho\_t^{-1}B\_t}^n \langle A | \rho\_t^{-1} B\_t \rangle \right)^k d^3 \mathbf{x}\_t \tag{467}$$

*over the domain Dt* = *ηt*(*D*), *generated by corresponding to the flow (454) diffeomorphism subgroup ηt* ∈ Diff(*M*), *t* ∈ R, *and arbitrary domain D* ⊂ *M*, *are for all integers n* ∈ Z+, *k* ∈ Z, *the MHD invariants of the superfluid flow (438). In particular, the following relationships* {*<sup>H</sup>*, *I*(*B*) *<sup>n</sup>*,*k* } = 0 *hold for all n* ∈ Z+.

It is natural here to mention [194,196] that the specific entropy functional *γ*0 = *σt* : *M* → *C*∞(*M* × R; R) satisfies the sufficient condition L*d*/*dtσt* = 0, *t* ∈ R, a priori generates for the superfluid flow (438) the infinite hierarchy

$$I\_{n,k}^{(\sigma)} = \int\_{D\_t} \rho\_t \left( \mathcal{L}\_{\rho\_t^{-1}B\_t}^n \sigma\_t(\mathbf{x}\_t) \right)^k d^3 \mathbf{x}\_{t\_\prime} \tag{468}$$

*n* ∈ Z+, *k* ∈ Z, of the MHD invariants over the domain *Dt* = *ηt*(*D*), generated by the corresponding to the flow (454) diffeomorphism subgroup *ηt* ∈ Diff(*M*), *t* ∈ R, and arbitrary domain *D* ⊂ *M*.

To construct other MHD invariants, depending on the superfluid velocity *vt* ∈ <sup>Γ</sup>(*T*(*M*)), *t* ∈ R, let us consider, following [81], two differential one-forms *<sup>α</sup>t*|*dxt* , *βt*|*dxt* ∈ <sup>Λ</sup><sup>1</sup>(*M*), *xt* := *ηt*(*X*), *X* ∈ *M*, satisfying for all *t* ∈ R the following identity:

$$
\mathcal{L}\_{d/dt} \langle \mathbf{u}\_t | d\mathbf{x}\_t \rangle = d\mathbf{h}\_t \quad + \mathcal{L}\_{\mathbf{u}\_t} \langle \beta\_t | d\mathbf{x}\_t \rangle,\tag{469}
$$

for some function *ht* ∈ <sup>Λ</sup><sup>0</sup>(*M*), where the vector field

$$d\mathbf{x}\_{l}/d\tau = \ u\_{l}(\mathbf{x}\_{l})\tag{470}$$

is uniform with respect to the evolution parameter *τ* ∈ R and satisfies the following constraints:

$$\left[\mathcal{L}\_{d/\text{dt}}, \mathcal{L}\_{u\_{\text{t}}}\right] = 0, \quad \langle \nabla | \rho\_{\text{t}} u\_{\text{t}} \rangle = 0 \tag{471}$$

and *ut ∂M* at almost all points *xt* ∈ *∂M* for all evolution parameters *t*, *τ* ∈ R. Then one can formulate the following general proposition.

**Proposition 19.** *The following integral expressions*

$$\begin{aligned} I\_0^{(a,\emptyset)} &= \int\_M \rho\_t \langle u\_t | u\_t \rangle d^3 \mathbf{x}\_{t\_\prime} I\_1^{(a,\emptyset)} = \int\_M \rho\_t [\langle u\_t | \mathbf{v}\_t \rangle + h\_t] d^3 \mathbf{x}\_{t\_\prime} \\ I\_2^{(a,\emptyset)} &= \int\_M \rho\_t \langle \mathcal{L}\_{d/dt} u\_t | u\_t \rangle d^3 \mathbf{x}\_t \end{aligned} \tag{472}$$

*over the whole domain M* ⊂ R<sup>3</sup> *are for all integers n* ∈ Z+, *k* ∈ Z, *the global MHD invariants.*

**Proof.** Consider, for example, a proof that *I* (*<sup>α</sup>*,*β*) 0 : G → R is an invariant: taking into account that <sup>L</sup>*d*/*dt*(*ρtd*<sup>3</sup>*xt*) = 0, one obtains the expression:

$$\begin{array}{lcl}dI\_{0}^{(a,\beta)}/dt &= \int\_{M} \rho\_{t} \mathcal{L}\_{d/dt} \langle a\_{t}|u\_{t}\rangle d^{3}x\_{t} = \\ &= \int\_{M} \rho\_{t} i\_{u\_{t}}(d\|t\_{t} + \mathcal{L}\_{u\_{t}}\langle \beta\_{t}|d\mathbf{x}\_{t}\rangle) d^{3}x\_{t} = \\ &= \int\_{M} \rho\_{t} (\dot{i}\_{u\_{t}} d\|t\_{t} + \dot{i}\_{u\_{t}}(\dot{u}\_{t}d + \dot{i}\dot{u}\_{u\_{t}})\langle \beta\_{t}|d\mathbf{x}\_{t}\rangle) d^{3}x\_{t} = \\ &= \int\_{M} \rho\_{t} i\_{u\_{t}} d(h\_{t} + \langle \beta\_{t}|u\_{t}\rangle) d^{3}x\_{t} = \\ &= \int\_{M} \langle \nabla|\dot{h}\_{t}\rho\_{t} u\_{t}\rangle d^{3}x\_{t} = \int\_{\partial M} \langle \ddot{h}\_{t}\rho\_{t} u\_{t}|dS\_{t}^{2}\rangle = 0 \end{array} \tag{473}$$

for all *t* ∈ R, where we put, by definition, *ht* := (*ht* + *βt*|*ut* ), denoted *dS*<sup>2</sup> *t* the surface measure on the boundary *∂M*, used the Cartan representation L*ut* = *iutd* + *diut* and the natural boundary tangency condition *ρtut ∂M*, thus proving the proposition. Exactly similar calculations ensue for the next two invariant *I* (*<sup>α</sup>*,*β*) *k* : G → R, *k* = 1, 2, on which we will not stop here.

˜

As a simple example, one can put *α*(0) *t* := (*vt*) - *vt*, *βt* := *Bt*, the vector field *ut* = *ρ*<sup>−</sup><sup>1</sup> *t Bt* : *M* → *<sup>T</sup>*(*M*), *t* ∈ R, and show by easy calculations, using the variational equality (439) that

$$\mathcal{L}\_{d/dt} \langle \upsilon\_l | d\mathbf{x}\_l \rangle = d(|\upsilon\_l|^2/2 \ -\mathsf{h}\_l - |\mathsf{B}\_l|^2/\rho\_l) + \mathcal{L}\_{\mathsf{U}\_l} \langle \mathsf{B}\_l | d\mathbf{x}\_l \rangle + T\_l d\sigma\_l,\tag{474}$$

where, we have denoted the specific enthalpy [55,198,199] function *ht* := *et* + *ptρ*<sup>−</sup><sup>1</sup> *t* . As a consequence of equality (474), under the spatial temperature constancy ∇ *Tt* = 0 condition for all *t* ∈ R, one obtains the following MHD superfluid invariant:

$$I\_0^{(v,B)} := \int\_M \langle v\_t | B\_t \rangle d^3 \mathbf{x}\_t = \int\_M \langle m\_t | \rho\_t^{-1} B\_t \rangle,\tag{475}$$

where *mt* - *mt*(*xt*)|*dxt* ⊗ *d*3*xt* ∈ *di f f* ∗(*M*) and *ρ*<sup>−</sup><sup>1</sup> *t Bt* - *ρ*<sup>−</sup><sup>1</sup> *t* (*x*)*Bt*|*∂*/*∂x* ∈ *<sup>T</sup>*(*M*), coinciding with the MHD invariant, presented before in [81,194]. If the above temperature condition does not hold, the equality (474) reduces to the differential relationship

$$
\partial \langle \upsilon\_t | B\_t \rangle / \partial t + \langle \nabla | [\upsilon\_t \langle \upsilon\_t | B\_t \rangle + B\_t (h\_t - |\upsilon\_t|^2 / 2]] \rangle + \rho\_t T\_t \langle \rho\_t^{-1} B\_t | \nabla \upsilon\_t \rangle,\tag{476}
$$

satisfied for all *xt* ∈ *M* and *t* ∈ R.

> *Remark.* It is worth remarking here that the following baroclinic relationship

$$
\nabla \rho\_t^{-1} \times \nabla p\_t = -\nabla T\_t \times \nabla \sigma\_t \tag{477}
$$

holds for all *xt* ∈ *M* and *t* ∈ R.

 Similarly, we also easily obtain the following invariant

$$I\_1^{(v,B)} = \int\_M \rho\_t [|m\_t|^2/\left(2\rho\_t^2\right) + w\_t^{(0)}(\rho\_t, \sigma\_t) + |B\_t|^2/\left(2\rho\_t\right)]d^3 \mathbf{x}\_t = H,\tag{478}$$

coinciding exactly with the Hamiltonian function for the flow (438) on the phase space G∗. The third invariant is, eventually, closely related to the vorticity vector *ξt* := ∇ × *vt* : *M* → E3, *t* ∈ R, and needs a more detailed analysis.

It is instructive now to analyze the existence of integral invariants for the pure hydrodynamic case when the magnetic field *Bt* = 0, *t* ∈ R, following the approach, devised before in [81]. In particular, owing to the relationship (477), there holds the following integral expression for the vorticity *ξt* := ∇ × *vt*, *t* ∈ R :

$$\mathcal{L}\_{d/dt}\mathfrak{z}\_t - \langle \mathfrak{z}\_t | \nabla \rangle \upsilon\_t = \nabla T\_t \times \nabla \sigma\_t \tag{479}$$

and define the vector field

$$\mu\_t := \rho\_t^{-1} \vec{\xi}\_t \exp f\_t(\mathbf{x}\_t) \tag{480}$$

for some scalar smooth mapping *ft* : *M* → R, which we will choose from the assumed commutation condition

$$[\mathcal{L}\_{d/dt}, \mathcal{L}\_{u\_t}] = 0.\tag{481}$$

The latter gives rise to the equality *ξt*L*d*/*dt ft*(*xt*) = −∇*Tt* × ∇*<sup>σ</sup>t* at any *xt* := *ηt*(*X*) ∈ *M*, *X* ∈ *M*, or

$$
\dot{f}\_t \left(\nabla \times v\_t\right) + \nabla T\_t \times \nabla \sigma\_t = 0,\tag{482}
$$

where we took into account that L*d*/*dt ft*(*xt*) = *d ft*(*xt*)/*dt* := ˙ *ft*(*xt*), *xt* ∈ *M*, with respect to the temporal parameter *t* ∈ R. From (482), one obtains that the mapping *ft* : *M* → R should satisfy the following constraints:

$$
\nabla f\_t = k\_t \upsilon\_{t\prime} \quad f\_t \upsilon\_t = \rho\_t^{-1} \nabla p(t) + \nabla \omega\_t \tag{483}
$$

for some scalar smooth functions *kt* and *ωt* : *M* → R, *t* ∈ R. It is easy to check that the system (483) is compatible, i.e., the quasi-stationary thermodynamic relationship *p*(0) *t* = *ρ*2*t ∂<sup>w</sup>*0(*ρ<sup>t</sup>*, *<sup>σ</sup>t*)/*∂ρt* jointly with the Euler Equation (393) make it possible to determine these unknown scalar smooth functions *kt* and *ωt* : *M* → R for all *t* ∈ R.

Consider now, following [81], a slightly modified expression (474) at the magnetic field *Bt* = 0:

$$\mathcal{L}\_{d/dt} \langle v\_t \exp f\_t | dx\_t \rangle = \exp f\_t d(\omega\_t + |v\_t|^2/2) \tag{484}$$

and calculate the related integral expression:

$$\begin{array}{lcl}\frac{d}{dt}\int\_M \rho\_t(i\_{\boldsymbol{u}\_l}\langle\boldsymbol{v}\_t|d\mathbf{x}\_t\rangle)d^3\mathbf{x}\_t &= \int\_M \rho\_t \mathcal{L}\_{d/dt}(i\_{\boldsymbol{u}\_l}\langle\boldsymbol{v}\_t|d\mathbf{x}\_t\rangle)d^3\mathbf{x}\_t = \\ &= \int\_M \rho\_t(i\_{\boldsymbol{u}\_l}\mathcal{L}\_{d/dt}\langle\boldsymbol{v}\_t|d\mathbf{x}\_t\rangle)d^3\mathbf{x}\_t = \int\_M \rho\_t(i\_{\boldsymbol{u}\_l}d\tilde{h})d^3\mathbf{x}\_t = \\ &= \int\_M \langle i\_{\boldsymbol{\rho}\_l\boldsymbol{u}\_l}d\tilde{h}\rangle d^3\mathbf{x}\_t = \int\_M \langle \nabla \tilde{h}\_l|\rho\_l\boldsymbol{u}\_l\rangle d^3\mathbf{x}\_t = \int\_M \langle \nabla \tilde{h}\_l|\tilde{\boldsymbol{\xi}}\_t \cdot \exp f\_l(\mathbf{x}\_t)\rangle d^3\mathbf{x}\_t,\end{array} \tag{485}$$

where we put, by definition, the function ˜ *ht* := *ωt* + |*vt*|2/2.

If now to put that the mapping *ft* : *M* → R satisfies for all *t* ∈ R the constraint ∇ *ft*|*ξt* = 0, the integral expression (485) reduces to

$$\begin{cases} \frac{d}{dt} \int\_M \rho\_t \left( i\_{\rm u\_l} \langle \boldsymbol{v}\_l | d\mathbf{x}\_t \rangle \right) d^3 \mathbf{x}\_t = \int\_M \langle \nabla | \left( \exp f\_t(\mathbf{x}\_t) \tilde{h}\_t \boldsymbol{\xi}\_t \right) \rangle d^3 \mathbf{x}\_t = \\ \quad = \int\_{\partial M} \langle \exp f\_t(\mathbf{x}\_t) \tilde{h}\_t \boldsymbol{\xi}\_t | d^2 \mathbf{S}\_t \rangle = 0, \end{cases} \tag{486}$$

where the vorticity vector tangency *ξt*||*∂<sup>M</sup>* constraint is assumed. Thus, under conditions assumed above, the following vortex functional

$$I = \int\_M \langle v\_t | \nabla \times v\_t \rangle d^3 x\_t \tag{487}$$

persists to be conserved for all *t* ∈ R.

If the function *ft* : *M* → R, being defined by relationships (483), satisfies for all *t* ∈ R the scalar constraint ∇ *ft*|*ξt* = 0, one easily derives the following differential relationship:

$$\begin{split} \mathcal{L}\_{d/dt} \langle \nabla f\_t | \xi\_t \rangle &= k\_t \langle v\_t | \xi\_t \rangle + \langle \nabla | f\_t \nabla T\_t \times \nabla \sigma\_t \rangle = \\ &= \langle \nabla \dot{f}\_t | \xi\_t \rangle + \langle \nabla | f\_t \nabla T\_t \times \nabla \sigma\_t \rangle = 0, \end{split} \tag{488}$$

or, equivalently, in the integral form

$$\begin{split} \frac{d}{dt} \int\_{D\_t} \langle \nabla f\_t | \tilde{\xi}\_t \rangle \rho\_t d^3 \mathbf{x}\_t &= \int\_{D\_t} \mathcal{L}\_{d/dt} \langle \nabla f\_t | \tilde{\xi}\_t \rangle \rho\_t d^3 \mathbf{x}\_t = \\ &= \int\_{D\_t} \left[ \langle \nabla f\_t | \tilde{\xi}\_t \rangle + \langle \nabla | f\_t \nabla T\_t \times \nabla \sigma\_t \rangle \right] \rho\_t d^3 \mathbf{x}\_t = \\ &= \int\_{D\_t} \left[ \langle \nabla \dot{f}\_t | \tilde{\xi}\_t \rangle - \langle \nabla f\_t | \nabla \times \rho\_t^{-1} \nabla p\_t^{(0)} \rangle \right] \rho\_t d^3 \mathbf{x}\_t \\ &= \int\_{D\_t} \left[ \langle \nabla f\_t | \tilde{\xi}\_t \rangle \rho\_t - \rho\_t \langle \nabla \rho\_t^{-1} | \nabla \times p\_t^{(0)} \nabla f\_t \rangle \right] d^3 \mathbf{x}\_t = \\ &= \int\_{D\_t} \left[ \langle \nabla \dot{f}\_t | \tilde{\xi}\_t \rangle \rho\_t + \langle \nabla \ln \rho\_t | \nabla \times p\_t^{(0)} \nabla f\_t \rangle \right] d^3 \mathbf{x}\_t = \\ &= \int\_{D\_t} \langle \nabla f\_t | \tilde{\xi}\_t \rangle \rho\_t d^3 \mathbf{x}\_t. \end{split} \tag{489}$$

where we took into account that for the isentropic fluid flow under regard there holds the tangency ∇*ρt*||*∂Dt* condition for all *t* ∈ R. If the right hand side of (489) proves to be zero, i.e., ∇ ˙ *ft*|*ξ t* = 0, *t* ∈ R, this will mean that the constraint ∇ *ft*|*ξt* = 0 for all *t* ∈ R, if ∇ *ft*|*ξt* |*<sup>t</sup>*=<sup>0</sup> = 0 at *t* = 0, thus producing the vortex conservation functional (487).

#### **11. A Modified Current Lie Algebra Symmetry on Torus, Its Lie-Algebraic Structure and Related Integrable Heavenly Type Dynamical Systems**

#### *11.1. Introductory Notes*

The main object of our study is integrable multidimensional dispersionless differential equations, which possess modified Lax–Sato type representations, related with their hidden Hamiltonian structures. Equations of this type arise and are widely applied in mechanics, general relativity, differential geometry and the theory of integrable systems. Among the most mentioned are the Boyer–Finley equations, heavenly type Pleba ´nski equations, which are descriptive of a class of self-dual 4-manifolds, as well as the dispersionless Kadomtsev– Petviashvili (dKP) equation, also known as the Khokhlov–Zabolotskaya equation, which arises in non-linear acoustics and the theory of Einstein–Weyl structures. Their integrability have been investigated by a whole variety of modern techniques including symmetry analysis, differential-geometric and algebrogeometric methods, dispersionless ¯*∂*-dressing, factorization techniques, Virasoro constraints, hydrodynamic reductions, etc. The first important examples of the related Hamiltonian structures were previously demonstrated in [202–206] and later were developed in [207–214] , where many examples of dispersionless differential equations were analyzed in detail as flows on orbits of the coadjoint action of loop vector field algebras diff : (T*<sup>n</sup>*), generated by specially chosen seed elements ˜ *l* ∈ diff : (T*<sup>n</sup>*)<sup>∗</sup>. In these works, it was observed that many integrable multidimensional dispersionless differential equations are generated by seed elements of a very special structure, namely for them

there exist such analytical functional elements *η*˜, *ρ*˜ ∈ Ω<sup>0</sup>(T*n*) ⊗ C that ˜ *l* = *η*˜*dρ*˜. As the latter naturally generates the symplectic structure *ω*˜ (2) := T*n dη*˜ ∧ *dρ*˜ ∈ Ω<sup>2</sup>(T*n*) ⊗ C on the moduli space [215,216] of flat connections on T*<sup>n</sup>*, related to coadjoint actions of the corresponding Casimir functionals, the geometric nature of many integrable multidimensional dispersionless differential equations can be also studied using cohomological techniques, devised in [215,217] for the case of Riemannian surfaces. It is also worth mentioning that in [207–209] a deep connection of the related Hamiltonian flows on diff :(T*n*)∗ was revealed with the well known in classical mechanics Lagrange–d'Alembert principle.

In this section, developing the approach, devised in [202,203,218], we will describe a Lie algebraic structure and integrability properties of a generalized hierarchy of the Lax-Sato type compatible systems of Hamiltonian flows and related integrable multidimensional dispersionless differential equations. Such systems are called the heavenly type equations and were first introduced by Pleba ´nski in [219]. The heavenly type equations were analyzed in many articles (see, e.g., [203,218,220–227]) using several different approaches. In [131,207,209,228] the heavenly type equations were analyzed by using non-associative and non-commutative current algebras on the torus T*<sup>m</sup>*, *m* ∈ N. We also mention that [229,230] B. Szablikowski and A. Sergyeyev developed some generalizations of the classical AKS-algebraic and related *R*-structures [11,17–19]. In [203,218] and recently in [207,231], these ideas were applied to a semi-direct Lie algebra G˜ := diff :(T*<sup>n</sup>*)diff :(T*n*)∗ of the loop Lie algebra diff :(T*n*) := *Vect* :(T*n*) of vector fields on the torus T*<sup>n</sup>*, *n* ∈ Z+, and its dual space diff :(T*<sup>n</sup>*)<sup>∗</sup>. Several interesting and deep results about the orbits of the corresponding coadjoint actions on the space G˜∗ - G˜ and the classical Lie–Poisson type structures on them were presented. It is worth especially remarking here that the AKSalgebraic scheme is naturally embedded into the classical *R*-structure approach via the following construction.

Let a (G˜; [·, ·]) denote a Lie algebra over C and G˜∗ be its natural adjoint space. Take some tensor element *r* ∈ G ⊗˜ G -˜ Hom(G˜<sup>∗</sup>; G˜) and consider its splitting into symmetric and antisymmetric parts

> *r*

$$
\sigma = k \oplus \sigma\_\prime \tag{490}
$$

respectively, and assume that the symmetric tensor *k* ∈ G ⊗˜ G˜ does not degenerate. That allows the definition on the Lie algebra G˜ of a symmetric non-degenerate bi-linear product (·|·) : G ⊗˜ G →˜ C via the expression

$$(a|b) := k^{-1}a(b) \tag{491}$$

for any *a*, *b* ∈ G˜. The composed mapping *R* := *σ* ◦ *k*−<sup>1</sup> : G →˜ G˜, following the scheme

$$
\tilde{\mathcal{G}} \stackrel{k^{-1}}{\rightarrow} \tilde{\mathcal{G}}^\* \stackrel{\sigma}{\rightarrow} \tilde{\mathcal{G}}\_\* \tag{492}
$$

defines the following *R*-structure on the Lie algebra G˜:

$$[a,b]\_R := [Ra,b] + [a,Rb] \tag{493}$$

for all elements *a*, *b* ∈ G˜. The following theorem, defining the related Poissonian structure [19,121,207,217,232,233] on the adjoint space G˜ holds.

**Theorem 9.** *Let α*, *β* ∈ G˜∗ *be arbitrary and define the bracket*

$$\{\alpha,\beta\} := ad\_{ra}^\*\beta - ad\_{r\beta}^\*\alpha. \tag{494}$$

*Then the bracket (494) is Poissonian if the R-structure on the Lie algebra* G˜ *defines the Lie structure on* G˜, *that is there holds the Yang–Baxter equation*

$$[Ra, Rb] - R[a, b]\_R = -[a, b] \tag{495}$$

*for any a*, *b* ∈ G ˜ .

**Remark 10.** *The above theorem makes it possible to consider the Hamiltonian flows on the coadjoint space* G ˜ ∗ *as those determined on the Lie algebra* G ˜ . *The latter is exceptionally useful if for the scalar product (491) there exists such a trace-type* Tr(·) *symmetric and ad-invariant functional (of Killing type) that*

$$\text{Tr}(ab) := (a|b), \quad (a|[b,c]) = (([a,b]|,c) \tag{496}$$

*for any a*, *b and c* ∈ G ˜ . *Then any Hamiltonian flow of an element a* ∈ G ˜ *is representable in the standard Lax type form*

$$da/dt = \left[\text{grad}(h), a\right].\tag{497}$$

*where* grad(*h*) ∈ G˜ *is generated by the corresponding smooth Hamiltonian function h* ∈ <sup>D</sup>(G˜). :

Concerning the loop Lie algebra G ˜ := diff (T*n*) on the torus T*<sup>n</sup>*, it is well known that such a trace-type functional on G ˜ does not exist, thus we need to study the Hamiltonian flows on the adjoint loop space G ˜ ∗ - Ω<sup>1</sup>(T*n*) of meromorphic differential forms on the torus T*n* and obtain, as a result, integrable dispersionless differential equations as compatibility conditions for the related loop vector fields, generated by Casimir functionals on G ˜ ∗. This procedure is much more complicated for analysis than the standard one and employs more geometrical tools and considerations about the orbit space structure of the seed elements ˜ *l* ∈ G ˜ ∗, generating a hierarchy of integrable Hamiltonian flows. The latter, in part, is deeply related to its reduction properties, guaranteeing the existence of nontrivial Casimir invariants on its coadjoint orbits. By applying and extending these ideas to central extensions of Lie algebras, we construct new classes of commuting Hamiltonian flows on an extended adjoint space G ¯ := G ˜ ∗ ⊕ C. These Hamiltonian flows are generated by seed elements (*a*˜ - ˜ *l*; *α*) ∈ G¯∗ and specially constructed Casimir invariants on the corresponding orbits of G ˜ ∗. In most cases, these seed elements appeared to be represented as specially factorized differential objects, whose real geometric nature is still much hidden and not clear. Moreover, we found that the corresponding compatibility condition of constructed Hamiltonian flows coincides exactly with the compatibility condition for a system of related three Lax–Sato type linear vector field equations. As examples, we found and described new multidimensional generalizations of the Mikhalev–Pavlov and Alonso–Shabat type integrable dispersionless equation, whose seed elements possess a special factorized structure, allowing to extend them to the multidimensional case of arbitrary dimension.

#### *11.2. Differential-Geometric Setting: The Diffeomorphism Group* Diff(T*n*) *and Its Description*

Consider the *n*-dimensional torus T*n* and call points *X* ∈ T*n* as the Lagrangian variables of a configuration *η* ∈ Diff(T*<sup>n</sup>*). The manifold T*<sup>n</sup>*, thought of as the target space of a configuration *η* ∈ Diff(T*<sup>n</sup>*), is called the spatial or Eulerian configuration, whose points, called spatial or Eulerian points, will be denoted by small letters *x* ∈ T*<sup>n</sup>*. Then any oneparametric configuration of Diff(T*n*) is a time *t* ∈ R dependent family [41,122,181,192,193] of diffeomorphisms written as

$$\mathbb{T}^{\mathfrak{n}} \ni \mathfrak{x} = \eta(X, t) := \eta\_t(X) \in \mathbb{T}^{\mathfrak{n}} \tag{498}$$

for any initial configuration *X* ∈ T*n* and some mappings *ηt* ∈ Diff(T*<sup>n</sup>*), *t* ∈ R.

Being interested in studying flows on the space of Lagrangian configurations *η* ∈ Diff(T*n*) with respect to the temporal variable *t* ∈ R, generated by group diffeomorphisms *ηt* ∈ Diff(T*<sup>n</sup>*), *t* ∈ R, let us proceed to describing the structure of tangent *<sup>T</sup>ηt*(Diff(T*<sup>n</sup>*)) and cotangent *<sup>T</sup>*<sup>∗</sup>*ηt*(Diff(T*<sup>n</sup>*)) spaces to the diffeomorphism group Diff(T*n*) at the points *ηt* ∈ Diff(T*n*) for any *t* ∈ R. Determine first the tangent space *<sup>T</sup>ηt*(Diff(T*<sup>n</sup>*)) to the diffeomorphism group manifold Diff(T*n*) at point *η* ∈ Diff(T*n*) for which we will make use of the construction, devised before in [122,181,194]. Namely, let *η* ∈ Diff(T*n*) be a Lagrangian configuration and try to determine the tangent space *<sup>T</sup>η*(Diff(T*<sup>n</sup>*)) at *η* ∈ Diff(T*n*) as

the collection of vectors *ξη* := *dητ*/*dτ*|*<sup>τ</sup>*=0, where R →*ητ* ∈ Diff(T*<sup>n</sup>*), *ητ*|*<sup>τ</sup>*=<sup>0</sup> = *η*, is a smooth curve on Diff(T*<sup>n</sup>*), and for arbitrary reference point *X* ∈ T*n* there holds *ξη*(*X*) = *dητ*(*X*)/*dτ*|*<sup>τ</sup>*=0. The latter equivalently means that the vectors *ξη*(*X*) ∈ *<sup>T</sup>η*(*X*)(T*<sup>n</sup>*), *X* ∈ T*<sup>n</sup>*, represent a vector field *ξ* : T*n* → *T*(T*n*) on the manifold T*n* for any *η* ∈ Diff(T*<sup>n</sup>*). Thus, the tangent space *<sup>T</sup>η*(Diff(T*<sup>n</sup>*)) coincides with the set of vector fields on T*<sup>n</sup>*:

$$T\_{\mathcal{V}}(\text{Diff}(\mathbb{T}^{\text{ul}})) \simeq \{ \mathbb{X}\_{\mathcal{V}} \in \Gamma(T(\mathbb{T}^{\text{ul}})): \mathbb{X}\_{\mathcal{V}}(X) \in T\_{\mathcal{\tilde{V}}(X)}(\mathbb{T}^{\text{ul}}) \}\tag{499}$$

and similarly, the cotangent space *T*∗ *η* (Diff(T*n*)) consists of all one-form densities on T*n* over *η* ∈ Diff(T*<sup>n</sup>*):

$$T^\*\_{\eta}(\text{Diff}(\mathbb{T}^{\eta})) = \{ \mathfrak{a}\_{\eta} \in \Omega^1(\mathbb{T}^{\eta}) \otimes \Omega^3(\mathbb{T}^{\eta}) : \mathfrak{a}\_{\eta}(X) \in T^\*\_{\eta(X)}(\mathbb{T}^{\eta}) \otimes |\Omega^3(\mathbb{T}^{\eta})| \}\tag{500}$$

subject to the canonical non-degenerate pairing (·|·)*c* on *T*∗ *η* (Diff(T*n*)) × *<sup>T</sup>η*(Diff(T*<sup>n</sup>*)) : if *αη* ∈ *T*∗ *η* (Diff(T*<sup>n</sup>*)), *ξη* ∈ *<sup>T</sup>η*(Diff(T*<sup>n</sup>*)), where *αη*|*X* = *αη*(*X*)|*dx* ⊗ *d*3*X*, *ξη*|*X* = *ξη*(*X*)|*∂*/*∂x* , then

$$(a\_{\eta}|\xi\_{\eta})\_{\mathfrak{c}} := \int\_{\mathbb{T}^n} \langle a\_{\eta}(X)|\xi\_{\eta}(X)\rangle d^3X. \tag{501}$$

The construction above makes it possible to identify the cotangent bundle *T*∗ *η* (Diff(T*n*)) at the fixed Lagrangian configuration *η* ∈ Diff(T*n*) to the tangent space *<sup>T</sup>η*(Diff(T*<sup>n</sup>*)), as the tangent space *T*(T*n*) is endowed with the natural internal tangent bundle metric ·| · at any point *η*(*X*) ∈ T*<sup>n</sup>*, identifying *T*(T*n*) with *T*∗(T*n*) via the related metric isomorphism : *T*∗(T*n*) → *<sup>T</sup>*(T*<sup>n</sup>*). The latter can be also naturally lifted to *T*∗ *η* (Diff(T*n*)) at *η* ∈ Diff(T*<sup>n</sup>*), namely: for any elements *αη*, *βη* ∈ *T*∗ *η* (Diff(T*<sup>n</sup>*)), *αη*|*X* = *αη*(*X*)|*dx* ⊗ *d*3*X* and *βη*|*X* = *βη*(*X*)|*dx* ⊗ *d*3*X* ∈ *T*∗ *η* (Diff(T*n*)) we can define the metric

$$(\mathfrak{a}\_{\eta}|\mathfrak{f}\_{\eta}) := \int\_{\mathbb{T}^n} \langle a\_{\eta}^\sharp(X)|\mathfrak{f}\_{\eta}^\sharp(X)\rangle d^3X,\tag{502}$$

where, by definition, *α η*(*X*) := *αη*(*X*)|*dx* ), *β η*(*X*) := *βη*(*X*)|*dx* ∈ *<sup>T</sup>η*(*X*)(T*<sup>n</sup>*) for any *X* ∈ T*<sup>n</sup>*. Based on the construction above, one can proceed to constructing smooth invariant functionals on the cotangent bundle *T*∗(Diff(T*n*)) subject to the corresponding coadjoint actions of the diffeomorphism group Diff(T*<sup>n</sup>*). Moreover, as the cotangen<sup>t</sup> bundle *T*∗(Diff(T*n*)) is *a priori* endowed with the canonical symplectic structure, equivalent [11,18,19,26,28,41,122,181,195] to the corresponding Poisson bracket on the space of smooth functionals on *<sup>T</sup>*<sup>∗</sup>(Diff(T*<sup>n</sup>*)), one can study both the related Hamiltonian flows on it and their adjoint symmetries and complete integrability.

Consider now the cotangent bundle *T*∗(Diff(T*n*)) as a smooth manifold endowed with the canonical symplectic structure [26,122] on it, equivalent to the corresponding canonical Poisson bracket on the space of smooth functionals on it. Taking into account that the cotangent space *T*∗ *η* (Diff(T*n*)) at *η* ∈ Diff(T*<sup>n</sup>*), shifted by the right *<sup>R</sup>η*−<sup>1</sup> - action to the space *<sup>T</sup>*<sup>∗</sup>*Id*(Diff(T*<sup>n</sup>*)), *Id* ∈ Diff(T*<sup>n</sup>*), becomes diffeomorphic to the adjoint space diff<sup>∗</sup>(T*n*) to the Lie algebra diff(T*n*) - Γ(*T*(T*n*)) of vector fields on T*<sup>n</sup>*, as there was stated [31–33,41] still by S. Lie in 1887 this canonical Poisson bracket on *T*∗ *η* (Diff(T*n*)) transforms [26,31,32,41,181,195] into the classical Lie-Poisson bracket on the adjoint space G∗. Moreover, the orbits of the diffeomorphism group Diff(T*n*) on *T*∗(Diff(T*n*)) respectively transform into the coadjoint orbits on the adjoint space G∗, generated by suitable elements of the Lie algebra G. To construct in detail this Lie–Poisson bracket, we formulate the following preliminary simple lemma.

**Lemma 2.** *The Lie algebra* diff(T*n*) - Γ(*T*(T*n*)) *is determined by the following Lie commutator relationships:*

$$
\langle a\_1, a\_2 \rangle = \langle a\_1 | \nabla \rangle a\_2 - \langle a\_2 | \nabla \rangle a\_1 \tag{503}
$$

*for any vector fields a*1, *a*2 ∈ Γ(*T*(T*n*)) *on the manifold* T*<sup>n</sup>*.

**Proof.** Proof of the commutation relationships (503) easily follows from the group multiplication

$$(\varphi\_{1,t} \circ \varphi\_{2,t})(X) = \varphi\_{2,t}(\varphi\_{1,t}(X))\tag{504}$$

for any local group diffeomorphisms *ϕ*1,*t*, *ϕ*2,*t* ∈ Diff(T*<sup>n</sup>*), *t* ∈ R, and *X* ∈ T*n* under condition that *aj*(*X*) := *<sup>d</sup>ϕj*,*<sup>t</sup>*(*X*)/*dt*|*<sup>t</sup>*=<sup>0</sup> and *<sup>ϕ</sup>j*,*<sup>t</sup>*|*<sup>t</sup>*=<sup>0</sup> = *Id* ∈ Diff(T*<sup>n</sup>*), *j* = 1, 2.

To calculate the Poisson bracket on the cotangent space *T*∗*η* (Diff(T*n*)) at any *η* ∈ Diff(T*<sup>n</sup>*), let us consider the cotangent space *T*∗*η* (Diff(T*n*)) - diff<sup>∗</sup>(T*<sup>n</sup>*), the adjoint space to the tangent space *<sup>T</sup>η*(Diff(T*<sup>n</sup>*)) of left invariant vector fields on Diff(T*n*) at any *η* ∈ Diff(T*<sup>n</sup>*), and take the canonical symplectic structure on *T*∗*η* (Diff(T*n*)) in the form *<sup>ω</sup>*(2)(*<sup>μ</sup>*, *η*) := *δα*(*μ*, *η*), where the canonical Liouville form *<sup>α</sup>*(*μ*, *η*) := (*μ*|*δη*)*c* ∈ <sup>Ω</sup><sup>1</sup>(*μ*,*η*)(*T*<sup>∗</sup>*<sup>η</sup>* (Diff(T*n*))) at a point (*μ*, *η*) ∈ *T*∗*η* (Diff(T*n*)) is defined *a priori* on the tangent space *<sup>T</sup>η*(Diff(T*<sup>n</sup>*)) - Γ(*T*(*M*)) of right-invariant vector fields on the torus manifold T*<sup>n</sup>*. Having calculated the corresponding Poisson bracket of smooth functions (*μ*|*a*)*<sup>c</sup>*,(*μ*|*b*)*<sup>c</sup>* ∈ *<sup>C</sup>*<sup>∞</sup>(*T*<sup>∗</sup>*η* (Diff(T*<sup>n</sup>*)); R) on *T*∗*η* (Diff(T*n*)) - diff<sup>∗</sup>(T*<sup>n</sup>*), *η* ∈ Diff(T*<sup>n</sup>*), one can formulate the following proposition.

**Proposition 20.** *The Lie–Poisson bracket on the coadjoint space T*∗*η* (Diff(T*n*)) - diff<sup>∗</sup>(T*<sup>n</sup>*), *η* ∈ *M*, *is equal to the expression*

$$(\{f, \emptyset\}(\mu) = (\mu | [\delta g(\mu) / \delta \mu, \delta f(\mu) / \delta \mu])\_{\mathbb{C}} \tag{505}$$

*for any smooth functionals f* , *g* ∈ *<sup>C</sup>*<sup>∞</sup>(G∗; <sup>R</sup>).

**Proof.** By definition [26,122] of the Poisson bracket of smooth functions (*μ*|*a*)*<sup>c</sup>*,(*μ*|*b*)*<sup>c</sup>* ∈ *<sup>C</sup>*<sup>∞</sup>(*T*<sup>∗</sup>*η* (Diff(T*<sup>n</sup>*)); R) on the symplectic space *T*∗*η* (Diff(T*<sup>n</sup>*)), it is easy to calculate that

$$\begin{aligned} \{\mu(a), \mu(b)\} &:= \delta a (X\_{a\prime} X\_b) = \\ \hline = X\_{\mathfrak{a}}(a \vert X\_{\mathfrak{b}})\_{\mathfrak{c}} - X\_{\mathfrak{b}}(a \vert X\_{\mathfrak{a}})\_{\mathfrak{c}} - (a \vert [X\_{\mathfrak{a}}, X\_{\mathfrak{b}}])\_{\mathfrak{c}}. \end{aligned} \tag{506}$$

where *Xa* := *<sup>δ</sup>*(*μ*|*a*)*c*/*δμ* = *a* ∈ diff(T*<sup>n</sup>*), *Xb* := *<sup>δ</sup>*(*μ*|*b*)*c*/*δμ* = *b* ∈ diff(T*<sup>n</sup>*). Since the expressions *Xa*(*α*|*Xb*)*c* = 0 and *Xb*(*α*|*Xa*)*c* = 0 owing the right-invariance of the vector fields *Xa*, *Xb* ∈ *<sup>T</sup>η*(Diff(T*<sup>n</sup>*)), the Poisson bracket (506) transforms into

$$\begin{array}{lcl} \{\ (\mu|a)\_{\varepsilon}, (\mu|b)\_{\varepsilon}\} = -(a|[X\_{\mu}, X\_{b}])\_{\varepsilon} = \\ = (\mu|[b, a])\_{\varepsilon} = (\mu|[\delta(\mu|b)\_{\varepsilon}/\delta\mu, \delta(\mu|a)\_{\varepsilon}/\delta\mu])\_{\varepsilon} \end{array} \tag{507}$$

for all (*μ*, *η*) ∈ *T*∗*η* (Diff(T*n*)) - diff<sup>∗</sup>(T*<sup>n</sup>*), *η* ∈ Diff(T*n*) and any *a*, *b* ∈ diff(T*<sup>n</sup>*). The Poisson bracket (506) is easily generalized to

$$(\{f, \emptyset\}(\mu) = (\mu | [\delta g(\mu) / \delta \mu, \delta f(\mu) / \delta \mu])\_\mathfrak{c} \tag{508}$$

for any smooth functionals *f* , *g* ∈ *<sup>C</sup>*<sup>∞</sup>(G∗; <sup>R</sup>), finishing the proof.

Based on the Lie–Poisson bracket (505), one can naturally construct Hamiltonian flows on the adjoint space diff<sup>∗</sup>(T*n*) via the expressions

$$
\partial \mathcal{l} / \partial t = -ad\_{\text{grad }h(l)}^\* l \tag{509}
$$

for any element *l* ∈ diff<sup>∗</sup>(T*<sup>n</sup>*), *t* ∈ R, where, by definition, *ddε h*(*l* + *<sup>ε</sup>m*)|*<sup>ε</sup>*=<sup>0</sup> := (*m*| grad *h*(*l*))*<sup>c</sup>*, for some smooth Hamiltonian function *h* ∈ *<sup>C</sup>*<sup>∞</sup>(diff<sup>∗</sup>(T*<sup>n</sup>*); <sup>R</sup>). If the system possesses enough additional invariants except the Hamiltonian function, one can expect its simplification often reducing to its complete integrability. Below, we proceed to developing an effective enough analytical scheme, before suggested in [207,209,234] for suitably constructed holomorphic loop diffeomorphism groups on tori, allowing to

generate infinite hierarchies of such completely integrable Hamiltonian systems on related functional phase spaces.

#### *11.3. A Modified Current Lie Algebra and Related Symmetry Analysis on Functional Manifolds*

Consider a smooth manifold *M* ⊂ R*<sup>n</sup>*, *n* ∈ N, endowed with the generalized quantum current group [26,181,216] group *G* as the semidirect product Diff(*M*) - (Λ<sup>0</sup>(*M*) × Λ<sup>1</sup>(*M*)) of the diffeomorphism group Diff(*M*) with the Abelian groups Ω<sup>0</sup>(*M*) and <sup>Ω</sup><sup>1</sup>(*M*), defined by the natural Diff(*M*)—group action Diff(*M*) × *G* → *G*:

$$\begin{array}{rcl}(\eta \circ \varrho)(X) := \varrho(\eta(X)), (\eta \circ r)(X) := r(\eta(X)),\\ \eta \circ \langle b(X)|dX\rangle := \eta^\* \langle b(X)|dX\rangle\end{array} \tag{510}$$

for *η* ∈ Diff(*M*), *X* ∈ *M*, and any (*ϕ*;*r*, *b*) ∈ Diff(*M*) × (Ω<sup>0</sup>(*M*) × <sup>Ω</sup><sup>0</sup>(*M*). The semidirect product group *G* is endowed with the following internal right group multiplication subject to the Eulerian variable *x* := *η*(*X*) ∈ *M*:

$$\begin{aligned} \left(\begin{aligned} \left(\boldsymbol{\varrho}\_{1}; r\_{1}, \langle b\_{1}|d\boldsymbol{x}\rangle\right) \circ \left(\boldsymbol{\varrho}\_{2}; r\_{2}, \langle b\_{2}|d\boldsymbol{x}\rangle\right) = \\ = \left(\boldsymbol{\varrho}\_{2} \cdot \boldsymbol{\varrho}\_{1}; r\_{1} + r\_{2} \cdot \boldsymbol{\varrho}\_{1}, \langle b\_{1}|d\boldsymbol{x}\rangle + \langle b\_{2}|d\boldsymbol{x}\rangle \circ \boldsymbol{\varrho}\_{1}\right) \end{aligned} \tag{511}$$

at a fixed point *η* ∈ Diff(*M*) and arbitrary elements *ϕ*1, *ϕ*2 ∈ Diff(*M*), *r*1,*r*2 ∈ Ω<sup>0</sup>(*M*) and *b*1 - *b*1|*dx* , *b*2 - *b*2|*dx* ∈ <sup>Ω</sup><sup>1</sup>(*M*).

Let G - *TId*(*G*) = *di f f*(*M*) - (Ω<sup>0</sup>(*M*) × <sup>Ω</sup><sup>1</sup>(*M*)), *Id* ∈ *G*, denote the Lie algebra of the current group *G*, where we took into account that *T*(Ω<sup>0</sup>(*M*)) - <sup>Ω</sup><sup>0</sup>(*M*), *T*(Ω<sup>1</sup>(*M*)) - <sup>Ω</sup><sup>1</sup>(*M*), and proceed to studying its coadjoint action on the adjoint space G∗. Using (511), one can easily write down that

$$[(a\_1; r\_1, b\_1), (a\_2; r\_2, b\_2)] = (\mathcal{L}\_{a\_1} a\_2; \mathcal{L}\_{a\_2} r\_1 - \mathcal{L}\_{a\_1} r\_2, \mathcal{L}\_{a\_2} \langle b\_1 | dx \rangle - \mathcal{L}\_{a\_1} \langle b\_2 | dx \rangle),\tag{512}$$

where L*a* denotes the standard [122,123,181] Lie derivative with respect to a vector field *a* ∈*di f f*(*M*). From (512) one easily ensues the following current Lie algebra G commutation relationships:

$$\begin{split} \left[ \left( a\_1; r\_1, b\_1 \right), \left( a\_2; r\_2, b\_2 \right) \right] &= \left( \langle \left( a\_1 | \frac{\partial}{\partial x} \rangle a\_2 - \langle a\_2 | \frac{\partial}{\partial x} \rangle a\_1 \right) | \frac{\partial}{\partial x} \rangle; \langle a\_2 | \frac{\partial}{\partial x} r\_1 \rangle - \\ - \langle a\_1 | \frac{\partial}{\partial x} r\_2 \rangle, \ \langle \langle a\_2 | \frac{\partial}{\partial x} \rangle b\_1 | dx \rangle - \langle \langle a\_1 | \frac{\partial}{\partial x} \rangle b\_2 | dx \rangle + \langle b\_1 | \langle dx | \frac{\partial}{\partial x} \rangle a\_2 \rangle - \langle b\_2 | \langle dx | \frac{\partial}{\partial x} \rangle a\_1 \rangle \rangle, \end{split} \tag{513}$$

for any elements *a*1, *a*2 ∈ *di f f*(*M*) - *<sup>T</sup>*(*M*),*r*1,*r*<sup>2</sup> ∈ Ω<sup>0</sup>(*M*) and *b*1, *b*2 ∈ <sup>Ω</sup><sup>1</sup>(*M*), where we have also denoted the gradient vector *∂∂x* := *∂∂x*1 , *∂∂x*2 , ..., *∂∂xn* - at *x* ∈ *M*. The adjoint space G∗ to the semidirect product Lie algebra G =*di f f*(*M*) - (Ω<sup>0</sup>(*M*) ⊕ Ω<sup>1</sup>(*M*)) can be written symbolically as G∗ = (Ω<sup>1</sup>(*M*) ⊗ Ω*n*(*M*)) × (∗Ω<sup>0</sup>(*M*) ⊕ ∗Ω<sup>1</sup>(*M*)) =*di f f* ∗(*M*) × (Ω*<sup>n</sup>*(*M*)⊕ <sup>Ω</sup>*<sup>n</sup>*−<sup>1</sup>(*M*)), where ∗ : Ω(*M*) → Ω(*M*) denotes the corresponding Hodge isomorphism with respect to the natural scalar product

$$(a^{(k)} | \gamma^{(s)}) := \delta\_{sk} \int\_M (a^{(k)} \wedge \*\gamma^{(s)}) \tag{514}$$

for any forms *α*(*k*) ∈ Ω*<sup>k</sup>*(*M*) and *γ*(*s*) ∈ Ω*s*(*M*) , *k*,*<sup>s</sup>* = 1, *n*. Then, taking into account that the adjoint space G∗ is endowed [27,28,41,177,194,200] with the canonical Lie– Poisson bracket

$$\begin{split} \{f,h\}(l) &:= \left(l|\left|\nabla f(l),\nabla h(l)\right|\right) = \int\_{M} \left(\left<\mu\right| \left<\frac{\delta f}{\delta \mu} \middle| \frac{\partial h}{\partial x}\right> \frac{\delta h}{\delta \mu} - \left<\frac{\delta h}{\delta \mu} \middle| \frac{\partial}{\partial x}\right> \frac{\delta f}{\delta \mu}\right) d^{n}x + \\ &+ \int\_{M} \rho \left(\left<\frac{\delta f}{\delta \mu} \middle| \frac{\partial}{\partial x}\frac{\delta h}{\delta \rho}\right> - \left<\frac{\delta h}{\delta \mu} \middle| \frac{\partial}{\partial x}\frac{\delta f}{\delta \rho}\right>\right) d^{n}x + \\ &+ \int\_{M} \left(\left<\beta\middle| \left<\frac{\delta f}{\delta \mu} \middle| \frac{\partial}{\partial x}\right> \frac{\delta h}{\delta \beta} - \left<\frac{\delta h}{\delta \mu} \middle| \frac{\partial}{\partial x}\right> \frac{\delta f}{\delta \beta}\right> + \\ &+ \left<\frac{\delta f}{\delta \beta}\middle| \frac{\partial}{\partial x}\left<\frac{\delta h}{\delta \mu}\right> - \left<\frac{\delta h}{\delta \mu}\right>\left<\beta\middle| \frac{\partial}{\partial x}\right> \frac{\delta f}{\delta \mu}\right>\right) d^{n}x \end{split} \tag{515}$$

for any smooth functionals *f* , *g* ∈ D(G∗) on the G∗, where we have denoted by *l* := ( *μ*|*dx* ⊗ *dnx*; *ρdnx*, ∗ *β*|*dx* ⊗ *dnx*) ∈ G∗ and by ∇(◦)(*l*) := *<sup>δ</sup>*(◦) *δμ* | *∂∂x* ; *<sup>δ</sup>*(◦) *δρ* , *<sup>δ</sup>*(◦) *δρ* |*dx* the corresponding functional gradient.

**Remark 11.** *We remark here that the bracket (515) naturally derives, as it was demonstrated in [29,31,32,41], from the canonical symplectic structure on the cotangent phase space <sup>T</sup>*<sup>∗</sup>(*G*).

Based on the Lie–Poisson bracket, one can construct the Hamiltonian system

$$\frac{\partial}{\partial t}(\mu,\rho,\beta)^{\mathsf{T}} = \{H, (\mu,\rho,\beta)^{\mathsf{T}}\},\tag{516}$$

:

where *t* ∈ R is the related evolution parameter and *H* ∈ D(G∗) is some suitably constructed Hamiltonian function. For the evolution flow (516) to be integrable, it should possess [11,122,181,235] enough commuting to each of the other invariant functionals *Hj* ∈ <sup>D</sup>(G∗), *j* ∈ N, which is in most cases a very complicated problem. Thereby, taking this into account, we will proceed the following way: we will construct a set *a priori* commuting to each of the other invariants *hj* ∈ <sup>D</sup>(G˜∗), *j* ∈ N, defined on the coadjoint space G˜∗ to a suitably generalized Lie algebra G ˜ .

Namely, let us consider a group *G* ˜ := *G* ˜ + × *G* ˜ −,where *G* ˜ ± := Diff ±(*M*)- (Ω0±(*M*) × <sup>Ω</sup>1±(*M*)) are subgroups of the smooth loop mappings {C ⊃ S1 → *<sup>G</sup>*}, holomorphically extended, respectively, on the interior <sup>D</sup>1+ ⊂ C and on the exterior D<sup>1</sup>− ⊂ C domains of the unit centrally located disk D<sup>1</sup> ⊂ C<sup>1</sup> and such that for any *g*˜(*λ*) ∈ *G*˜−, *λ* ∈ D<sup>1</sup>−, *g*˜(∞) = *Id* ∈ *G*. The corresponding Lie subalgebras G ˜ ± - diff : ±(*M*)- (Ω0±(*M*) × <sup>Ω</sup>0±(*M*)) of the loop current subgroups *G* ˜ ± consist, in general, of vector fields on S1 × T*<sup>n</sup>*, holomorphically extended, respectively, on regions <sup>D</sup>1± ⊂ C1, where for any p˜(*λ*) ∈ G˜− the value p˜(∞) = 0. The loop current Lie algebra splitting G ˜ = G ˜ + ⊕ G ˜ −, where

$$\mathcal{G}\_{+} = \bigcup\_{m \in \mathbb{Z}\_{+}} \left\{ \sum\_{j=0}^{m} \lambda^{j} \langle a\_{-j}(\mathbf{x}) | \frac{\partial}{\partial \mathbf{x}} \rangle \otimes d^{n} \mathbf{x}; \sum\_{j=0}^{m} \lambda^{j} \rho\_{-j}(\mathbf{x}), \sum\_{j=0}^{m} \lambda^{j} \langle b\_{-j}(\mathbf{x}) | d\mathbf{x} \rangle \right\}, \tag{517}$$
 
$$\mathcal{G}\_{-} = \left\{ \sum\_{j \in \mathbb{N}} \lambda^{-j} \langle a\_{j}(\mathbf{x}) | \frac{\partial}{\partial \mathbf{x}} \rangle \otimes d^{n} \mathbf{x}; \sum\_{j \in \mathbb{N}} \lambda^{-j} \rho\_{j}(\mathbf{x}), \sum\_{j \in \mathbb{N}} \lambda^{-j} \langle b\_{j}(\mathbf{x}) | d\mathbf{x} \rangle \right\},$$

can be naturally identified with a dense subspace of the dual space G ˜ ∗ through the pairing

$$(\tilde{l}|\vec{a}) := \mathop{\rm res}\_{\lambda \in \mathbb{C}} (l(\mathbf{x}; \lambda) | p(\mathbf{x}; \lambda))\_{\mathcal{H}^0} \tag{518}$$

with respect to the scalar product

$$(l(\mathbf{x};\lambda)|p(\mathbf{x};\lambda))\_{H^0} := \int\_M d^n \mathbf{x} [\langle \mu(\mathbf{x};\lambda)|a(\mathbf{x};\lambda)\rangle + \rho(\mathbf{x};\lambda)r(\mathbf{x};\lambda) + \langle \beta(\mathbf{x};\lambda)|b(\mathbf{x};\lambda)\rangle]. \tag{519}$$

on the usual Hilbert space *H*<sup>0</sup> := *<sup>L</sup>*2(*<sup>M</sup>*; C*n*+<sup>1</sup> × C<sup>1</sup> × C*n*+<sup>1</sup>) for any elements ˜*l* := (*μ*˜; *ρ*˜, *β*˜) ∈ G˜∗ and *p*˜ := (*a*˜;*r*˜, ˜*b*) ∈ G˜, naturally represented in their component wise canonical form as

$$\begin{split} \vec{\sigma} := (\vec{a}; \vec{r}, \vec{b}) &= \left( \left< \mathbf{a}(\mathbf{x}; \boldsymbol{\lambda}) | \frac{\partial}{\partial \mathbf{x}} \right>; \, r(\mathbf{x}; \boldsymbol{\lambda}), (\mathbf{b}(\mathbf{x}; \boldsymbol{\lambda}) | d\mathbf{x}) \right), \boldsymbol{I} := (\vec{\mu}; \vec{\rho}, \vec{\rho}) = \mathbf{0} \\ &= (\langle \mu(\mathbf{x}; \boldsymbol{\lambda}) | d\mathbf{x} \rangle \otimes d^{\mathbf{n}} \mathbf{x}; \rho(\mathbf{x}; \boldsymbol{\lambda}) d^{\mathbf{3}} \mathbf{x}, \ast \langle \beta(\mathbf{x}; \boldsymbol{\lambda}) | d\mathbf{x} \rangle \otimes d^{\mathbf{n}} \mathbf{x}), \end{split} \tag{520}$$

where for any x := (*x*; *λ*) ∈ C×*M* we have denoted, for brevity, the gradient operator *∂∂*x := ( *∂∂λ* ; *∂∂x* ) = *∂∂λ* ; *∂∂x*1 , *∂∂x*2 , ..., *∂∂xn* - in the Euclidean space (E*<sup>n</sup>*; ·, · ) and *a* ˜ := %a(*x*; *λ*)| *∂∂*x& := *a*(0)(*x*; *λ*) *∂∂λ* + %*a*(*x*; *λ*)| *∂∂x*&, ˜*b* := b(*x*; *<sup>λ</sup>*)|*d*x := *<sup>b</sup>*(0)(*x*; *λ*)*dλ* + *b*(*x*; *λ*)|*dx* , *μ*˜ := *μ*(*x*; *<sup>λ</sup>*)|*d*x := *μ*(0)(*x*; *λ*)*dλ* + *μ*(*x*; *λ*)|*dx* . The corresponding Lie commutator [*p*˜1, *p*˜2] ∈ G˜ of any vectors *p*˜1 = (*a*˜1;*r*˜1, ˜*b*1), *p*˜2 = (*a*˜2;*r*˜2, ˜*b*2) ∈ G˜ is calculated the standard way, using (513), and equals

$$\begin{split} \left[ \left( \mathbf{\bar{a}}\_{1}; \mathbf{\bar{r}}\_{1}, \mathbf{\bar{b}}\_{1} \right), \left( \mathbf{\bar{a}}\_{2}; \mathbf{\bar{r}}\_{2}, \mathbf{\bar{b}}\_{2} \right) \right] &= \begin{pmatrix} \left( \left( \langle \mathbf{a}\_{1} | \frac{\partial}{\partial \mathbf{x}} \rangle \mathbf{a}\_{2} - \langle \mathbf{a}\_{2} | \frac{\partial}{\partial \mathbf{x}} \rangle \mathbf{a}\_{1} \right) | \frac{\partial}{\partial \mathbf{x}} \right); \\ \langle \mathbf{a}\_{2} | \frac{\partial}{\partial \mathbf{x}} \mathbf{r}\_{1} \rangle - \langle \mathbf{a}\_{1} | \frac{\partial}{\partial \mathbf{x}} \mathbf{r}\_{2} \rangle, \langle \mathbf{a}\_{2} | \frac{\partial}{\partial \mathbf{x}} \rangle \langle \mathbf{b}\_{1} | d\mathbf{x} \rangle - \\ - \langle \mathbf{a}\_{1} | \frac{\partial}{\partial \mathbf{x}} \rangle \langle \mathbf{b}\_{2} | d\mathbf{x} \rangle + \langle \mathbf{b}\_{1} | \langle d\mathbf{x} | \frac{\partial}{\partial \mathbf{x}} \rangle \mathbf{a}\_{2} \rangle - \langle \mathbf{b}\_{2} | \langle d\mathbf{x} | \frac{\partial}{\partial \mathbf{x}} \rangle \mathbf{a}\_{1} \rangle \end{pmatrix}. \end{split} \tag{521}$$

The expression (521) makes it possible to construct the related Lie–Poisson bracket on the adjoint space G ˜ ∗, modifying that of (515):

$$\begin{split} \{f,h\} := \operatorname{res}\_{\lambda} \int\_{M} \langle \mu \vert \langle \frac{\delta f}{\delta \mu} \vert \frac{\partial}{\partial \mathbf{x}} \rangle \frac{\delta h}{\delta \mu} - \langle \frac{\delta h}{\delta \mu} \vert \frac{\partial}{\partial \mathbf{x}} \rangle \frac{\delta f}{\delta \mu} \rangle d^{n} \mathbf{x} + \\ \quad + \operatorname{res}\_{\lambda} \int\_{M} \rho \left( \langle \frac{\delta f}{\delta \mu} \vert \frac{\partial}{\partial \mathbf{x}} \frac{\delta h}{\delta \rho} \rangle - \langle \frac{\delta h}{\delta \mu} \vert \frac{\partial}{\partial \mathbf{x}} \frac{\delta f}{\delta \rho} \rangle \right) d^{n} \mathbf{x} + \\ \quad + \operatorname{res}\_{\lambda} \int\_{M} \left( \langle \beta \vert \langle \frac{\delta f}{\delta \mu} \vert \frac{\partial}{\partial \mathbf{x}} \rangle \frac{\delta h}{\delta \beta} \rangle - \langle \frac{\delta h}{\delta \mu} \vert \frac{\partial}{\partial \mathbf{x}} \rangle \frac{\delta f}{\delta \rho} \rangle \right) \\ \quad + \langle \frac{\delta f}{\delta \beta} \vert \langle \beta \vert \frac{\partial}{\partial \mathbf{x}} \rangle \frac{\delta h}{\delta \mu} \rangle - \langle \frac{\delta h}{\delta \beta} \vert \langle \beta \vert \frac{\partial}{\partial \mathbf{x}} \rangle \frac{\delta f}{\delta \mu} \rangle \right) d^{n} \mathbf{x} \end{split} \tag{522}$$

for any smooth functionals *f* , *h* ∈ <sup>D</sup>(G˜∗).

{

The Lie–Poisson bracket (522) is strongly degenerate and possesses a lot of Casimir invariants *hj* ∈ <sup>D</sup>(G˜∗), *j* ∈ Z+, satisfying the condition

> {

$$\{f\_{\prime}h\_{\dot{\jmath}}\} = 0\tag{523}$$

for all smooth functionals *f* ∈ D(G˜∗) and *j* ∈ Z+. As the Lie algebra G˜ acts on its adjoint space G ˜ ∗ for any *p*˜ = (*a*˜;*r*˜, ˜ *b*) ∈ G˜ and ˜*l* = (*μ*˜; *ρ*˜, *β*˜) ∈ G˜∗ as *ad*∗ : G ×˜ G˜∗ → G˜∗, where

$$\begin{split} \boldsymbol{\rho} \boldsymbol{d}\_{\boldsymbol{\beta}}^{\*} \boldsymbol{\tilde{I}} &= \left( - \langle \frac{\partial}{\partial \mathbf{x}} \diamond | \mathbf{a} \rangle \langle \boldsymbol{\mu} | d\mathbf{x} \rangle \otimes d^{\mathrm{n}} \mathbf{x} - \langle \boldsymbol{\mu} | \langle d\mathbf{x} | \frac{\partial}{\partial \mathbf{x}} \rangle \mathbf{a} \rangle \otimes d^{\mathrm{n}} \mathbf{x} + \\ &+ \rho \langle d\mathbf{x} | \frac{\partial \mathbf{r}}{\partial \mathbf{x}} \rangle \otimes d^{\mathrm{n}} \mathbf{x} + \langle \beta | \langle d\mathbf{x} | \frac{\partial}{\partial \mathbf{x}} \mathbf{x} \rangle \otimes d^{\mathrm{n}} \mathbf{x} - \langle \frac{\partial}{\partial \mathbf{x}} \circ | \beta \rangle \langle \mathbf{x} | d\mathbf{x} \rangle \otimes d^{\mathrm{n}} \mathbf{x}; \\ & \langle \frac{\partial}{\partial \mathbf{x}} | \rho \mathbf{a} \rangle \otimes d^{\mathrm{n}} \mathbf{x}, \* \langle \langle \frac{\partial}{\partial \mathbf{x}} \circ | \mathbf{x} \rangle \beta | d\mathbf{x} \rangle \otimes d^{\mathrm{n}} \mathbf{x} - \* \langle \langle \beta | \frac{\partial}{\partial \mathbf{x}} \rangle \mathbf{a} | d\mathbf{x} \rangle \otimes d^{\mathrm{n}} \mathbf{x} \end{split} \tag{5.24}$$

the latter condition (523) is easily rewritten as

$$ad^\*\_{\nabla h(\vec{l})} \, l = 0,\tag{525}$$

where ∇*h*(˜ *l*) := *δhδμ* | *∂ ∂*x ; *δhδρ* , *δhδβ* |*dx* - ∈ G˜, being equivalent, owing to (524), to the following three differential-functional relationships:

$$\begin{split} \langle \frac{\partial}{\partial \mathbf{x}} \diamond | \frac{\delta \mathbf{h}}{\delta \mu} \rangle \mu + \langle \mu | \diamond \frac{\partial}{\partial \mathbf{x}} \frac{\delta \mathbf{h}}{\delta \mu} \rangle - \langle \beta | \diamond \frac{\partial}{\partial \mathbf{x}} \frac{\delta \mathbf{h}}{\delta \beta} \rangle + \langle \frac{\partial}{\partial \mathbf{x}} \diamond | \beta \rangle \frac{\delta \mathbf{h}}{\delta \beta} - \\ -\rho \frac{\partial}{\partial \mathbf{x}} \frac{\delta \mathbf{h}}{\delta \rho} = 0, \quad \langle \frac{\partial}{\partial \mathbf{x}} | \rho \frac{\delta \mathbf{h}}{\delta \mu} \rangle = 0, \ \langle \frac{\partial}{\partial \mathbf{x}} \diamond | \frac{\delta \mathbf{h}}{\delta \mu} \rangle \beta - \langle \beta | \frac{\partial}{\partial \mathbf{x}} \rangle \frac{\delta \mathbf{h}}{\delta \mu} = 0 \end{split} \tag{526}$$

for any (*μ*˜; *ρ*˜, *β*) ∈ G˜∗. Recall now that the constructed above loop Lie algebra G˜ = G˜+ ⊕ G˜ −, as the direct sum of its subalgebras, possesses the additional Lie commutator

$$\mathbb{E}\left[\vec{p}\_1,\vec{p}\_2\right]\_R := \left[R\vec{p}\_1,\vec{p}\_2\right] + \left[\vec{p}\_1,R\vec{p}\_2\right] = \left[\vec{p}\_{1,+},\vec{p}\_{2,+}\right] - \left[\vec{p}\_{1,-},\vec{p}\_{2,-}\right] \tag{527}$$

for any *p*˜1,*p*˜2 ∈ G˜, where, by definition, the linear homomorphism *R* := ( *P*+ − *<sup>P</sup>*−)/2, projectors *P*± : G →˜ G˜±, and *p*˜*j*,<sup>±</sup> := *P*± *p*˜*j* ∈ G˜ ±, *j* = 1, 2. Based on the second Lie commutator (527) we can construct, in the same way as above, the second Lie–Poisson bracket on the adjoint space G˜∗ as

{ *f* , *h*}*R* := *resλ M μ*| *R<sup>δ</sup> f δμ*| *∂ ∂*x *δh δμ* − *R δhδμ*| *∂ ∂*x *δ f δμ dnx*+ + *resλ M μ*| *δ f δμ*| *∂ ∂*x *R δhδμ* − *δhδμ*| *∂ ∂*x *Rδ f δμ dnx*+ + *resλ M ρ Rδ f δμ*| *∂ ∂*x *δh δρ* − *R δhδμ*| *∂ ∂*x *δ f δρ dnx*+ + *resλ M ρ δ f δμ*| *∂ ∂*x *Rδhδρ* − *δhδμ*| *∂ ∂*x *Rδ f δρ dnx*+ + *resλ M β*| *R<sup>δ</sup> f δμ*| *∂ ∂*x *δh δβ* − ; *R δhδμ*| *∂ ∂*x < *δ f δβ* + (528) + *β*| *δ f δμ*| *∂ ∂*x *R δhδβ* − ; *δhδμ*| *∂ ∂*x < *Rδ f δβ* + + *δ f δβ* | *β*| *∂ ∂*x *R δhδμ* − *δhδβ* | *β*| *∂ ∂*x *Rδ f δμ* + + *Rδ f δβ* | *β*| *∂ ∂*x *δh δμ* − *R δhδβ* | *β*| *∂ ∂*x *δ f δμ dn*x

*11.4. A New Modified Spatially Four-Dimensional Mikhalev–Pavlov Heavenly Type Integrable System*

Let a seed element *a*˜ - *l* ∈ G˜∗ be chosen as

˜

$$
\tilde{a} \times \tilde{l} = ((u\_{\mathbf{x}} + v\_{\mathbf{x}}\lambda - \lambda^2)\partial / \partial \mathbf{x} \times (w\_{\mathbf{x}} + \zeta\_{\mathbf{x}}\lambda)dx,\tag{529}
$$

where *u*, *v*, *w*, *ζ* ∈ *C*<sup>2</sup>(R<sup>2</sup> × (S<sup>1</sup> × <sup>T</sup><sup>1</sup>); <sup>R</sup>). The asymptotic splits for the components of the gradient of the corresponding Casimir functional *h* ∈ <sup>I</sup>(G˜∗), as |*λ*| → ∞ have the following forms:

$$\begin{split} \nabla h\_{\mathbb{f}} &\sim 1 - \upsilon\_{\mathbf{x}} \lambda^{-1} - \mu\_{\mathbf{x}} \lambda^{-2} - \upsilon\_{z} \lambda^{-3} - (\mu\_{z} + \upsilon\_{\mathbf{x}} \upsilon\_{z} - 2(\partial\_{\mathbf{x}}^{-1} \upsilon\_{\mathbf{x}\mathbf{x}} \upsilon\_{z})) \lambda^{-4} + \\ &+ \upsilon\_{\mathbf{y}} \lambda^{-5} - (-\mu\_{\mathbf{y}} - \upsilon\_{\mathbf{x}} \upsilon\_{\mathbf{y}} + 2(\partial\_{\mathbf{x}}^{-1} \upsilon\_{\mathbf{x}\mathbf{x}} \upsilon\_{\mathbf{y}})) \lambda^{-6} + \dots, \\ \nabla h\_{\mathbb{f}} &\sim \zeta\_{\mathbf{x}} \lambda^{-1} + \upsilon\_{\mathbf{x}} \lambda^{-2} + \zeta\_{\mathbf{z}} \lambda^{-3} + (\mu\_{z} - \zeta\_{\mathbf{x}} \upsilon\_{z} + 2\upsilon\_{\mathbf{x}} \zeta\_{\mathbf{z}} - (\partial\_{\mathbf{x}}^{-1} \upsilon\_{\mathbf{x}} \zeta\_{\mathbf{x}})\_{\mathbf{z}}) \lambda^{-4} - 1 \\ &- \zeta\_{\mathbf{y}} \lambda^{-5} + (-\upsilon\_{\mathbf{y}} + \zeta\_{\mathbf{x}} \upsilon\_{\mathbf{y}} - 2\upsilon\_{\mathbf{x}} \zeta\_{\mathbf{y}} + (\partial\_{\mathbf{x}}^{-1} \upsilon\_{\mathbf{x}} \zeta\_{\mathbf{x}})\_{\mathbf{y}}) \lambda^{-6} + \dots. \end{split}$$

In the case when

$$\begin{aligned} \nabla h\_{l\_r+}^{(y)} &:= \lambda^4 - v\_x \lambda^3 - \mu\_x \lambda^2 - v\_z \lambda - (\mu\_z + v\_x v\_z - 2(\partial\_x^{-1} v\_{xx} v\_z)), \\\\ \nabla h\_{l\_r+}^{(y)} &:= \zeta\_x \lambda^3 + w\_x \lambda^2 + \zeta\_z \lambda + (w\_z - \zeta\_x v\_z + 2v\_x \zeta\_z - (\partial\_x^{-1} v\_x \zeta\_x)\_z), \end{aligned}$$

and

$$\begin{split} \nabla h\_{\hat{l},+}^{(t)} &= \lambda^6 - v\_{\text{x}}\lambda^5 - u\_{\text{x}}\lambda^4 - v\_{\text{z}}\lambda^3 - \left(u\_{\text{z}} + v\_{\text{x}}v\_{\text{z}} - 2(\partial\_{\text{x}}^{-1}v\_{\text{xx}}v\_{\text{z}})\right)\lambda^2 + \\ &+ v\_{\text{y}}\lambda - \left(-u\_{\text{y}} - v\_{\text{x}}v\_{\text{y}} + 2(\partial\_{\text{x}}^{-1}v\_{\text{xx}}v\_{\text{y}})\right)\_{\text{s}} \\\\ &(t) &= -\lambda + v\_{\text{y}} + \frac{1}{2}\lambda + \frac{1}{2}v\_{\text{z}} + \frac{1}{2}\lambda + \frac{1}{2}v\_{\text{y}} + \frac{1}{2}\lambda + \frac{1}{2}\lambda + \frac{1}{2}v\_{\text{z}} \end{split} \tag{530}$$

$$\begin{aligned} \nabla h\_{\mathfrak{d},+}^{(t)} &= \zeta\_x \lambda^5 + w\_x \lambda^4 + \zeta\_z \lambda^3 + (w\_z - \zeta\_x v\_z + 2v\_x \zeta\_z - (\partial\_x^{-1} v\_x \zeta\_x)\_z) \lambda^2 - \zeta\_y \\ &- \zeta\_y \lambda + (-w\_y + \zeta\_x v\_y - 2v\_x \zeta\_y + (\partial\_x^{-1} v\_x \zeta\_x)\_y)\_t \end{aligned}$$

the compatibility condition of the Hamiltonian vector flows leads to the system of new integrable evolution equations:

*uzt* + *uyy* = −*uyuxz* + *uzuxy* − *vyvxy* + *vzvxt* − *uzvyvxx* + *uyvzvxx*− (531) − *<sup>v</sup>*2*xvzvxy* + *<sup>v</sup>*2*xvyvxz* − <sup>2</sup>*exuxy* − 2*sxuxz* + 2*ext* − <sup>2</sup>*sxy* + <sup>2</sup>*exvyvxx* + 2*sxvzvxx*, *vzt* + *vyy* = −*uyvxz* + *uzvxy* − *vyuxz* + *vzuxy* − <sup>2</sup>*exvxy* − 2*sxvxz* − <sup>2</sup>*vxvyvxz* + <sup>2</sup>*vxvzvxy*, − *uxy* − *uzz* = *uxuxz* − *uzuxx* − *uxxvxvz* + *uxvxzvx* − *uxvxxvz* + (*vxvz*)*z* + 2*uxxex* − 2*exz*, − *vxy* − *vzz* = *uxzvx* − *uzvxx* − *uxxvz* + *uxvxz* − 2*vxxvxvz* + *<sup>v</sup>*2*xvxz* + 2*vxxex*, − *uxt* + *uyz* = −*uxuxy* + *uyuxx* + *uxxvxvy* − *uxvxyvx* + *uxvxxvy* − (*vxvy*)*z* + 2*uxxsx* − 2*sxz*, − *vxt* + *vyz* = −*uxyvx* + *uyvxx* + *uxxvy* − *uxvxy* + <sup>2</sup>*vxxvxvy* − *<sup>v</sup>*2*xvxy* + 2*vxxsx*,

where

$$e\_{\mathfrak{X}} = \upsilon\_{\mathfrak{X}} \upsilon\_{\mathfrak{Z}\_{\prime}} \quad s\_{\mathfrak{X}} = -\upsilon\_{\mathfrak{X}} \upsilon\_{\mathfrak{Y}}.$$

Under the constraint *v* = 0, one obtains a new spatially four-dimensional system

$$
\mu\_{zt} + \mu\_{yy} = -\mu\_y \mu\_{xz} + \mu\_z \mu\_{xy} \tag{532}
$$

$$
$$

$$
$$

which reduces to the Mikhalev–Pavlov [204,208,223] integrable heavenly type equation, if to put *z* = *x* ∈ R.

Here, we can observe that the seed element (529) can be presented in the following special compact form:

$$\vec{u} \times \vec{I} := \frac{d\vec{\eta}}{dx} \partial / \partial \mathbf{x} \ltimes d\vec{\rho}, \ \vec{\eta} = u + v\lambda - \lambda^2 \mathbf{x}, \ \vec{\rho} = w + \zeta \lambda,\tag{533}$$

deeply connected with the geometry of the related moduli space of flat connections, related to the coadjoint actions of the corresponding Casimir functionals. Its possible generalization to multidimensional Mikhalev–Pavlov type equations can be done by the seed element

$$
\vec{a} \bowtie \vec{I} := \langle \nabla \vec{\eta} | \nabla \rangle \bowtie d\vec{\rho} \tag{534}
$$

for some elements *η*˜, *ρ*˜ ∈ Ω<sup>0</sup>(T*n*) ⊗ C, *n* ∈ N. An analysis of the case (534) and corresponding systems of multidimensional Mikhalev–Pavlov type equations is planned to be done in a separate study.

#### *11.5. A Modified Martinez Alonso-Shabat Heavenly Type Integrable System* If the seed element *a*˜ - ˜ *l* ∈ G ˜ ∗ is chosen as

$$\begin{split} \vec{u} \times \vec{l} &= (((u\_{x\_1} + c u\_{x\_2}) + \lambda) \partial / \partial x\_1 + ((v\_{x\_1} + c v\_{x\_2}) + c \lambda) \partial / \partial x\_2) \times \\ &\quad \times ((w\_{x\_1} + c w\_{x\_2}) dx\_1 + (\zeta\_{x\_1} + c \zeta\_{x\_2}) dx\_2), \end{split} \tag{535}$$

where *u*, *v*, *w*, *ζ* ∈ *C*<sup>2</sup>(R<sup>2</sup> × S1 × T2; <sup>R</sup>), *c* ∈ R \ {0}, one has the following asymptotic splits for the components of the gradients of the corresponding Casimir functionals *<sup>h</sup>*(1), *h*(2) ∈ I(G˜∗) as |*λ*| → ∞:

$$\begin{aligned} \nabla h\_{\tilde{l}}^{(1)} &\sim \left( \begin{array}{c} 1 + (\mu\_{\mathcal{X}\_1} + c\mu\_{\mathcal{X}\_2})\lambda^{-1} - \mu\_z\lambda^{-2} + \dots \\ c + (\upsilon\_{\mathcal{X}\_1} + c\upsilon\_{\mathcal{X}\_2})\lambda^{-1} - \upsilon\_z\lambda^{-2} + \dots \end{array} \right), \\ \nabla h\_{\tilde{a}}^{(1)} &\sim \left( \begin{array}{c} (w\_{\mathcal{X}\_1} + cw\_{\mathcal{X}\_2})\lambda^{-1} - w\_z\lambda^{-2} + \dots \\ (\tilde{\zeta}\_{\mathcal{X}\_1} + c\tilde{\zeta}\_{\mathcal{X}\_2})\lambda^{-1} - \tilde{\zeta}\_z\lambda^{-2} + \dots \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} \nabla h\_{\boldsymbol{I}}^{(2)} &\simeq \begin{pmatrix} 1 + (\boldsymbol{u}\_{\boldsymbol{x}\_{1}} - c\boldsymbol{u}\_{\boldsymbol{x}\_{2}})\lambda^{-1} + \chi\lambda^{-2} + \dots \\ -c + (\boldsymbol{v}\_{\boldsymbol{x}\_{1}} - c\boldsymbol{v}\_{\boldsymbol{x}\_{2}})\lambda^{-1} + \omega\lambda^{-2} + \dots \end{pmatrix} \prime \\ \nabla h\_{\boldsymbol{d}}^{(2)} &\simeq \begin{pmatrix} (\boldsymbol{w}\_{\boldsymbol{x}\_{1}} - c\boldsymbol{w}\_{\boldsymbol{x}\_{2}})\lambda^{-1} + \varrho\lambda^{-2} + \dots \\ (\zeta\_{\boldsymbol{x}\_{1}} - c\zeta\_{\boldsymbol{x}\_{2}})\lambda^{-1} + \chi\lambda^{-2} + \dots \end{pmatrix} \prime \end{aligned}$$

where

$$\begin{aligned} \chi\_{\overline{x}\_1} + \mathfrak{c}\chi\_{\overline{x}\_2} &= -(\mathfrak{u}\_{\overline{x}\underline{x}\_1} - \mathfrak{c}\mathfrak{u}\_{\overline{x}\underline{x}\_2}) + 2\mathfrak{c}(\mathfrak{u}\_{\overline{x}\_1}\mathfrak{u}\_{\overline{x}\_1\overline{x}\_2} - \mathfrak{u}\_{\overline{x}\_2}\mathfrak{u}\_{\overline{x}\_1\overline{x}\_1} + \mathfrak{v}\_{\overline{x}\_1}\mathfrak{u}\_{\overline{x}\_2\overline{x}\_2} - \mathfrak{v}\_{\overline{x}\_2}\mathfrak{u}\_{\overline{x}\_1\overline{x}\_2}), \\ \mathfrak{w}\_{\overline{x}\_1} + \mathfrak{c}\mathfrak{w}\_{\overline{x}\_2} &= -(\mathfrak{v}\_{\overline{x}\underline{x}\_1} - \mathfrak{c}\mathfrak{v}\_{\overline{x}\underline{x}\_2}) + 2\mathfrak{c}(\mathfrak{u}\_{\overline{x}\_1}\mathfrak{v}\_{\overline{x}\_1\overline{x}\_2} - \mathfrak{u}\_{\overline{x}\_2}\mathfrak{v}\_{\overline{x}\_1\overline{x}\_2} + \mathfrak{v}\_{\overline{x}\_1}\mathfrak{v}\_{\overline{x}\_2\overline{x}\_2} - \mathfrak{v}\_{\overline{x}\_2}\mathfrak{v}\_{\overline{x}\_1\overline{x}\_2}), \end{aligned} \tag{536}$$

and

*ρ<sup>x</sup>*1 + *<sup>c</sup>ρ<sup>x</sup>*2 = <sup>−</sup>(*wzx*1 − *cwzx*2 ) + <sup>2</sup>*c*(*ux*1*wx*1*x*2 − *ux*2*wx*1*x*1 + <sup>2</sup>*wx*2*ux*1*x*1<sup>−</sup> − <sup>2</sup>*wx*1*ux*1*x*2 + *vx*1*wx*2*x*2 − *vx*2*wx*1*x*2 + *wx*2 *vx*1*x*2 − *wx*2 *vx*2*x*2 + *ζ<sup>x</sup>*2 *vx*1*x*1 − *ζ<sup>x</sup>*1 *vx*1*x*2 ), *χ<sup>x</sup>*1 + *<sup>c</sup>χ<sup>x</sup>*2 = <sup>−</sup>(*ζzx*1 − *<sup>c</sup>ζzx*2 ) + <sup>2</sup>*c*(*vζ<sup>x</sup>*2*x*2 − *vx*2 *ζ<sup>x</sup>*1*x*2 + <sup>2</sup>*ζ<sup>x</sup>*2 *vx*1*x*2<sup>−</sup> − <sup>2</sup>*ζ<sup>x</sup>*1 *vx*2*x*2 + *ux*1 *ζ<sup>x</sup>*1*x*2 − *ux*2 *ζ<sup>x</sup>*1*x*1 + *ζ<sup>x</sup>*2*ux*1*x*1 − *ζ<sup>x</sup>*1*ux*1*x*2 + *wx*2*ux*1*x*2 − *wx*1*ux*2*x*2 ).

In the case when the reduced Casimir gradients are equal to the expressions

$$
\nabla h\_{\vec{l},+}^{(y)} = \begin{pmatrix}
\lambda^2 + (u\_{x\_1} + c u\_{x\_2})\lambda - u\_z \\
c\lambda^2 + (v\_{x\_1} + c v\_{x\_2})\lambda - v\_z
\end{pmatrix},
\nabla h\_{\vec{l},+}^{(y)} = \begin{pmatrix}
(w\_{x\_1} + c w\_{x\_2})\lambda - w\_z \\
(\zeta\_{x\_1} + c \zeta\_{x\_2})\lambda - \zeta\_z
\end{pmatrix},
$$

and

$$
\nabla h\_{l,+}^{(t)} = \begin{pmatrix}
\lambda^2 + (u\_{x\_1} - \varepsilon u\_{x\_2})\lambda + \chi \\
\end{pmatrix},
\nabla h\_{l,+}^{(t)} = \begin{pmatrix}
(w\_{x\_1} - \varepsilon w\_{x\_2})\lambda + \rho \\
(\zeta\_{x\_1} - \varepsilon \zeta\_{x\_2})\lambda + \chi \\
\end{pmatrix},
$$

the Lax–Sato compatibility condition of the Hamiltonian vector flows leads to the system of evolution equations:

*uzt* + *χy* = <sup>−</sup>*uzx*1*χ* − *uzx*2*<sup>ω</sup>* + *uzx*1 + *vzx*2 , (537) *vzt* + *<sup>ω</sup>y* = <sup>−</sup>*vzx*1*χ* − *vzx*2*<sup>ω</sup>* + *uzωx*1 + *vzωx*2 , *uyx*1 + *cuyx*2 = <sup>−</sup>(*ux*1 + *cux*2 )*uzx*1 − (*vx*1 + *cvx*2 )*uzx*2 + (*ux*1*x*1 + *cux*1*x*2 )*uz*<sup>+</sup> + (*ux*1*x*2 + *cux*2*x*2 )*vz* − *uzz*, *vyx*1 + *cvyx*2 = <sup>−</sup>(*ux*1 + *cux*2 )*vzx*1 − (*vx*1 + *cvx*2 )*vzx*2 + (*vx*1*x*1 + *cvx*1*x*2 )*uz*<sup>+</sup> + (*vx*1*x*2 + *cvx*2*x*2 )*vz* − *vzz*, *utx*1 + *cutx*2 = (*ux*1 + *cux*2 )*<sup>χ</sup><sup>x</sup>*1 + (*vx*1 + *cvx*2 )*<sup>χ</sup><sup>x</sup>*2 − (*ux*1*x*1 + *cux*1*x*2 )*<sup>χ</sup>*− − (*ux*1*x*2 + *cux*2*x*2 )*ω* + *χ<sup>z</sup>*, *vtx*1 + *cvtx*2 = (*ux*1 + *cux*2 )*<sup>ω</sup>x*1 + (*vx*1 + *cvx*2 )*<sup>ω</sup>x*2 − (*vx*1*x*1 + *cvx*1*x*2 )*<sup>χ</sup>*− − (*vx*1*x*2+ *cvx*2*x*2)*ω* + *ωz*,

generalizing the Martinez Alonso–Shabat heavenly type integrable system. Thus, the following proposition holds.

**Proposition 21.** *The constructed system of heavenly type Equations (536) and (537) has the Lax– Sato vector field representation with the "spectral" parameter λ* ∈ C*, which is related to the element a* ˜ - ˜ *l* ∈ G ˜ ∗ *in the form (535).*

The system of Equations (536) and (537) admits the reduction when *u* = *v*. In this case, under *c* = 1 one obtains such a system:

$$\begin{aligned} \mu\_{\overline{z}t} + \chi\_{\overline{y}} &= - (\mathfrak{u}\_{\overline{z}x\_1} + \mathfrak{u}\_{\overline{z}x\_2})\chi + \mathfrak{u}\_z(\chi\_{\overline{x}\_1} + \chi\_{\overline{x}\_2}), \\ \chi\_{\overline{x}\_1} + \chi\_{\overline{x}\_2} &= - (\mathfrak{u}\_{\overline{z}x\_1} - \mathfrak{u}\_{\overline{z}x\_2}) - 2(\mathfrak{u}\_{\overline{x}\_1}\mathfrak{u}\_{\overline{x}\_2})\chi\_1 - 2(\mathfrak{u}\_{\overline{x}\_1}\mathfrak{u}\_{\overline{x}\_2})\chi\_2. \end{aligned} \tag{538}$$

The additional constraint *uz* = *ux*1 + *ux*2 transforms the system (538) into the following interesting integro-differential equation:

(*<sup>u</sup>*˜*tx*1 + *<sup>u</sup>*˜*tx*2 ) − (*uyx*˜ 1 − *uyx*˜ 2 ) = *ux*1*x*2 (*ux*1 − *ux*2 ) − *ux*1*x*1*ux*2 + *ux*2*x*2*ux*1<sup>−</sup> − *ux*1*x*2 (*u*2*x*1 − *u*2*x*2 ) − *ux*1*x*1*ux*2 (*ux*1 + *ux*2 ) + *ux*2*x*2*ux*1 (*ux*1 + *ux*2 )− − <sup>2</sup>(P(*ux*1*ux*2 )*y*˜)+(*ux*1*x*1 + <sup>2</sup>*ux*1*x*2 + *ux*2*x*2 )(P*ux*1*ux*2 ), P = (*∂*/*∂<sup>x</sup>*1+ *<sup>∂</sup>*/*∂<sup>x</sup>*2)−<sup>1</sup>(*∂*/*∂<sup>x</sup>*1− *∂*/*∂<sup>x</sup>*2),

where ˜*t* = 2*t* and *y*˜ = 2*y*. Thus, the Equation (538) is integrable and can be considered as some multi-dimensional generalization of the Martinez Alonso–Shabat system [236].

#### *11.6. A Modified Current Loop Algebra and Multidimensional Heavenly Type Integrable Equations: The Generalized Lie-Algebraic Structures*

A further generalization of the multi-dimensional case related to the loop group Diff : (T*n*) on the torus T*<sup>n</sup>*, *n* ∈ Z+ can be developed [207–209] by the following approach. Since the Lie algebra diff : (T*n*) consists of the loop group elements, analytically continued from the circle S1 := *∂*D1, being the boundary of the disk D<sup>1</sup> ⊂ C, by means of the complex "spectral"variable *λ* ∈ C both into the interior <sup>D</sup>1+ ⊂ C and the exterior D<sup>1</sup>− ⊂ C parts of the disk D<sup>1</sup> ⊂ C, one can take into account its analytical invariance to the circle diffeomorphism group Diff(S<sup>1</sup>). The latter gives rise to the naturally extended holomorphic Lie algebra diff(T*n* × C) = diff : (T*n* × D<sup>1</sup>+) ⊕ diff :(T*n* × D<sup>1</sup>−) on the torus T*n* × C, whose elements are representable as *a*¯(*x*; *λ*) := %*a*(*x*; *<sup>λ</sup>*), *∂∂*x& = *n*∑*j*=1 *aj*(*x*; *λ*) *∂∂xj* + *<sup>a</sup>*0(*<sup>x</sup>*; *λ*) *∂∂λ* for some holomorphic in *λ* ∈ <sup>D</sup>1± vectors *a*(*x*; *λ*) ∈ E × E*n* for all *x* ∈ T*<sup>n</sup>*, and where we

denoted by *∂∂*x := ( *∂∂λ* , *∂∂x*1 , *∂∂x*2 , ..., *∂∂xn* )- the generalized Euclidean vector gradient with respect to the vector variable x := (*<sup>λ</sup>*, *x*) ∈ C × T*<sup>n</sup>*.

Let us construct a modified current loop Lie algebra G ¯ as the semi-direct sum G ¯ := diff( T*n* ×C)diff(T*n* ×C)<sup>∗</sup> of the Lie algebra diff(T*n* ×C) and its adjoint space diff(T*n* × C)<sup>∗</sup>, taking into account their natural pairing

$$(\bar{l}|d) := \mathop{\rm res}\_{\lambda \in \mathbb{C}} (l(\infty)|a(\infty))\_{H^0} \tag{539}$$

for any ¯ *l* ∈ diff(T*n* × C)∗ and *a*¯ ∈ diff( T*n* × C). The corresponding Lie commutator on the loop Lie algebra G ¯ is given for any *a*¯1 - ¯ *l*1, *a*¯2 - ¯ *l*2 ∈ G ¯ by

$$\left[\left[\vec{a}\_1 \times \vec{l}\_1, \vec{a}\_2 \times \vec{l}\_2\right] \right.\\ \left. := \left[\vec{a}\_1, a\_2\right] \ltimes ad\_{a\_1}^\* \vec{l}\_2 - ad\_{a\_2}^\* \vec{l}\_1. \tag{540}$$

The Lie algebra G ¯ also splits into the direct sum of two subalgebras:

$$
\hat{\mathcal{G}} = \hat{\mathcal{G}}\_{+} \oplus \hat{\mathcal{G}}\_{-},\tag{541}
$$

allowing the introduction of the classical *R*-structure:

$$[\mathbb{\vec{a}}\_1 \ltimes \bar{I}\_1, \mathbb{\vec{a}}\_2 \ltimes \bar{I}\_2]\_R := [R(\mathbb{\vec{a}}\_1 \ltimes \bar{I}\_1), \ \mathbb{\vec{a}}\_2 \ltimes \bar{I}\_2] + [\ \mathbb{\vec{a}}\_1 \ltimes \bar{I}\_1, R(\mathbb{\vec{a}}\_2 \ltimes \bar{I}\_2)] \tag{542}$$

for any *a*¯1 - ¯ *l*1, *a*¯2 - ¯ *l*2 ∈ G ¯ , where, by definition,

$$R := (P\_+ - P\_-) / \mathcal{2},\tag{543}$$

and

$$P\_{\pm} \vec{\mathcal{G}} := \vec{\mathcal{G}}\_{\pm} \subset \vec{\mathcal{G}}.\tag{544}$$

The space G ¯ ∗ (adjoint to the Lie algebra G ¯ ) can be identified with the space G ¯ by using the symmetric and non-degenerate form

$$(\mathbb{d} \ltimes \mathbb{I} | \overline{r} \ltimes \mathbb{m}) := \mathop{\rm res}\_{\lambda \in \mathbb{C}} (\mathbb{d} \ltimes \mathbb{I} | \overline{r} \ltimes \mathbb{m})\_{H^{0} \ltimes} \tag{545}$$

where, by definition,

$$(\vec{a} \times \vec{l} | \vec{r} \times \vec{m})\_{H^0} = (\vec{m} | \vec{a})\_{H^0} + (\vec{l} | \vec{r})\_{H^0} \tag{546}$$

for any pair of elements *a*¯ - ¯ *l*, *r*¯ - *m*¯ ∈ G ¯ .

**Remark 12.** *The above constructed Lie algebra* G ¯ , *being metrized by means of the symmetric, nondegenerate bilinear form (545), is owing to the construction described in the introduction, to uniquely represent the coadjoint orbits on* G ¯ ∗ - G ¯ *in the standard Lax type form on* G ¯ , *that will be used further.*

Owing to the convolution (546), the Lie algebra G ¯ becomes metrized. For arbitrary smooth functions *f* , *g* ∈ D(G¯∗) one can naturally determine two Lie–Poisson brackets

$$\{f, \emptyset\} := (\vec{a} \ltimes \vec{l} | [\nabla f(\vec{l}, \vec{a}), \nabla \lg(\vec{l}, \vec{a})])\tag{547}$$

and

$$\{f, \mathcal{G}\}\_{\mathbb{R}} := (\mathfrak{d} \ltimes \bar{l} | [\nabla f(\vec{l}, \mathfrak{d}), \nabla \mathcal{g}(\vec{l}, \mathfrak{d})]\_{\mathbb{R}})\,. \tag{548}$$

where at any seed element *a*¯ - ¯ *l* ∈ G ¯ ∗ - G ¯ the gradient element ∇ *f*( ¯ *l*, *a*¯) := ∇ *f*¯*l* - ∇ *fa*¯ - ∇ *f*(*l*, *a*)|(*∂*/*∂*x, *d*x)- ∈ G¯ and ∇ *f*¯*l* = ∇ *fl*|*∂*/*∂*x , ∇ *fa*¯ = ∇ *fa*|*d*x , and, similarly, the gradient element ∇*g*( ¯ *l*, *a*¯) := ∇*g*¯*l* - ∇*ga*¯ - ∇*g*(*l*, *a*)|(*∂*/*∂*x, *d*x)- ∈ G¯∗ and ∇*g*¯*l* = ∇*gl*|*∂*/*∂*x , ∇*ga*¯ = ∇*ga*|*d*x are calculated with respect to the metric (546).

Let us now assume that a smooth function *h* ∈ I(G¯∗) is a Casimir invariant, that is

$$ad^\*\_{\nabla h(\bar{l}, \bar{a})} (\mathfrak{a} \ltimes \mathfrak{l}) = 0 \tag{549}$$

for a chosen seed element *a*¯ - ¯ *l* ∈ G¯∗ - G¯. Since for an element *a*¯ - ¯ *l* ∈ G¯∗ - G¯ and an arbitrary *f* ∈ D(G¯∗) the adjoint mapping is

$$ad^\*\_{\nabla f(\mathbb{I}, \mathbb{I})} (\bar{a} \times \bar{l}) = (\left[\nabla h\_{\bar{l}}, \tilde{a}\right] \ltimes (ad^\*\_{\nabla h\_{\bar{l}}} \tilde{l} + ad^\*\_{\bar{u}} \nabla h\_{\bar{u}})),\tag{550}$$

the condition (549) can be rewritten as

$$\left[\nabla h\_{\vec{l}}, \vec{a}\right] = 0, \quad \left.ad^\*\_{\nabla h\_{\vec{l}}}\right| \vec{l} + \left.ad^\*\_{\vec{a}}\nabla h\_{\vec{a}} = 0,\tag{551}$$

and one can easily obtain that the Casimir functional *h* ∈ I(G¯∗) satisfies the system of determining equations

$$
\langle \nabla h\_l | \partial / \partial \mathbf{x} \rangle a - \langle a | \partial / \partial \mathbf{x} \rangle \nabla h\_l = 0,
$$

$$
\langle \partial / \partial \mathbf{x} | \diamond \nabla h\_l \rangle l + \langle l | (\partial / \partial \mathbf{x} \nabla h\_l) \rangle + \tag{552}
$$

$$
+ \langle \partial / \partial \mathbf{x} | \diamond a \rangle \nabla h\_a + \langle a | (\partial / \partial \mathbf{x} \nabla h\_d) \rangle = 0.
$$

For the Casimir functional *h* ∈ D(G¯∗) the Equation (552) should be be solved analytically. In the case when an element ¯ *l* - *a*¯ ∈ G¯∗ is singular as |*λ*| → <sup>∞</sup>, one can consider the general asymptotic expansion

$$\nabla h^{(p)}(l, a) \sim \lambda^p \sum\_{j \in \mathbb{Z}\_+} (\nabla h^{(p)}\_{l, j}; \nabla h^{(p)}\_{a, j}) \lambda^{-j} \tag{553}$$

for some suitably chosen *p* ∈ Z+, which is substituted into the Equation (552). The latter is then solved recurrently giving rise to a set of gradient expressions for the Casimir functionals *h*(*p*) ∈ D(G¯∗) at the specially found integers *p* ∈ Z+.

Assume now that *<sup>h</sup>*(*y*), *h*(*t*) ∈ I(G¯∗) are such Casimir functionals for which the Hamiltonian vector field generators

$$
\nabla h^{(y)}(l,\mathfrak{d})\_+ := (\,\,\nabla h^{(p\_y)}(l,\mathfrak{d}))\_+, \quad \nabla h^{(t)}(l,\mathfrak{d})\_+ := (\,\,\nabla h^{(p\_t)}(l,\mathfrak{d}))\_+,\tag{554}
$$

are, respectively, defined at some specially found integers *py*, *pt* ∈ Z+. These invariants generate owing to the Lie–Poisson bracket (548) the following commuting to each other Hamiltonian flows:

$$\frac{\partial}{\partial y}(\vec{a} \times \vec{l}) = -ad^\*\_{\nabla h^{(y)}(\vec{l}, \mathfrak{a})\_+}(\vec{a} \ltimes \bar{l}), \tag{555}$$

$$\frac{\partial}{\partial t}(\vec{a} \ltimes \bar{l}) = -ad^\*\_{\nabla h^{(t)}(\vec{l}, \mathfrak{a})\_+}(\vec{a} \ltimes \bar{l}),$$

on an element *a*¯ - ¯ *l* ∈ G¯∗ - G¯ with respect to the corresponding evolution parameters *t*, *y* ∈ R. Owing to the construction, the flows (554) can be rewritten equivalently as

$$
\begin{split}
\partial l/\partial \boldsymbol{l} &= -\left< \frac{\partial}{\partial \mathbf{x}} \vert \boldsymbol{\circ} \nabla h\_{l}^{(p\_{l})} \right> \boldsymbol{l} - \left< l \vert (\frac{\partial}{\partial \mathbf{x}} \nabla h\_{l}^{(p\_{l})}) \right> - \left< \frac{\partial}{\partial \mathbf{x}} \vert \boldsymbol{\circ} a \right> \nabla h\_{\mathbf{z}}^{(p\_{l})} - \left< a \vert (\frac{\partial}{\partial \mathbf{x}} \nabla h\_{\mathbf{z}}^{(p\_{l})}) \right>, \\
\partial l/\partial \boldsymbol{y} &= -\left< \frac{\partial}{\partial \mathbf{x}} \vert \boldsymbol{\circ} \nabla h\_{l}^{(p\_{l})} \right> \boldsymbol{l} - \left< l \vert (\frac{\partial}{\partial \mathbf{x}} \nabla h\_{l}^{(p\_{l})}) \right> - \left< \frac{\partial}{\partial \mathbf{x}} \vert \boldsymbol{\circ} a \right> \nabla h\_{\mathbf{z}}^{(p\_{l})} - \left< a \vert (\frac{\partial}{\partial \mathbf{x}} \nabla h\_{\mathbf{z}}^{(p\_{l})}) \right>, \\
\partial a/\partial t &= -\left< \nabla h\_{l}^{(p\_{l})} \vert \frac{\partial}{\partial \mathbf{x}} \rangle a + \left< a \vert \frac{\partial}{\partial \mathbf{x}} \right> \nabla h\_{l}^{(p\_{l})}, \ \partial a/\partial y &= -\left< \nabla h\_{l}^{(p\_{l})} \vert \frac{\partial}{\partial \mathbf{x}} \right> a + \left< a \vert \frac{\partial}{\partial \mathbf{x}} \right> \nabla h\_{l}^{(p\_{l})}.
\end{split} \tag{6.56}
$$

where *y*, *t* ∈ R are the corresponding evolution parameters. Since the invariants *<sup>h</sup>*(*y*), *h*(*t*) ∈ I(G¯∗) are commuting to each other with respect to the Lie–Poisson bracket (548), the flows (556) are commuting too, meaning equivalently that the corresponding Hamiltonian vector field generators

$$
\nabla h\_+^{(t)} := \left< \nabla h\_I^{(p\_t)}(l)\_+ | \frac{\partial}{\partial \mathbf{x}} \right>\_{\prime} \quad \nabla h\_+^{(y)} := \left< \nabla h\_I^{(p\_y)}(l)\_+ | \frac{\partial}{\partial \mathbf{x}} \right> \tag{557}
$$

satisfy the Lax type compatibility condition

$$\frac{\partial}{\partial y}\nabla h\_+^{(t)} - \frac{\partial}{\partial t}\nabla h\_+^{(y)} = [\nabla h\_+^{(t)}, \nabla h\_+^{(y)}] \tag{558}$$

for all *y*, *t* ∈ R. On the other hand, the condition (558) is equivalent to the compatibility condition of two linear equations

$$\left(\frac{\partial}{\partial t} + \nabla h\_+^{(t)}\right)\psi = 0, \ \langle a|\frac{\partial}{\partial \mathbf{x}}\rangle \psi = 0, \ \left(\frac{\partial}{\partial y} + \nabla h\_+^{(y)}\right)\psi = 0\tag{559}$$

for a function *ψ* ∈ *C*<sup>2</sup>(R<sup>2</sup> × T*n* × C; C), all *y*, *t* ∈ R and any x ∈ T*n* × C. The results obtained above can be formulated as the following proposition.

**Proposition 22.** *Let a seed element a*¯ - ¯ *l* ∈ G ¯ ∗ *and <sup>h</sup>*(*y*), *h*(*t*) ∈ I(G¯∗) *are some Casimir functionals subject to the metric* (·|·) *on the holomorphic current loop algebra* G¯ *and the natural coadjoint action on the co-algebra* G ¯ ∗ - G ¯ . *Then the following dynamical systems*

$$\frac{\partial}{\partial y}(\mathfrak{a}\ltimes\bar{\varGamma}) = -ad^\*\_{\nabla h^{(\mathfrak{g})}(\varGamma,\mathfrak{a})\_+}(\mathfrak{a}\ltimes\bar{\varGamma}), \quad \frac{\partial}{\partial t}(\mathfrak{a}\ltimes\bar{\varGamma}) = -ad^\*\_{\nabla h^{(\mathfrak{f})}(\varGamma,\mathfrak{a})\_+}(\mathfrak{a}\ltimes\bar{\varGamma})\tag{560}$$

*are commuting to each other Hamiltonian flows for evolution parameters y*, *t* ∈ R. *Moreover, the compatibility condition of these flows is equivalent to the vector field representation*

$$(\partial / \partial t + \nabla h\_+^{(t)})\psi = 0, \ \langle a|\partial / \partial \mathbf{x} \rangle \psi = 0, \ (\partial / \partial y + \nabla h\_+^{(y)})\psi = 0,\tag{561}$$

*where ψ* ∈ *C*<sup>2</sup>(R<sup>2</sup> × T*n* × C; C) *and the vector fields* ∇*h*(*t*) + , ∇*h*(*y*) + ∈ diff(T*n* × C) *are given by the expressions (557).*

**Remark 13.** *As it was mentioned above, the expansion (553) is effective if a chosen seed element a* ¯ - ¯ *l* ∈ G ¯ ∗ *is singular as* |*λ*| → ∞. *In the case when it is singular as* |*λ*| → 0, *the expression (553) should be respectively replaced by the expansion*

$$\nabla h^{(p)}\left(\overline{l}, \overline{a}\right) \sim \lambda^{-p} \sum\_{j \in \mathbb{Z}\_+} \nabla h^{(p)}\_{\overline{j}}\left(\overline{l}, \overline{a}\right) \lambda^{\overline{j}} \tag{562}$$

*for suitably chosen integers p* ∈ Z+, *and the reduced Casimir function gradients are then given by the Hamiltonian vector field generators*

$$
\nabla h^{(y)}(\vec{l}, \vec{a}) \\_ = \lambda (\lambda^{-p\_g - 1} \nabla h^{(p\_g)}(\vec{l}, \vec{a})) \\_ \quad \nabla h^{(t)}(\vec{l}, \vec{a}) \\_ = \lambda (\lambda^{-p\_l - 1} \nabla h^{(p\_l)}(\vec{l}, \vec{a})) \\_ \tag{563}
$$

*for suitably chosen positive integers py*, *pt* ∈ Z+ *and the corresponding Hamiltonian flows are, respectively, written as*

$$\frac{\partial}{\partial t}(\vec{a}\times\boldsymbol{I}) = ad^\*\_{\vcorner\boldsymbol{h}^{(\boldsymbol{I})}(\boldsymbol{I},\boldsymbol{\upbeta})\_{-}}(\vec{a}\times\boldsymbol{I}), \quad \frac{\partial}{\partial y}(\vec{a}\times\boldsymbol{I}) = ad^\*\_{\vcorner\boldsymbol{h}^{(\boldsymbol{I})}(\boldsymbol{I},\boldsymbol{\upbeta})\_{-}}(\vec{a}\times\boldsymbol{I})\tag{564}$$

*for evolution parameters y*, *t* ∈ R.

As it was demonstrated above, the presented construction of Hamiltonian flows on the adjoint space G ¯ ∗ can be generalized proceeding to the point product G ¯ := G ¯S<sup>1</sup> = ∏ *z*∈S<sup>1</sup>G ¯

of the holomorphic current Lie algebra G ¯ , endowed with the central extension, generated by a two-cocycle *ω*2 : G ¯ ×G ¯ → C, where

$$
\omega\_2(\vec{a}\_1 \times \vec{l}\_1, \vec{a}\_2 \times \vec{l}\_2) := \int\_{\mathbb{S}^1} [(\vec{l}\_1, \partial \vec{a}\_2 / \partial z)\_1 - (\vec{l}\_2, \partial \vec{a}\_1 / \partial z)\_1] dz \tag{565}
$$

for any pair of elements *a*¯1 - ¯ *l*1, *a*¯2 - ¯ *l*2 ∈ G. The resulting *R*-deformed Lie–Poisson bracket for any smooth functionals *h*, *f* ∈ D(G ∗) on the adjoint space G ∗ to the centrally extended loop Lie algebra G := G ¯ ⊕ C becomes equal to

$$\begin{split} \{h\_{\prime}f\}\_{R} &:= (\mathbb{d} \ltimes \mathbb{I})[\nabla h(\mathbb{I}, \mathbb{d}), \nabla f(\mathbb{I}, \mathbb{d})]\_{R}) + \\ &+ \omega\_{2}(R\nabla h(\mathbb{\bar{I}}, \mathbb{d}), \nabla f(\mathbb{\bar{I}}, \mathbb{d})) + \omega\_{2}(\nabla h(\mathbb{\bar{I}}, \mathbb{d}), R\nabla f(\mathbb{\bar{I}}, \mathbb{d})). \end{split} \tag{566}$$

The corresponding Casimir functionals *h*(*p*) ∈ I(G∗) for specially chosen *p* ∈ Z+, are defined with respect to the standard Lie–Poisson bracket as

$$\{h^{(p)}, f\} := (\mathfrak{d} \ltimes \bar{l})[\nabla h^{(p)}(\vec{l}, \mathfrak{d}), \nabla f(\vec{l}, \mathfrak{d})]) + \ \omega\_2(\nabla h^{(p)}(\vec{l}, \mathfrak{d}), \nabla f(\vec{l}, \mathfrak{d})) = 0 \tag{567}$$

for all smooth functionals *f* ∈ D(G ∗). Based on the equality one easily finds that the gradients ∇*h*(*p*) ∈ G of the Casimir functionals *h*(*p*) ∈ <sup>I</sup>(<sup>G</sup><sup>∗</sup>), *p* ∈ Z+, satisfy the following equations:

$$
\frac{1}{2} \left[ \nabla h\_{\varGamma} \vec{a} \right] - \frac{\partial}{\partial z} \nabla h\_{\varGamma} = 0, \text{ and} \\
\stackrel{\ast}{\nabla h\_{\varGamma}} \vec{l} + a d\_{\varwidetilde{a}}^{\*} \nabla h\_{\varGamma} - \frac{\partial}{\partial z} \nabla h\_{\varGamma} = 0 \tag{568}
$$

for a chosen element *a*¯ - ¯ *l* ∈ G ∗. Making use of the suitable Casimir functionals *<sup>h</sup>*(*y*), *h*(*t*) ∈ I(G <sup>∗</sup>), one can construct, making use of (566), the following commuting Hamiltonian flows on the adjoint space G ∗:

$$\frac{\partial}{\partial y}(\mathfrak{a}\ltimes\bar{\varGamma}) = \{h^{(y)}, \mathfrak{a}\ltimes\bar{\varGamma}\}\_{R\succ} \\ \frac{\partial}{\partial t}(\mathfrak{a}\ltimes\bar{\varGamma}) = \{h^{(t)}, \mathfrak{a}\ltimes\bar{\varGamma}\}\_{R\succ} \\ \tag{569}$$

which are equivalent to the evolution equations

$$\frac{\partial}{\partial y}\mathbb{d} = -[\nabla h\_{\mathbb{L},+'}^{(y)}, \mathbb{d}] + \frac{\partial}{\partial z}\nabla h\_{\mathbb{L},+'}^{(y)} \quad \frac{\partial}{\partial t}\mathbb{d} = -[\nabla h\_{\mathbb{L},+'}^{(t)}, \mathbb{d}] + \frac{\partial}{\partial z}\nabla h\_{\mathbb{L},+}^{(t)} \tag{570}$$

and

$$\frac{\partial}{\partial y}\bar{I} = -ad^\*\_{\nabla h^{(y)}\_{l\_{\prec}}}\bar{I} - ad^\*\_{\mathbb{A}}(\nabla h^{(y)}\_{\mathbb{A}\_{r+}}) + \frac{\partial}{\partial z}\nabla h^{(y)}\_{\mathbb{A}\_{r+}}.\tag{571}$$

$$\frac{\partial}{\partial t}I = -ad^\*\_{\nabla h^{(t)}\_{l\_{\prec}}}\bar{I} - ad^\*\_{\mathbb{A}}(\nabla h^{(t)}\_{\mathbb{A}\_{r+}}) + \frac{\partial}{\partial z}\nabla h^{(t)}\_{\mathbb{A}\_{r+}}.$$

The results obtained above are summarized as

**Proposition 23.** *The Hamiltonian flows (569) on the adjoint space* G ∗ *generate the separately commuting evolution flows (570) and (571), giving rise to the following unique Lax type compatibility condition:*

$$
\left[\nabla h\_{l,+\prime}^{(y)} \nabla h\_{l,+}^{(t)}\right] - \frac{\partial}{\partial t} \nabla h\_{l,+}^{(y)} + \frac{\partial}{\partial y} \nabla h\_{l,+}^{(t)} = 0,\tag{572}
$$

*being equivalent to some system of nonlinear heavenly type equations in partial derivatives. Moreover, the system of evolution flows (570) and (571) can be considered as the compatibility condition for the following set of linear vector equations*

$$\frac{\partial \psi}{\partial y} + \nabla h\_{l,+}^{(y)} \psi = 0, \ \frac{\partial \psi}{\partial z} + \langle a|\partial \slash \partial \chi \rangle \psi = 0, \ \frac{\partial \psi}{\partial t} + \nabla h\_{l,+}^{(t)} \psi = 0 \tag{573}$$

*for all* (*y*, *t*, *z*; x) ∈ (R<sup>2</sup> × S<sup>1</sup>) × T*n* × C *and a function ψ* ∈ *C*<sup>2</sup>((R<sup>2</sup> × S<sup>1</sup>) × T*n* × C; C).

**Remark 14.** *The Lie-algebraic scheme of constructing heavenly type integrable equations on respectively chosen smooth functional manifolds, applied above for the modified current loop Lie algebra* G ¯ := diff : (T*n* × C) - diff : (T*n* × C)∗ *as the semi-direct sum of the Lie algebra* diff : (T*n* × C) *and its dual space* diff : (T*n* × C)<sup>∗</sup>, *can be naturally reformulated within a respectively generalized Lagrange–d'Alembert mechanical principle, as was done in the work [214], and which will be analyzed in a separate work under preparation.*

#### *11.7. A New Modified Spatially Four-Dimensional Mikhalev-Pavlov type Heavenly Equation* Letaseedelement *a*˜ - ˜ *l* ∈ G ∗ bechosenas

$$\vec{u} \times \vec{l} = ((\mu\_x - \lambda)\partial / \partial \mathbf{x} + \upsilon\_x \partial / \partial \lambda) \times (w\_x d\mathbf{x} + \eta\_x d\lambda),\tag{574}$$

where *u*, *v*, *w*, *η* ∈ *C*<sup>2</sup>(R<sup>2</sup> × (S<sup>1</sup> × C); <sup>R</sup>). The asymptotic expressions for the components of the gradients (562) of the corresponding Casimir functionals *h*(*p*) ∈ <sup>I</sup>(<sup>G</sup><sup>∗</sup>), *p* ∈ Z+, as |*λ*| → ∞ have the following forms:

$$
\nabla h\_{\rm I} \sim \lambda^{p} \begin{pmatrix} 1 - u\_{\rm x} \lambda^{-1} + (-u\_{\rm z} + (p-1)v)\lambda^{-2} + (u\_{\rm y} + (p-2)(u\_{\rm x}v + \chi\_{\rm x})) \lambda^{-3} + \dots \\ -v\_{\rm x} \lambda^{-1} - v\_{\rm z} \lambda^{-2} + (v\_{\rm y} - (p-2)v\_{\rm x}v) \lambda^{-3} + \dots \end{pmatrix},
$$

$$
\nabla h\_{\rm B} \sim \lambda^{p} \begin{pmatrix} w\_{\rm x} \lambda^{-1} + w\_{\rm z} \lambda^{-2} + (-w\_{\rm y} + (p-2)(wv)\_{\rm x}) \lambda^{-3} + \dots \\ \eta\_{\rm x} \lambda^{-1} + (\eta\_{\rm z} + (p-1)w) \lambda^{-2} + (-\eta\_{\rm y} + (p-2)\omega\_{\rm x}) \lambda^{-3} + \dots \end{pmatrix},
$$

*p* ∈ Z+, where

$$
\chi\_{xx} = \upsilon\_z + \mu\_x \upsilon\_{x\prime} \quad \omega\_{xx} = \upsilon\_z - \mu\_x \upsilon\_x - \upsilon\_x \eta\_x + \upsilon \eta\_x.
$$

In the case when

$$\begin{aligned} \nabla h\_{I,+}^{(y)} &:= \begin{pmatrix} \lambda^2 - \mu\_{\mathcal{X}}\lambda + (-\mu\_z + \upsilon) \\ -\upsilon\_{\mathcal{X}}\lambda - \upsilon\_z \end{pmatrix}, \\\\ \nabla h\_{\mathcal{R},+}^{(y)} &:= \begin{pmatrix} \upsilon\_{\mathcal{X}}\lambda + \upsilon\_z \\ \eta\_{\mathcal{X}}\lambda + (\eta\_z + \upsilon) \end{pmatrix}, \end{aligned}$$

and

$$\begin{aligned} \nabla h\_{l\_r+}^{(t)} &= \begin{pmatrix} \lambda^3 - u\_\mathbf{x}\lambda^2 + (-u\_z + 2\upsilon)\lambda + (u\_\mathbf{y} + u\_\mathbf{x}\upsilon + \chi\_\mathbf{x}) \\ -\upsilon\_\mathbf{x}\lambda^2 - \upsilon\_z\lambda + (\upsilon\_\mathbf{y} - \upsilon\_\mathbf{x}\upsilon) \end{pmatrix}, \\\\ \nabla h\_{l\_r+}^{(t)} &= \begin{pmatrix} w\_\mathbf{x}\lambda^2 + w\_z\lambda + (-w\_\mathbf{y} + (\upsilon\upsilon)\_\mathbf{x}) \\ \eta\_\mathbf{x}\lambda^2 + (\eta\_z + 2\upsilon)\lambda + (-\eta\_\mathbf{y} + \omega\_\mathbf{x}) \end{pmatrix}, \end{aligned}$$

the compatibility condition of the Hamiltonian vector flows (569) leads to the system of evolution equations:

$$\begin{aligned} u\_{zl} + u\_{yy} &= -u\_y u\_{zx} + u\_z u\_{xy} - u\_{xy} \upsilon - u\_{zz} \upsilon - \chi\_x u\_{xz}, \\ \upsilon\_{zl} + \upsilon\_{yy} &= \upsilon \upsilon\_x^2 - \upsilon\_z^2 - \upsilon \upsilon\_{xy} - \upsilon \upsilon\_{zz} - u\_y \upsilon\_{xx} + u\_z \upsilon\_{xy} - u\_z \upsilon\_x^2 - \chi\_x \upsilon\_{xx}, \\ -u\_{xy} - u\_{zz} &= u\_x u\_{xx} - u\_z u\_{xx} + u\_{xx} \upsilon\_r \\ -\upsilon\_{xy} - \upsilon\_{zz} &= \upsilon\_x^2 + \upsilon\_{xx} \upsilon + u\_x \upsilon\_{xz} - u\_z \upsilon\_{xx}, \\ -u\_{xt} + u\_{yz} &= -u\_x u\_{xy} + u\_y u\_{xx} + u\_{xz} \upsilon + u\_{xx} \chi\_{x}, \\ -\upsilon\_{xt} + \upsilon\_{yz} &= -u\_x \upsilon\_{xy} + u\_y \upsilon\_{xx} + u\_x \upsilon\_x^2 + \upsilon\_{xx} \upsilon + 2 \upsilon\_{x} \upsilon\_z. \end{aligned}$$

Under the constraint *v* = 0 one obtains the modified Michalev–Pavlov type integrable system (532).

Here, we can also observe that the seed element (574) can also be presented in the compact form:

$$
\tilde{a} \ltimes \tilde{l} := \left( \frac{\partial \tilde{\eta}\_1}{\partial \mathbf{x}} \frac{\partial}{\partial \mathbf{x}} + \frac{\partial \tilde{\eta}\_0}{\partial \lambda} \frac{\partial}{\partial \lambda} \right) \ltimes d\tilde{\rho}, \tag{576}
$$

$$
\vec{\eta}\_0 = \lambda \upsilon\_{\mathbf{x}}, \vec{\eta}\_1 = \mu \, -\lambda \, \mathbf{x}, \,\, \vec{\rho} = w + \eta\_{\mathbf{x}} \lambda\_{\mathbf{x}}.
$$

being closely connected with the geometry of the related moduli space of flat connections, related to the coadjoint actions of the corresponding Casimir functionals. Its suitable generalization to multidimensional Mikhalev–Pavlov type equations can be chosen as

$$\vec{a} \times \vec{I} := \left( \langle \nabla\_x \vec{\eta} | \nabla\_x \rangle + \nabla\_\lambda \vec{\eta}\_0 \nabla\_\lambda \right) \ltimes d\vec{\rho} \tag{577}$$

for some elements *η*˜, *η*˜0, *ρ*˜ ∈ Ω<sup>0</sup>(T*n*) ⊗ C, *n* ∈ N. The analysis of corresponding systems of integrable multidimensional Mikhalev–Pavlov type equations is planned to be presented in a separate study.
