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9 pages, 250 KiB

Open AccessArticle

On the Damped Pinney Equation from Noether Symmetry Principles

by Fernando Haas

Symmetry 2024, 16(10), 1310; https://doi.org/10.3390/sym16101310 - 4 Oct 2024

Viewed by 398

There are several versions of the damped form of the celebrated Pinney equation, which is the natural partner of the undamped time-dependent harmonic oscillator. In this work, these dissipative versions of the Pinney equation are briefly reviewed. We show that Noether’s theorem for [...] Read more.

There are several versions of the damped form of the celebrated Pinney equation, which is the natural partner of the undamped time-dependent harmonic oscillator. In this work, these dissipative versions of the Pinney equation are briefly reviewed. We show that Noether’s theorem for the usual time-dependent harmonic oscillator as a guiding principle for derivation of the Pinney equation also works in the damped case, selecting a Noether symmetry-based damped Pinney equation. The results are extended to general nonlinear damped Ermakov systems. A certain time-rescaling always allows to remove the damping from the final equations. Full article

(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)

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15 pages, 269 KiB

Open AccessArticle

Quantum Stability of Hamiltonian Evolution on a Finsler Manifold

by Gil Elgressy and Lawrence Horwitz

Symmetry 2024, 16(8), 1077; https://doi.org/10.3390/sym16081077 - 20 Aug 2024

Viewed by 648

Abstract

This paper is a study of a generalization of the quantum Riemannian Hamiltonian evolution, previously analyzed by us, in the geometrization of quantum mechanical evolution in a Finsler geometry. We find results with dynamical equations governing the evolution of the trajectories defined by [...] Read more.

This paper is a study of a generalization of the quantum Riemannian Hamiltonian evolution, previously analyzed by us, in the geometrization of quantum mechanical evolution in a Finsler geometry. We find results with dynamical equations governing the evolution of the trajectories defined by the expectation values of the position. The analysis appears to provide an underlying geometry described by a geodesic equation, with a connection form with a second term which is an essentially quantum effect. These dynamical equations provide a new geometric approach to the quantum evolution where we suggest a definition for “local instability” in the quantum theory. Full article

(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)

18 pages, 314 KiB

Open AccessArticle

Geometric Linearization for Constraint Hamiltonian Systems

by Andronikos Paliathanasis

Symmetry 2024, 16(8), 988; https://doi.org/10.3390/sym16080988 - 4 Aug 2024

Viewed by 875

Abstract

This study investigates the geometric linearization of constraint Hamiltonian systems using the Jacobi metric and the Eisenhart lift. We establish a connection between linearization and maximally symmetric spacetimes, focusing on the Noether symmetries admitted by the constraint Hamiltonian systems. Specifically, for systems derived [...] Read more.

This study investigates the geometric linearization of constraint Hamiltonian systems using the Jacobi metric and the Eisenhart lift. We establish a connection between linearization and maximally symmetric spacetimes, focusing on the Noether symmetries admitted by the constraint Hamiltonian systems. Specifically, for systems derived from the singular Lagrangian

L (N, q^{k}, {\dot{q}}^{k}) = \frac{1}{2 N} g_{i j} {\dot{q}}^{i} {\dot{q}}^{j} - N V (q^{k}),

where N and

q^{i}

are dependent variables and

dim g_{i j} = n

, the existence of

\frac{n (n + 1)}{2}

Noether symmetries is shown to be equivalent to the linearization of the equations of motion. The application of these results is demonstrated through various examples of special interest. This approach opens new directions in the study of differential equation linearization. Full article

(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)

16 pages, 1199 KiB

Open AccessArticle

A Reappraisal of Lagrangians with Non-Quadratic Velocity Dependence and Branched Hamiltonians

by Bijan Bagchi, Aritra Ghosh and Miloslav Znojil

Symmetry 2024, 16(7), 860; https://doi.org/10.3390/sym16070860 - 7 Jul 2024

Viewed by 1170

Abstract

Time and again, non-conventional forms of Lagrangians with non-quadratic velocity dependence have received attention in the literature. For one thing, such Lagrangians have deep connections with several aspects of nonlinear dynamics including specifically the types of the Liénard class; for another, very often, [...] Read more.

Time and again, non-conventional forms of Lagrangians with non-quadratic velocity dependence have received attention in the literature. For one thing, such Lagrangians have deep connections with several aspects of nonlinear dynamics including specifically the types of the Liénard class; for another, very often, the problem of their quantization opens up multiple branches of the corresponding Hamiltonians, ending up with the presence of singularities in the associated eigenfunctions. In this article, we furnish a brief review of the classical theory of such Lagrangians and the associated branched Hamiltonians, starting with the example of Liénard-type systems. We then take up other cases where the Lagrangians depend on velocity with powers greater than two while still having a tractable mathematical structure, while also describing the associated branched Hamiltonians for such systems. For various examples, we emphasize the emergence of the notion of momentum-dependent mass in the theory of branched Hamiltonians. Full article

(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)

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16 pages, 1031 KiB

Open AccessArticle

Symmetry Breaking and Dynamic Transition in the Negative Mass Term Klein–Gordon Equations

by Ferenc Márkus and Katalin Gambár

Symmetry 2024, 16(2), 144; https://doi.org/10.3390/sym16020144 - 26 Jan 2024

Cited by 2 | Viewed by 956

Abstract

Through the discussion of three physical processes, we show that the Klein–Gordon equations with a negative mass term describe special dynamics. In the case of two classical disciplines—mechanics and thermodynamics—the Lagrangian-based mathematical description is the same, even though the nature of the investigated [...] Read more.

Through the discussion of three physical processes, we show that the Klein–Gordon equations with a negative mass term describe special dynamics. In the case of two classical disciplines—mechanics and thermodynamics—the Lagrangian-based mathematical description is the same, even though the nature of the investigated processes seems completely different. The unique feature of this type of equation is that it contains wave propagation and dissipative behavior in one framework. The dissipative behavior appears through a repulsive potential. The transition between the two types of dynamics can be specified precisely, and its physical meaning is clear. The success of the two descriptions inspires extension to the case of electrodynamics. We reverse the suggestion here. We create a Klein–Gordon equation with a negative mass term, but first, we modify Maxwell’s equations. The repulsive interaction that appears here results in a charge spike. However, the Coulomb interaction limits this. The charge separation is also associated with the high-speed movement of the charged particle localized in a small space domain. As a result, we arrive at a picture of a fast vibrating phenomenon with an electromagnetism-related Klein–Gordon equation with a negative mass term. The calculated maximal frequency value

ω = 1.74 \times 10^{21}

1/s. Full article

(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)

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16 pages, 672 KiB

Open AccessFeature PaperArticle

Detecting Phase Transitions through Non-Equilibrium Work Fluctuations

by Matteo Colangeli, Antonio Di Francesco and Lamberto Rondoni

Symmetry 2024, 16(1), 125; https://doi.org/10.3390/sym16010125 - 20 Jan 2024

Viewed by 1174

Abstract

We show how averages of exponential functions of path-dependent quantities, such as those of Work Fluctuation Theorems, detect phase transitions in deterministic and stochastic systems. State space truncation—the restriction of the observations to a subset of state space with prescribed probability—is introduced to [...] Read more.

We show how averages of exponential functions of path-dependent quantities, such as those of Work Fluctuation Theorems, detect phase transitions in deterministic and stochastic systems. State space truncation—the restriction of the observations to a subset of state space with prescribed probability—is introduced to obtain that result. Two stochastic processes undergoing first-order phase transitions are analyzed both analytically and numerically: a variant of the Ehrenfest urn model and the 2D Ising model subject to a magnetic field. In the presence of phase transitions, we prove that even minimal state space truncation makes averages of exponentials of path-dependent variables sensibly deviate from full state space values. Specifically, in the case of discontinuous phase transitions, this approach is strikingly effective in locating the transition value of the control parameter. As this approach works even with variables different from those of fluctuation theorems, it provides a new recipe to identify order parameters in the study of non-equilibrium phase transitions, profiting from the often incomplete statistics that are available. Full article

(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)

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12 pages, 710 KiB

Open AccessArticle

Adiabatic Manipulation of a System Interacting with a Spin Bath

by Benedetto Militello and Anna Napoli

Symmetry 2023, 15(11), 2028; https://doi.org/10.3390/sym15112028 - 8 Nov 2023

Cited by 2 | Viewed by 1114

Abstract

The Stimulated Raman Adiabatic Passage, a very efficient technique for manipulating a quantum system based on the adiabatic theorem, is analyzed in the case where the manipulated physical system is interacting with a spin bath. The exploitation of the rotating wave approximation allows [...] Read more.

The Stimulated Raman Adiabatic Passage, a very efficient technique for manipulating a quantum system based on the adiabatic theorem, is analyzed in the case where the manipulated physical system is interacting with a spin bath. The exploitation of the rotating wave approximation allows for the identification of a constant of motion, which simplifies both the analytical and the numerical treatment, which allows for evaluating the total unitary evolution of the system and bath. The efficiency of the population transfer process is investigated in several regimes, including the weak and strong coupling with the environment and the off-resonance. The formation of appropriate Zeno subspaces explains the lowering of the efficiency in the strong damping regime. Full article

(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)

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17 pages, 319 KiB

Open AccessFeature PaperArticle

Complex Quantum Hydrodynamics in Momentum Space with Broken Time-Reversal Symmetry

by Dieter Schuch and Moise Bonilla-Licea

Symmetry 2023, 15(7), 1347; https://doi.org/10.3390/sym15071347 - 1 Jul 2023

Viewed by 1107

Abstract

Shortly after Schrödinger’s wave mechanics in terms of complex wave functions was published, Madelung formulated this theory in terms of two real hydrodynamic-like equations. This version is also the formal basis of Bohmian mechanics, albeit with a different ontological interpretation. A point of [...] Read more.

Shortly after Schrödinger’s wave mechanics in terms of complex wave functions was published, Madelung formulated this theory in terms of two real hydrodynamic-like equations. This version is also the formal basis of Bohmian mechanics, albeit with a different ontological interpretation. A point of criticism raised by Pauli against Bohmian mechanics is its missing symmetry between position and momentum that is present in classical phase space as well as in the quantum mechanical position and momentum representations. Both Madelung’s quantum hydrodynamics formulation and Bohmian mechanics are usually expressed only in position space. Recently, with the use of complex quantities, we were able to provide a hydrodynamic formulation also in momentum space. In this paper, we extend this formalism to include dissipative systems with broken time-reversal symmetry. In classical Hamiltonian mechanics and conventional quantum mechanics, closed systems with reversible time-evolution are usually considered. Extending the discussion to include open systems with dissipation, another form of symmetry is broken, that under time-reversal. There are different ways of describing such systems; for instance, Langevin and Fokker–Planck-type equations are commonly used in classical physics. We now investigate how these aspects can be incorporated into our complex hydrodynamic description and what modifications occur in the corresponding equations, not only in position, but particularly in momentum space. Full article

(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)

19 pages, 857 KiB

Open AccessFeature PaperArticle

Finite Reservoirs Corrections to Hamiltonian Systems Statistics and Time Symmetry Breaking

by Matteo Colangeli, Antonio Di Francesco and Lamberto Rondoni

Symmetry 2023, 15(6), 1268; https://doi.org/10.3390/sym15061268 - 15 Jun 2023

Cited by 1 | Viewed by 1207

Abstract

We consider several Hamiltonian systems perturbed by external agents that preserve their Hamiltonian structure. We investigate the corrections to the canonical statistics resulting from coupling such systems with possibly large but finite reservoirs and from the onset of processes breaking the time-reversal symmetry. [...] Read more.

We consider several Hamiltonian systems perturbed by external agents that preserve their Hamiltonian structure. We investigate the corrections to the canonical statistics resulting from coupling such systems with possibly large but finite reservoirs and from the onset of processes breaking the time-reversal symmetry. We analyze exactly solvable oscillator systems and perform simulations of relatively more complex ones. This indicates that the standard statistical mechanical formalism needs to be adjusted in the ever more investigated nano-scale science and technology. In particular, the hypothesis that heat reservoirs be considered infinite and be described by the classical ensembles is found to be critical when exponential quantities are considered since the large size limit may not coincide with the infinite size canonical result. Furthermore, process-dependent emergent irreversibility affects ensemble averages, effectively frustrating, on a statistical level, the time reversal invariance of Hamiltonian dynamics that are used to obtain numerous results. Full article

(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)

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36 pages, 506 KiB

Open AccessFeature PaperArticle

On Rational Solutions of Dressing Chains of Even Periodicity

by Henrik Aratyn, José Francisco Gomes, Gabriel Vieira Lobo and Abraham Hirsz Zimerman

Symmetry 2023, 15(1), 249; https://doi.org/10.3390/sym15010249 - 16 Jan 2023

Cited by 2 | Viewed by 1485

Abstract

We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalent Painlevé equations invariant under

A_{N - 1}^{(1)}

symmetry. This formalism identifies rational solutions (as well [...] Read more.

We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalent Painlevé equations invariant under

A_{N - 1}^{(1)}

symmetry. This formalism identifies rational solutions (as well as special function solutions) with points on orbits of fundamental shift operators of

A_{N - 1}^{(1)}

affine Weyl groups acting on seed configurations defined as first-order polynomial solutions of the underlying dressing chains. This approach clarifies the structure of rational solutions and establishes an explicit and systematic method towards their construction. For the special case of the

N = 4

dressing chain equations, the method yields all the known rational (and special function) solutions of the Painlevé V equation. The formalism naturally extends to

N = 6

and beyond as shown in the paper. Full article

(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)

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