On the Carathéodory Form in Higher-Order Variational Field Theory

Abstract

The Carathéodory form of the calculus of variations belongs to the class of Lepage equivalents of first-order Lagrangians in field theory. Here, this equivalent is generalized for second- and higher-order Lagrangians by means of intrinsic geometric operations applied to the well-known Poincaré–Cartan form and principal component of Lepage forms, respectively. For second-order theory, our definition coincides with the previous result obtained by Crampin and Saunders in a different way. The Carathéodory equivalent of the Hilbert Lagrangian in general relativity is discussed.

1. Introduction

In this note, we describe a generalization of the Carathéodory form [1] (cf. [2,3,4,5]) of the calculus of variations in second-order and, for specific Lagrangians, in higher-order field theory.

The theory of first-order multiple-integral variational functionals for unknown sections of fibered spaces over n-dimensional base manifolds can be described in a coordinate-independent way by means of the Poincaré–Cartan form [6] or its extensions, namely the Carathéodory form and the fundamental Lepage equivalent (also known as the Krupka–Betounes form; see [7,8]). For a comprehensive exposition of the theory as well as original references, see the monographs in [9,10].

The above-mentioned differential forms represent examples of Lepage forms—a far-going generalization of a 1-form, introduced by Cartan in the 1920s within the framework of the calculus of variations of simple integrals and classical mechanics. The generalization, given by Krupka [11,12], is motivated by the work of Lepage in the 1940s. We can briefly say that these objects define the same variational functional as it is prescribed by a given Lagrangian and, moreover, variational properties (as variations, extremals, or Noether’s type invariance) of the corresponding functional are globally characterized in terms of geometric operations (such as the exterior derivative and the Lie derivative) acting on integrands—the Lepage equivalents of a Lagrangian. In particular, the extremals are determined by means of the exterior derivative of the Lepage form. The meaning of Lepage forms for the calculus of variations and their basic properties, including examples of Lepage forms in geometric mechanics and field theory, have recently been summarized and further studied [13,14,15,16,17,18,19].

Our approach towards generalizing the Carathéodory form is based on a geometric relationship between the Poincaré–Cartan and Carathéodory forms, and analysis of the corresponding global properties. In [3], Crampin and Saunders obtained the Carathéodory form for second-order Lagrangians as a certain projection onto a sphere bundle. Here, we confirm this result by means of a different, straightforward method which furthermore allows higher-order generalization. It is a standard fact in the global variational field theory that the local expressions,

$${\Theta }_{\lambda }=\mathcal{L}{\omega }_0+\sum _{k=0}^{r-1}\left(\sum _{l=0}^{r-1-k}{(-1)}^l{d}_{p_1}\dots d_{p_l}\frac{\partial \mathcal{L}}{\partial y_{j_1\dots j_k{p}_1\dots p_{l}i}^{\sigma }}\right){\omega }_{j_1\dots j_k}^{\sigma }\wedge {\omega }_i,$$

(1)

which generalize the Poincaré–Cartan form of the calculus of variations, define (1), in general, globally for Lagrangians $\lambda =\mathcal{L}{\omega }_0$ of order $r=1$ and $r=2$ only; see the works of Krupka [12] (${\Theta }_{\lambda }$ is known as the principal component of a Lepage equivalent of Lagrangian $\lambda$) and Horák and Kolář [20] (for higher-order Poincaré–Cartan morphisms). We show that, if ${\Theta }_{\lambda }$ is a globally defined differential form for a class of Lagrangians of order $r\ge 3$, then a higher-order Carathéodory equivalent for Lagrangians belonging to this class naturally arises by means of geometric operations acting on ${\Theta }_{\lambda }$. To this purpose, for order $r=3$, we analyze conditions, which describe the obstructions for globally defined principal components of Lepage equivalents (1) (or, higher–order Poincaré–Cartan forms).

A concrete application of our result in second-order field theory includes the Carathéodory equivalent of the Hilbert Lagrangian in general relativity.

Basic underlying structures, well adapted to this paper, can be found in the work by Volná and Urban [18]. If $(U,\phi )$, $\phi =\left(x^i\right)$, is a chart on a smooth manifold X, we set

$${\omega }_0=d{x}^1\wedge \dots \wedge d{x}^n,{\omega }_j=i_{\partial /\partial x^j}{\omega }_0=\frac{1}{(n-1)!}{\epsilon }_{j{i}_2\dots i_n}d{x}^{i_2}\wedge \dots \wedge d{x}^{i_n},$$

where ${\epsilon }_{i_1{i}_2\dots i_n}$ is the Levi–Civita permutation symbol. If $\pi :Y\to X$ is a fibered manifold and W an open subset of Y, then there exists a unique morphism $h:{\Omega }^{r}W\to {\Omega }^{r+1}W$ of exterior algebras of differential forms such that for any fibered chart $(V,\psi )$, $\psi =(x^i,y^{\sigma })$, where $V\subset W$, and any differentiable function $f:W^r\to \mathbb{R}$, where $W^r={\left({\pi }^{r,0}\right)}^{-1}\left(W\right)$ and ${\pi }^{r,s}:J^{r}Y\to J^{s}Y$ the jet bundle projection,

$$hf=f\circ {\pi }^{r+1,r},hdf=\left(d_{i}f\right)d{x}^i,$$

where

$$d_i=\frac{\partial }{\partial x^i}+\sum _{j_1\le \dots \le j_k}\frac{\partial }{\partial y_{j_1\dots j_k}^{\sigma }}{y}_{j_1\dots j_{k}i}^{\sigma }$$

(2)

is the ith formal derivative operator associated with $(V,\psi )$. A differential form q-form $\rho \in {\Omega }_q^{r}W$ satisfying $h\rho =0$ is called contact, and $\rho$ is generated by contact 1-forms

$${\omega }_{j_1\dots j_k}^{\sigma }=d{y}_{j_1\dots j_k}^{\sigma }-y_{j_1\dots j_{k}s}^{\sigma }d{x}^s,0\le k\le r-1.$$

Throughout, we use the standard geometric concepts: the exterior derivative d, the contraction $i_{\Xi }\rho$ and the Lie derivative ${\partial }_{\Xi }\rho$ of a differential form $\rho$ with respect to a vector field $\Xi$, and the pull-back operation ∗ acting on differential forms.

2. Lepage Equivalents in First- and Second-Order Field Theory

By a Lagrangian $\lambda$ for a fibered manifold $\pi :Y\to X$ of order r we mean an element of the submodule ${\Omega }_{n,X}^{r}W$ of ${\pi }^r$-horizontal n-forms in the module of n-forms ${\Omega }_n^{r}W$, defined on an open subset $W^r$ of the rth jet prolongation $J^{r}Y$. In a fibered chart $(V,\psi )$, $\psi =(x^i,y^{\sigma })$, where $V\subset W$, the Lagrangian $\lambda \in {\Omega }_{n,X}^{r}W$ has an expression

$$\lambda =\mathcal{L}{\omega }_0,$$

(3)

where ${\omega }_0=d{x}^1\wedge d{x}^2\wedge \dots \wedge d{x}^n$ is the (local) volume element, and $\mathcal{L}:V^r\to \mathbb{R}$ is the Lagrange function associated to $\lambda$ and $(V,\psi )$.

An n-form $\rho \in {\Omega }_n^{s}W$ is called a Lepage equivalent of $\lambda \in {\Omega }_{n,X}^{r}W$, if the following two conditions are satisfied:

(i)

${\left({\pi }^{q,s+1}\right)}^{*}h\rho ={\left({\pi }^{q,r}\right)}^{*}\lambda$ (i.e., $\rho$ is equivalent with $\lambda$); and

(ii)

$h{i}_{\xi }d\rho =0$ for arbitrary ${\pi }^{s,0}$-vertical vector field $\xi$ on $W^s$ (i.e., $\rho$ is a Lepage form).

The following theorem describes the structure of the Lepage equivalent of a Lagrangian (see [9,12]).

Theorem 1.

Let $\lambda \in {\Omega }_{n,X}^{r}W$ be a Lagrangian of order r for Y, locally expressed by (3) with respect to a fibered chart $(V,\psi )$, $\psi =(x^i,y^{\sigma })$. An n-form $\rho \in {\Omega }_n^{s}W$ is a Lepage equivalent of λ if and only if

$${\left({\pi }^{s+1,s}\right)}^{*}\rho ={\Theta }_{\lambda }+d\mu +\eta ,$$

(4)

where the n-form ${\Theta }_{\lambda }$ is defined on $V^{2r-1}$ by (1), μ is a contact $(n-1)$-form, and an n-form η has the order of contactness $\ge 2$.

${\Theta }_{\lambda }$ is called the principal component of the Lepage form $\rho$ with respect to fibered chart $(V,\psi )$. In general, decomposition (4) is not uniquely determined with respect to contact forms $\mu$, $\eta$, and the principal component ${\Theta }_{\lambda }$ need not define a global form on $W^{2r-1}$. Nevertheless, the Lepage equivalent $\rho$ satisfying (4) is globally defined on $W^s$; moreover, $E_{\lambda }=p_{1}d\rho$ is a globally defined $(n+1)$-form on $W^{2r}$, called the Euler–Lagrange form associated to $\lambda$.

We recall the known examples of Lepage equivalents of first- and second-order Lagrangians, determined by means of additional requirements.

Lemma 1. (Principal Lepage form)

(a)

For every Lagrangian λ of order $r=1$, there exists a unique Lepage equivalent ${\Theta }_{\lambda }$ of λ on $W^1$, which is ${\pi }^{1,0}$-horizontal and has the order of contactness $\le 1$. In a fibered chart $(V,\psi )$, $\Theta {}_{\lambda }$ has an expression

$${\Theta }_{\lambda }=\mathcal{L}{\omega }_0+\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}{\omega }^{\sigma }\wedge {\omega }_j.$$

(5)

(b)

For every Lagrangian λ of order $r=2$, there exists a unique Lepage equivalent ${\Theta }_{\lambda }$ of λ on $W^3$, which is ${\pi }^{3,1}$-horizontal and has the order of contactness $\le 1$. In a fibered chart $(V,\psi )$, $\Theta {}_{\lambda }$ has an expression

$${\Theta }_{\lambda }=\mathcal{L}{\omega }_0+\left(\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{pj}^{\sigma }}\right){\omega }^{\sigma }\wedge {\omega }_j+\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}{\omega }_i^{\sigma }\wedge {\omega }_j.$$

(6)

For $r=1$ and $r=2$, the principal component ${\Theta }_{\lambda }$ (1) is a globally defined Lepage equivalent of $\lambda$. We point out that for $r\ge 3$ this is not true (see [12,20]). (5) is the well-known Poincaré–Cartan form (cf. García [6]), and it is generalized for second-order Lagrangians by the globally defined principal Lepage equivalent (6) on $W^3\subset J^{3}Y$.

Lemma 2 (Fundamental Lepage form).

Let $\lambda \in {\Omega }_{n,X}^{1}W$ be a Lagrangian of order 1 for Y, locally expressed by (3). There exists a unique Lepage equivalent $Z_{\lambda }\in {\Omega }_n^{1}W$ of λ, which satisfies $Z_{h\rho }={\left({\pi }^{1,0}\right)}^{*}\rho$ for any n-form $\rho \in {\Omega }_n^{0}W$ on W such that $h\rho =\lambda$. With respect to a fibered chart $(V,\psi )$, $Z_{\lambda }$ has an expression

$$\begin{array}{cc}\hfill Z_{\lambda }& =\mathcal{L}{\omega }_0+\sum _{k=1}^n\frac{1}{(n-k)!}\frac{1}{{(k!)}^2}\frac{{\partial }^k\mathcal{L}}{\partial y_{j_1}^{{\sigma }_1}\dots \partial y_{j_k}^{{\sigma }_k}}{\epsilon }_{j_1\dots j_k{i}_{k+1}\dots i_n}\hfill \\ \hfill & ·{\omega }^{{\sigma }_1}\wedge \dots \wedge {\omega }^{{\sigma }_k}\wedge d{x}^{i_{k+1}}\wedge \dots \wedge d{x}^{i_n}.\hfill \end{array}$$

(7)

$Z_{\lambda }$ (7) is known as the fundamental Lepage form [7,8]), and it is characterized by the equivalence: $Z_{\lambda }$ is closed if and only if $\lambda$ is trivial (i.e., the Euler–Lagrange expressions associated with $\lambda$ vanish identically). Recently, the form (7) was generalized in [21], and studied for variational problems for submanifolds in [5], as well as applied for studying symmetries and conservation laws in [22].

Lemma 3 (Carathéodory form).

Let $\lambda \in {\Omega }_{n,X}^{1}W$ be a non-vanishing Lagrangian of order 1 for Y (3). Then, a differential n-form ${\Lambda }_{\lambda }\in {\Omega }_n^{1}W$, locally expressed as

$$\begin{array}{cc}\hfill {\Lambda }_{\lambda }& =\frac{1}{{\mathcal{L}}^{n-1}}\underset{j=1}{\overset{n}{\bigwedge }}\left(\mathcal{L}d{x}^j+\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}{\omega }^{\sigma }\right)\hfill \\ \hfill & =\frac{1}{{\mathcal{L}}^{n-1}}\left(\mathcal{L}d{x}^1+\frac{\partial \mathcal{L}}{\partial y_1^{{\sigma }_1}}{\omega }^{{\sigma }_1}\right)\wedge \dots \wedge \left(\mathcal{L}d{x}^n+\frac{\partial \mathcal{L}}{\partial y_n^{{\sigma }_n}}{\omega }^{{\sigma }_n}\right),\hfill \end{array}$$

(8)

is a Lepage equivalent of λ.

${\Lambda }_{\lambda }$ (7) is the well-known Carathéodory form (cf. [1]), associated to Lagrangian $\lambda \in {\Omega }_{n,X}^{1}W$, which is nowhere zero. ${\Lambda }_{\lambda }$ is uniquely characterized by the following properties: ${\Lambda }_{\lambda }$ is: (i) a Lepage equivalent of $\lambda$; (ii) decomposable; and (iii) ${\pi }^{1,0}$-horizontal (i.e., semi-basic with respect to projection ${\pi }^{1,0}$).

3. The Carathéodory Form: Second-Order Generalization

Let $\lambda \in {\Omega }_{n,X}^{1}W$ be a non-vanishing, first-order Lagrangian on $W^1\subset J^{1}Y$. In the next lemma, we describe a new observation, showing that the Carathéodory form ${\Lambda }_{\lambda }$ (8) arises from the Poincaré–Cartan form ${\Theta }_{\lambda }$ (5) by means of contraction operations on differential forms with respect to the formal derivative vector fields $d_i$ (2).

Lemma 4.

The Carathéodory form ${\Lambda }_{\lambda }$ (8) and the Poincaré–Cartan form ${\Theta }_{\lambda }$ (5) satisfy

$$\begin{array}{cc}\hfill {\Lambda }_{\lambda }& =\frac{1}{{\mathcal{L}}^{n-1}}\underset{j=1}{\overset{n}{\bigwedge }}{(-1)}^{n-j}{i}_{d_n}\dots i_{d_{j+1}}{i}_{d_{j-1}}\dots i_{d_1}{\Theta }_{\lambda }.\hfill \end{array}$$

Proof. 

From the decomposable structure of ${\Lambda }_{\lambda }$, we see that what is needed to show is the formula

$$i_{d_n}\dots i_{d_{j+1}}{i}_{d_{j-1}}\dots i_{d_1}{\Theta }_{\lambda }={(-1)}^{n-j}\left(\mathcal{L}d{x}^j+\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}{\omega }^{\sigma }\right)$$

for every j, $1\le j\le n$. Since $d{x}^k\wedge {\omega }_j={\delta }_j^k{\omega }_0$, the Poincaré–Cartan form is expressible as

$$\begin{array}{cc}\hfill {\Theta }_{\lambda }& =\mathcal{L}{\omega }_0+\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}{\omega }^{\sigma }\wedge {\omega }_j=\mathcal{L}{\omega }_0+\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}d{y}^{\sigma }\wedge {\omega }_j-\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}{y}_k^{\sigma }d{x}^k\wedge {\omega }_j\hfill \\ \hfill & =\left(\mathcal{L}-\frac{\partial \mathcal{L}}{\partial y_1^{\sigma }}{y}_1^{\sigma }-\frac{\partial \mathcal{L}}{\partial y_2^{\sigma }}{y}_2^{\sigma }-\dots -\frac{\partial \mathcal{L}}{\partial y_n^{\sigma }}{y}_n^{\sigma }\right){\omega }_0+\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}d{y}^{\sigma }\wedge {\omega }_j.\hfill \end{array}$$

Applying the contraction operations to ${\Theta }_{\lambda }$, we obtain by means of a straightforward computation for every j,

$$\begin{array}{cc}\hfill & i_{d_{j-1}}\dots i_{d_1}{\Theta }_{\lambda }=\left(\mathcal{L}-\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}{y}_j^{\sigma }-\dots -\frac{\partial \mathcal{L}}{\partial y_n^{\sigma }}{y}_n^{\sigma }\right)d{x}^j\wedge \dots \wedge d{x}^n\hfill \\ \hfill & +{(-1)}^{j-1}\sum _{k=j}^n\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}d{y}^{\sigma }\wedge i_{d_{j-1}}\dots i_{d_1}{\omega }_k\hfill \\ \hfill & +\sum _{l=1}^{j-1}{(-1)}^{l-1}{y}_l^{\sigma }{i}_{d_{j-1}}\dots i_{d_{l+1}}{i}_{d_{l-1}}\dots i_{d_1}\left(\sum _{k=j}^n\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}{\omega }_k\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & i_{d_{j+1}}{i}_{d_{j-1}}\dots i_{d_1}{\Theta }_{\lambda }\hfill \\ \hfill & =-\left(\mathcal{L}-\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}{y}_j^{\sigma }-\sum _{k=j+2}^n\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}{y}_k^{\sigma }\right)d{x}^j\wedge d{x}^{j+2}\wedge \dots \wedge d{x}^n\hfill \\ \hfill & +{(-1)}^j\sum _{k=j}^n\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}d{y}^{\sigma }\wedge i_{d_{j+1}}{i}_{d_{j-1}}\dots i_{d_1}{\omega }_k\hfill \\ \hfill & +\sum _{l=1}^{j-1}{(-1)}^{l-1}{y}_l^{\sigma }{i}_{d_{j+1}}{i}_{d_{j-1}}\dots i_{d_{l+1}}{i}_{d_{l-1}}\dots i_{d_1}\left(\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}{\omega }_j+\sum _{k=j+2}^n\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}{\omega }_k\right)\hfill \\ \hfill & +{(-1)}^{j-1}{y}_{j+1}^{\sigma }{i}_{d_{j-1}}\dots i_{d_1}\left(\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}{\omega }_j+\sum _{k=j+2}^n\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}{\omega }_k\right).\hfill \end{array}$$

Following the inductive structure of the preceding expressions, we get after the next $n-j-1$ steps,

$$\begin{array}{cc}\hfill & i_{d_n}\dots i_{d_{j+1}}{i}_{d_{j-1}}\dots i_{d_1}{\Theta }_{\lambda }\hfill \\ \hfill & ={(-1)}^{n-j}\left(\mathcal{L}-\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}{y}_j^{\sigma }\right)d{x}^j+{(-1)}^{n-j}\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}d{y}^{\sigma }-{(-1)}^{n-j}\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}\sum _{k\ne j}{y}_k^{\sigma }d{x}^k\hfill \\ \hfill & ={(-1)}^{n-j}\left(\mathcal{L}d{x}^j+\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}{\omega }^{\sigma }\right),\hfill \end{array}$$

as required. □

An intrinsic nature of Lemma 4 indicates a possible extension of the Carathéodory form (8) for higher-order variational problems. We put

$${\Lambda }_{\lambda }=\frac{1}{{\mathcal{L}}^{n-1}}\underset{j=1}{\overset{n}{\bigwedge }}{(-1)}^{n-j}{i}_{d_n}\dots i_{d_{j+1}}{i}_{d_{j-1}}\dots i_{d_1}{\Theta }_{\lambda },$$

(9)

where ${\Theta }_{\lambda }$ in (9) denotes the principal Lepage equivalent (6) of a second-order Lagrangian $\lambda$, and verify that formula (9) defines a global form.

Theorem 2.

Let $\lambda \in {\Omega }_{n,X}^{2}W$ be a non-vanishing second-order Lagrangian on $W^2\subset J^{2}Y$. Then, ${\Lambda }_{\lambda }$ satisfies:

(a)

Formula (9) defines an n-form on $W^3\subset J^{3}Y$.

(b)

If $\lambda \in {\Omega }_{n,X}^{2}W$ has an expression (3) with respect to a fibered chart $(V,\psi )$, $\psi =(x^i,y^{\sigma })$, such that $V\subset W$, then ${\Lambda }_{\lambda }$ (9) is expressed by

$$\begin{array}{cc}\hfill {\Lambda }_{\lambda }& =\frac{1}{{\mathcal{L}}^{n-1}}\underset{j=1}{\overset{n}{\bigwedge }}\left(\mathcal{L}d{x}^j+\left(\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}-d_i\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}\right){\omega }^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}{\omega }_i^{\sigma }\right).\hfill \end{array}$$

(10)

(c)

${\Lambda }_{\lambda }\in {\Omega }_n^{3}W$ (9) associated to a second-order Lagrangian $\lambda \in {\Omega }_{n,X}^{2}W$ is a Lepage equivalent of λ, which is decomposable and ${\pi }^{3,1}$-horizontal.

Proof. 

• Suppose $(V,\psi )$, $\psi =(x^i,y^{\sigma })$, and $(\overline{V},\overline{\psi })$, $\overline{\psi }=({\overline{x}}^i,{\overline{y}}^{\sigma })$, are two overlapping fibered charts on W. For $\lambda \in {\Omega }_{n,X}^{2}W$, the corresponding chart expressions $\lambda =\mathcal{L}{\omega }_0$ and $\lambda =\overline{\mathcal{L}}{\overline{\omega }}_0$ satisfy

  $$\mathcal{L}=\left(\overline{\mathcal{L}}\circ {\overline{\psi }}^{-1}\circ \psi \right)det\frac{\partial {\overline{x}}^i}{\partial x^j}.$$

  (11)

  Since the push-forward vector field ${\overline{d}}_k$,

  $${\overline{d}}_k=\frac{\partial }{\partial {\overline{x}}^k}+{\overline{y}}_k^{\sigma }\frac{\partial }{\partial {\overline{y}}^{\sigma }}+{\overline{y}}_{kl}^{\sigma }\frac{\partial }{\partial {\overline{y}}_l^{\sigma }},$$

  of vector field $(\partial x^i/\partial {\overline{x}}^k)d_i$ with respect to the chart transformation ${\overline{\psi }}^{-1}\circ \psi$ satisfies

  $${({\overline{\psi }}^{-1}\circ \psi )}^{*}\left(i_{{\overline{d}}_k}{\Theta }_{\lambda }\right)=i_{\frac{\partial x^i}{\partial {\overline{x}}^k}{d}_i}\left({({\overline{\psi }}^{-1}\circ \psi )}^{*}{\Theta }_{\lambda }\right)=\frac{\partial x^i}{\partial {\overline{x}}^k}{i}_{d_i}\left({({\overline{\psi }}^{-1}\circ \psi )}^{*}{\Theta }_{\lambda }\right),$$

  and ${\Theta }_{\lambda }$ (6) is globally defined, we get

  $$\begin{array}{cc}\hfill & {({\overline{\psi }}^{-1}\circ \psi )}^{*}\frac{1}{{\overline{\mathcal{L}}}^{n-1}}\underset{j=1}{\overset{n}{\bigwedge }}{(-1)}^{n-j}{i}_{{\overline{d}}_n}\dots i_{{\overline{d}}_{j+1}}{i}_{{\overline{d}}_{j-1}}\dots i_{{\overline{d}}_1}{\Theta }_{\lambda }\hfill \\ \hfill & =\frac{1}{{\left(\overline{\mathcal{L}}\circ {\overline{\psi }}^{-1}\circ \psi \right)}^{n-1}}\underset{j=1}{\overset{n}{\bigwedge }}{(-1)}^{n-j}\frac{\partial x^{i_1}}{\partial {\overline{x}}^1}\dots \frac{\partial x^{i_{j-1}}}{\partial {\overline{x}}^{j-1}}\frac{\partial x^{i_{j+1}}}{\partial {\overline{x}}^{j+1}}\dots \frac{\partial x^{i_n}}{\partial {\overline{x}}^n}\hfill \\ \hfill & ·i_{d_{i_n}}\dots i_{d_{i_{j+1}}}{i}_{d_{i_{j-1}}}\dots i_{d_{i_1}}\left({({\overline{\psi }}^{-1}\circ \psi )}^{*}{\Theta }_{\lambda }\right)\hfill \\ \hfill & =\frac{1}{{\left(\overline{\mathcal{L}}\circ {\overline{\psi }}^{-1}\circ \psi \right)}^{n-1}}{\left(\frac{\partial x^{i_1}}{\partial {\overline{x}}^1}\dots \frac{\partial x^{i_n}}{\partial {\overline{x}}^n}{\epsilon }_{i_1\dots i_n}\right)}^n\hfill \\ \hfill & ·\underset{j=1}{\overset{n}{\bigwedge }}{(-1)}^{n-j}{i}_{d_n}\dots i_{d_{j+1}}{i}_{d_{j-1}}\dots i_{d_1}{({\overline{\psi }}^{-1}\circ \psi )}^{*}{\Theta }_{\lambda }\hfill \\ \hfill & =\frac{1}{{\left((\overline{\mathcal{L}}\circ {\overline{\psi }}^{-1}\circ \psi )det\frac{\partial {\overline{x}}^i}{\partial x^j}\right)}^{n-1}}\underset{j=1}{\overset{n}{\bigwedge }}{(-1)}^{n-j}{i}_{d_n}\dots i_{d_{j+1}}{i}_{d_{j-1}}\dots i_{d_1}{({\overline{\psi }}^{-1}\circ \psi )}^{*}{\Theta }_{\lambda }\hfill \\ \hfill & =\frac{1}{{\mathcal{L}}^{n-1}}\underset{j=1}{\overset{n}{\bigwedge }}{(-1)}^{n-j}{i}_{d_n}\dots i_{d_{j+1}}{i}_{d_{j-1}}\dots i_{d_1}{({\overline{\psi }}^{-1}\circ \psi )}^{*}{\Theta }_{\lambda }\hfill \\ \hfill & =\frac{1}{{\mathcal{L}}^{n-1}}\underset{j=1}{\overset{n}{\bigwedge }}{(-1)}^{n-j}{i}_{d_n}\dots i_{d_{j+1}}{i}_{d_{j-1}}\dots i_{d_1}{\Theta }_{\lambda },\hfill \end{array}$$

  as required.
• Analogously to the proof of Lemma 4, we find a chart expression of 1-form

  $$i_{d_n}\dots i_{d_{j+1}}{i}_{d_{j-1}}\dots i_{d_1}{\Theta }_{\lambda },$$

  where ${\Theta }_{\lambda }$ is the principal Lepage equivalent (6). Using $d{x}^k\wedge {\omega }_j={\delta }_j^k{\omega }_0$, we have

  $$\begin{array}{cc}\hfill {\Theta }_{\lambda }& =\left(\mathcal{L}-\left(\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{pj}^{\sigma }}\right)y_j^{\sigma }-\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}{y}_{ij}^{\sigma }\right){\omega }_0\hfill \\ \hfill & +\left(\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{pj}^{\sigma }}\right)d{y}^{\sigma }\wedge {\omega }_j+\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}d{y}_i^{\sigma }\wedge {\omega }_j.\hfill \end{array}$$

  Then,

  $$\begin{array}{cc}\hfill & i_{d_{j-1}}\dots i_{d_1}{\Theta }_{\lambda }\hfill \\ \hfill & =\left(\mathcal{L}-\left(\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{pk}^{\sigma }}\right)y_k^{\sigma }-\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}{y}_{ik}^{\sigma }\right)d{x}^j\wedge \dots \wedge d{x}^n\hfill \\ \hfill & +\sum _{l=1}^{j-1}{(-1)}^{l-1}\left(\left(\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{pk}^{\sigma }}\right)y_l^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}{y}_{il}^{\sigma }\right)i_{d_{j-1}}\dots i_{d_{l+1}}{i}_{d_{l-1}}\dots i_{d_1}{\omega }_k\hfill \\ \hfill & +{(-1)}^{j-1}\left(\left(\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{pk}^{\sigma }}\right)d{y}^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}d{y}_i^{\sigma }\right)\wedge \left(i_{d_{j-1}}\dots i_{d_1}{\omega }_k\right)\hfill \\ \hfill & =\left(\mathcal{L}-\sum _{k=j}^n\left(\left(\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{pk}^{\sigma }}\right)y_k^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}{y}_{ik}^{\sigma }\right)\right)d{x}^j\wedge \dots \wedge d{x}^n\hfill \\ \hfill & +\sum _{l=1}^{j-1}{(-1)}^{l-1}\sum _{k=j}^n\left(\left(\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{pk}^{\sigma }}\right)y_l^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}{y}_{il}^{\sigma }\right)i_{d_{j-1}}\dots i_{d_{l+1}}{i}_{d_{l-1}}\dots i_{d_1}{\omega }_k\hfill \\ \hfill & +{(-1)}^{j-1}\sum _{k=j}^n\left(\left(\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{pk}^{\sigma }}\right)d{y}^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}d{y}_i^{\sigma }\right)\wedge \left(i_{d_{j-1}}\dots i_{d_1}{\omega }_k\right),\hfill \end{array}$$

  and

  $$\begin{array}{cc}\hfill & i_{d_{j+1}}{i}_{d_{j-1}}\dots i_{d_1}{\Theta }_{\lambda }\hfill \\ \hfill & =\left(-\mathcal{L}+\sum _{\begin{array}{c}k=j\\ k\ne j+1\end{array}}^n\left(\left(\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{pk}^{\sigma }}\right)y_k^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}{y}_{ik}^{\sigma }\right)\right)d{x}^j\wedge \underset{l=j+2}{\overset{n}{\bigwedge }}d{x}^l\hfill \\ \hfill & +\sum _{l=1}^{j-1}{(-1)}^{l-1}\sum _{\begin{array}{c}k=j\\ k\ne j+1\end{array}}^n\left(\left(\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{pk}^{\sigma }}\right)y_l^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}{y}_{il}^{\sigma }\right)\hfill \\ \hfill & i_{d_{j+1}}{i}_{d_{j-1}}\dots i_{d_{l+1}}{i}_{d_{l-1}}\dots i_{d_1}{\omega }_k\hfill \\ \hfill & +{(-1)}^{j-1}\sum _{\begin{array}{c}k=j\\ k\ne j+1\end{array}}^n\left(\left(\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{pk}^{\sigma }}\right)y_{j+1}^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}{y}_{i,j+1}^{\sigma }\right)i_{d_{j-1}}\dots i_{d_1}{\omega }_k\hfill \\ \hfill & +{(-1)}^j\sum _{k=j}^n\left(\left(\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{pk}^{\sigma }}\right)d{y}^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}d{y}_i^{\sigma }\right)\wedge \left(i_{d_{j+1}}{i}_{d_{j-1}}\dots i_{d_1}{\omega }_k\right).\hfill \end{array}$$

  After the additional $n-j-1$ steps, we obtain

  $$\begin{array}{cc}\hfill & i_{d_n}\dots i_{d_{j+1}}{i}_{d_{j-1}}\dots i_{d_1}{\Theta }_{\lambda }\hfill \\ \hfill & ={(-1)}^{n-j}\left(\mathcal{L}d{x}^j+\left(\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{pj}^{\sigma }}\right){\omega }^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}{\omega }_i^{\sigma }\right).\hfill \end{array}$$
• From (10), it is evident that ${\Lambda }_{\lambda }$ (9) is decomposable, ${\pi }^{3,1}$-horizontal, and obeys $h{\Lambda }_{\lambda }=\lambda$. It is sufficient to verify that ${\Lambda }_{\lambda }$ is a Lepage form, that is $h{i}_{\xi }d{\Lambda }_{\lambda }=0$ for an arbitrary ${\pi }^{3,0}$-vertical vector field $\xi$ on $W^3\subset J^{3}Y$. This follows, however, by means of a straightforward computation using chart expression (10). Indeed, we have

  $$\begin{array}{cc}\hfill & d{\Lambda }_{\lambda }=(1-n)\frac{1}{{\mathcal{L}}^n}d\mathcal{L}\wedge \underset{j=1}{\overset{n}{\bigwedge }}\left(\mathcal{L}d{x}^j+\left(\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}-d_i\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}\right){\omega }^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}{\omega }_i^{\sigma }\right)\hfill \\ \hfill & +\frac{1}{{\mathcal{L}}^{n-1}}\sum _{k=1}^n{(-1)}^{k-1}d\left(\mathcal{L}d{x}^k+\left(\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}-d_i\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}\right){\omega }^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}{\omega }_i^{\sigma }\right)\hfill \\ \hfill & \wedge \underset{j\ne k}{\bigwedge }\left(\mathcal{L}d{x}^j+\left(\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}-d_i\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}\right){\omega }^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}{\omega }_i^{\sigma }\right),\hfill \end{array}$$

  and the contraction of $d{\Lambda }_{\lambda }$ with respect to a ${\pi }^{3,0}$-vertical vector field $\xi$ reads

  $$\begin{array}{cc}\hfill & i_{\xi }d{\Lambda }_{\lambda }=(1-n)\frac{1}{{\mathcal{L}}^n}{i}_{\xi }d\mathcal{L}\underset{j=1}{\overset{n}{\bigwedge }}\left(\mathcal{L}d{x}^j+\left(\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}-d_i\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}\right){\omega }^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}{\omega }_i^{\sigma }\right)\hfill \\ \hfill & -(1-n)\frac{1}{{\mathcal{L}}^n}d\mathcal{L}\wedge \sum _{l=1}^n{(-1)}^{l-1}\frac{\partial \mathcal{L}}{\partial y_{jl}^{\sigma }}{\xi }_j^{\sigma }\hfill \\ \hfill & \underset{j\ne l}{\bigwedge }\left(\mathcal{L}d{x}^j+\left(\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}-d_i\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}\right){\omega }^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}{\omega }_i^{\sigma }\right)\hfill \\ \hfill & +\frac{1}{{\mathcal{L}}^{n-1}}\sum _{k=1}^n{(-1)}^{k-1}\left(i_{\xi }d\mathcal{L}d{x}^k+i_{\xi }d\left(\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}-d_i\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}\right){\omega }^{\sigma }\right.\hfill \\ \hfill & \left.-\left(\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}-d_i\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}\right){\xi }_j^{\sigma }d{x}^j+i_{\xi }d\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}{\omega }_i^{\sigma }-{\xi }_i^{\sigma }d\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}-\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}{\xi }_{is}^{\sigma }d{x}^s\right)\hfill \\ \hfill & \wedge \underset{j\ne k}{\bigwedge }\left(\mathcal{L}d{x}^j+\left(\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}-d_i\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}\right){\omega }^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}{\omega }_i^{\sigma }\right)\hfill \\ \hfill & +\frac{1}{{\mathcal{L}}^{n-1}}\sum _{k=1}^n{(-1)}^{k-1}d\left(\mathcal{L}d{x}^k+\left(\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}-d_i\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}\right){\omega }^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}{\omega }_i^{\sigma }\right)\hfill \\ \hfill & \wedge \sum _{l<k}{(-1)}^{l-1}\frac{\partial \mathcal{L}}{\partial y_{il}^{\sigma }}{\xi }_i^{\sigma }\underset{j\ne k,l}{\bigwedge }\left(\mathcal{L}d{x}^j+\left(\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}-d_i\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}\right){\omega }^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}{\omega }_i^{\sigma }\right)\hfill \\ \hfill & +\frac{1}{{\mathcal{L}}^{n-1}}\sum _{k=1}^n{(-1)}^{k-1}d\left(\mathcal{L}d{x}^k+\left(\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}-d_i\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}\right){\omega }^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}{\omega }_i^{\sigma }\right)\hfill \\ \hfill & \wedge \sum _{l>k}{(-1)}^l\frac{\partial \mathcal{L}}{\partial y_{il}^{\sigma }}{\xi }_i^{\sigma }\underset{j\ne k,l}{\bigwedge }\left(\mathcal{L}d{x}^j+\left(\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}-d_i\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}\right){\omega }^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}{\omega }_i^{\sigma }\right).\hfill \end{array}$$

  Hence, the horizontal part of $i_{\xi }d{\Lambda }_{\lambda }$ satisfies

  $$\begin{array}{cc}\hfill h{i}_{\xi }d{\Lambda }_{\lambda }& =(1-n)i_{\xi }d\mathcal{L}{\omega }_0-(1-n)\frac{1}{\mathcal{L}}\sum _{l=1}^n\frac{\partial \mathcal{L}}{\partial y_{jl}^{\sigma }}{\xi }_j^{\sigma }{d}_l\mathcal{L}{\omega }_0+n{i}_{\xi }d\mathcal{L}{\omega }_0\hfill \\ \hfill & -\sum _{k=1}^n\left({\xi }_i^{\sigma }{d}_k\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}+\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}{\xi }_{ik}^{\sigma }\right){\omega }_0-\sum _{k=1}^n\left(\frac{\partial \mathcal{L}}{\partial y_k^{\sigma }}-d_i\frac{\partial \mathcal{L}}{\partial y_{ik}^{\sigma }}\right){\xi }_k^{\sigma }{\omega }_0\hfill \\ \hfill & +\frac{1}{\mathcal{L}}\sum _{k=1}^n{(-1)}^{k-1}{d}_s\mathcal{L}d{x}^s\wedge d{x}^k\wedge \sum _{l<k}{(-1)}^{l-1}\frac{\partial \mathcal{L}}{\partial y_{il}^{\sigma }}{\xi }_i^{\sigma }\underset{j\ne k,l}{\bigwedge }d{x}^j\hfill \\ \hfill & +\frac{1}{\mathcal{L}}\sum _{k=1}^n{(-1)}^{k-1}{d}_s\mathcal{L}d{x}^s\wedge d{x}^k\wedge \sum _{l>k}{(-1)}^l\frac{\partial \mathcal{L}}{\partial y_{il}^{\sigma }}{\xi }_i^{\sigma }\underset{j\ne k,l}{\bigwedge }d{x}^j\hfill \\ \hfill & =-(1-n)\frac{1}{\mathcal{L}}\sum _{l=1}^n\frac{\partial \mathcal{L}}{\partial y_{jl}^{\sigma }}{\xi }_j^{\sigma }{d}_l\mathcal{L}{\omega }_0-\frac{1}{\mathcal{L}}\sum _{k=1}^n\sum _{l\ne k}\frac{\partial \mathcal{L}}{\partial y_{il}^{\sigma }}{\xi }_i^{\sigma }{d}_s\mathcal{L}d{x}^s\wedge {\omega }_l\hfill \\ \hfill & =0,\hfill \end{array}$$

  where the identity $d{x}^k\wedge {\omega }_l={\delta }_l^k{\omega }_0$ is applied.

□

The Lepage equivalent ${\Lambda }_{\lambda }$ (9) is said to be the Carathéodory form associated to $\lambda \in {\Omega }_{n,X}^{2}W$.

4. The Carathéodory Form and Principal Lepage Equivalents in Higher-Order Theory

We point out that in the proof of Theorem 2, (a), the chart independence of Formula (9) is based on the principal Lepage equivalent ${\Theta }_{\lambda }$ (6) of a second-order Lagrangian, which is defined globally. Since for a Lagrangian of order $r\ge 3$, principal components of Lepage equivalents are, in general, local expressions (see the Introduction), we are allowed to apply the definition (9) for such class of Lagrangians of order r over a fibered manifold which assure the invariance of local expressions ${\Theta }_{\lambda }$ (1).

Consider now a third-order Lagrangian $\lambda \in {\Omega }_{n,X}^{3}W$. Then, the principal component ${\Theta }_{\lambda }$ of a Lepage equivalent of $\lambda$ reads

$$\begin{array}{cc}\hfill {\Theta }_{\lambda }& =\mathcal{L}{\omega }_0+\left(\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{pj}^{\sigma }}+d_p{d}_q\frac{\partial \mathcal{L}}{\partial y_{pqj}^{\sigma }}\right){\omega }^{\sigma }\wedge {\omega }_j\hfill \\ \hfill & +\left(\frac{\partial \mathcal{L}}{\partial y_{kj}^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{kpj}^{\sigma }}\right){\omega }_k^{\sigma }\wedge {\omega }_j+\frac{\partial \mathcal{L}}{\partial y_{klj}^{\sigma }}{\omega }_{kl}^{\sigma }\wedge {\omega }_j.\hfill \end{array}$$

(12)

In the following lemma, we describe conditions for invariance of (12).

Lemma 5.

The following two conditions are equivalent:

(a)

${\Theta }_{\lambda }$ satisfies

$${({\overline{\psi }}^{-1}\circ \psi )}^{*}{\overline{\Theta }}_{\lambda }={\Theta }_{\lambda }.$$

(b)

For arbitrary two overlapping fibered charts on Y, $(V,\psi )$, $\psi =(x^i,y^{\sigma })$, and $(\overline{V},\overline{\psi })$, $\overline{\psi }=({\overline{x}}^i,{\overline{y}}^{\sigma })$,

$$d_k\left(\left(\frac{\partial \mathcal{L}}{\partial y_{l_1{l}_{2}k}^{\tau }}\frac{\partial x^s}{\partial {\overline{x}}^p}-\frac{\partial \mathcal{L}}{\partial y_{l_1{l}_{2}s}^{\tau }}\frac{\partial x^k}{\partial {\overline{x}}^p}\right)\frac{{\partial }^2{\overline{x}}^p}{\partial x^{l_1}\partial x^{l_2}}\frac{\partial y^{\tau }}{\partial {\overline{y}}^{\sigma }}\right)=0.$$

(13)

Proof. 

The equivalence of Conditions (a) and (b) follows from the chart transformation

$$\begin{array}{cc}\hfill & {\overline{x}}^k={\overline{x}}^k\left(x^s\right),{\overline{y}}^{\sigma }={\overline{y}}^{\sigma }(x^s,y^{\nu }),\hfill \\ \hfill & {\overline{y}}_k^{\sigma }=\frac{d}{d{\overline{x}}^k}\left({\overline{y}}^{\sigma }\right)=\frac{\partial x^s}{\partial {\overline{x}}^k}\frac{\partial {\overline{y}}^{\sigma }}{\partial x^s}+\frac{\partial x^s}{\partial {\overline{x}}^k}\frac{\partial {\overline{y}}^{\sigma }}{\partial y^{\nu }}{y}_s^{\nu },\hfill \\ \hfill & {\overline{y}}_{ij}^{\sigma }=\frac{d}{d{\overline{x}}^j}\left({\overline{y}}_i^{\sigma }\right)=\frac{\partial x^t}{\partial {\overline{x}}^j}\frac{d}{d{x}^t}\left(\frac{\partial x^s}{\partial {\overline{x}}^i}\frac{\partial {\overline{y}}^{\sigma }}{\partial x^s}+\frac{\partial x^s}{\partial {\overline{x}}^i}\frac{\partial {\overline{y}}^{\sigma }}{\partial y^{\nu }}{y}_s^{\nu }\right),\hfill \\ \hfill & {\overline{y}}_{ijk}^{\sigma }=\frac{d}{d{\overline{x}}^k}\left({\overline{y}}_{ij}^{\sigma }\right)=\frac{\partial x^r}{\partial {\overline{x}}^k}\frac{d}{d{x}^r}\left(\frac{\partial x^t}{\partial {\overline{x}}^j}\frac{d}{d{x}^t}\left(\frac{\partial x^s}{\partial {\overline{x}}^i}\frac{\partial {\overline{y}}^{\sigma }}{\partial x^s}+\frac{\partial x^s}{\partial {\overline{x}}^i}\frac{\partial {\overline{y}}^{\sigma }}{\partial y^{\nu }}{y}_s^{\nu }\right)\right),\hfill \end{array}$$

applied to ${\Theta }_{\lambda }$, where the Lagrange function $\mathcal{L}$ is transformed by (11) and the following identities are employed,

$$\begin{array}{cc}\hfill & {\overline{\omega }}_j=\frac{\partial x^s}{\partial {\overline{x}}^j}det\left(\frac{\partial {\overline{x}}^p}{\partial x^q}\right){\omega }_s,\hfill \\ \hfill & {\overline{\omega }}^{\sigma }=\frac{\partial {\overline{y}}^{\sigma }}{\partial y^{\nu }}{\omega }^{\nu },{\overline{\omega }}_i^{\sigma }=\frac{\partial {\overline{y}}_i^{\sigma }}{\partial y^{\nu }}{\omega }^{\nu }+\frac{\partial {\overline{y}}_i^{\sigma }}{\partial y_p^{\nu }}{\omega }_p^{\nu },{\overline{\omega }}_{ij}^{\sigma }=\frac{\partial {\overline{y}}_{ij}^{\sigma }}{\partial y^{\nu }}{\omega }^{\nu }+\frac{\partial {\overline{y}}_{ij}^{\sigma }}{\partial y_p^{\nu }}{\omega }_p^{\nu }+\frac{\partial {\overline{y}}_{ij}^{\sigma }}{\partial y_{pq}^{\nu }}{\omega }_{pq}^{\nu }.\hfill \end{array}$$

□

Theorem 3.

Suppose that a fibered manifold $\pi :Y\to X$ and a non-vanishing third-order Lagrangian $\lambda \in {\Omega }_{n,X}^{3}W$ satisfy condition(13). Then, ${\Lambda }_{\lambda }$ (9), where ${\Theta }_{\lambda }$ is given by (12) and defines a differential n-form on $W^5\subset J^{5}Y$, which is a Lepage equivalent of λ, decomposable and ${\pi }^{5,2}$-horizontal. In a fibered chart $(V,\psi )$, $\psi =(x^i,y^{\sigma })$, on W, ${\Lambda }_{\lambda }$ has an expression

$$\begin{array}{cc}\hfill {\Lambda }_{\lambda }& =\frac{1}{{\mathcal{L}}^{n-1}}\underset{j=1}{\overset{n}{\bigwedge }}\left(\mathcal{L}d{x}^j+\left(\frac{\partial \mathcal{L}}{\partial y_j^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{pj}^{\sigma }}+d_p{d}_q\frac{\partial \mathcal{L}}{\partial y_{pqj}^{\sigma }}\right){\omega }^{\sigma }\right.\hfill \\ \hfill & +\left.\left(\frac{\partial \mathcal{L}}{\partial y_{ij}^{\sigma }}-d_p\frac{\partial \mathcal{L}}{\partial y_{ipj}^{\sigma }}\right){\omega }_i^{\sigma }+\frac{\partial \mathcal{L}}{\partial y_{ikj}^{\sigma }}{\omega }_{ik}^{\sigma }\right)\hfill \end{array}$$

(14)

Proof. 

This is an immediate consequence of Lemma 5 and the procedure given by Lemma 4 and Theorem 2. □

Note that, according to Lemma 5, characterizing obstructions for the principal component ${\Theta }_{\lambda }$ (12) to be a global form, the Carathéodory form (14) is well-defined for third-order Lagrangians on fibered manifolds, which satisfy condition (13). Trivial cases when (13) holds identically include: (i) Lagrangians independent of variables $y_{ijk}^{\sigma }$; and (ii) fibered manifolds with bases endowed by smooth structure with linear chart transformations. An example of (ii) are fibered manifolds over the two-dimensional open Möbius strip (for details, see [17]).

Suppose that a pair $(\lambda ,\pi )$, where $\pi :Y\to X$ is a fibered manifold and $\lambda \in {\Omega }_{n,X}^{r}W$ is a Lagrangian on $W^r\subset J^{r}Y$, induces the invariant principal component ${\Theta }_{\lambda }$ (1) of a Lepage equivalent of $\lambda$, with respect to fibered chart transformations on W. We call n-form ${\Lambda }_{\lambda }$ on $W^{2r-1}$,

$${\Lambda }_{\lambda }=\frac{1}{{\mathcal{L}}^{n-1}}\underset{j=1}{\overset{n}{\bigwedge }}{(-1)}^{n-j}{i}_{d_n}\dots i_{d_{j+1}}{i}_{d_{j-1}}\dots i_{d_1}{\Theta }_{\lambda },$$

(15)

where ${\Theta }_{\lambda }$ is given by (1), the Carathéodory form associated to Lagrangian $\lambda \in {\Omega }_{n,X}^{r}W$.

In a fibered chart $(V,\psi )$, $\psi =(x^i,y^{\sigma })$, on W, ${\Lambda }_{\lambda }$ has an expression

$$\begin{array}{cc}\hfill {\Lambda }_{\lambda }& =\frac{1}{{\mathcal{L}}^{n-1}}\underset{j=1}{\overset{n}{\bigwedge }}\left(\mathcal{L}d{x}^j+\sum _{s=0}^{r-1}\left(\sum _{k=0}^{r-1-s}{(-1)}^k{d}_{p_1}\dots d_{p_k}\frac{\partial \mathcal{L}}{\partial y_{i_1\dots i_s{p}_1\dots p_{k}j}^{\sigma }}\right){\omega }_{i_1\dots i_s}^{\sigma }\right).\hfill \end{array}$$

(16)

5. Example: The Carathéodory Equivalent of the Hilbert Lagrangian

Consider the fibered manifold $\mathrm{Met}X$ of metric fields over an n-dimensional manifold X (see [18] for geometry of $\mathrm{Met}X$). In a chart $(U,\phi )$, $\phi =\left(x^i\right)$, on X, section $g:X\supset U\to \mathrm{Met}X$ is expressed by $g=g_{ij}d{x}^i\otimes d{x}^j$, where $g_{ij}$ is symmetric and regular at every point $x\in U$. An induced fibered chart on the second jet prolongation $J^2\mathrm{Met}X$ reads $(V,\psi )$, $\psi =(x^i,g_{jk},g_{jk,l},g_{jk,lm})$.

The Hilbert Lagrangian is an odd-base n-form defined on $J^2\mathrm{Met}X$ by

$$\lambda =\mathcal{R}{\omega }_0,$$

(17)

where $\mathcal{R}=R\sqrt{|det(g_{ij}\left)\right|}$, $R=R(g_{ij},g_{ij,k},g_{ij,kl})$ is the scalar curvature on $J^2\mathrm{Met}X$, and $\mu =\sqrt{|det(g_{ij}\left)\right|}{\omega }_0$ is the Riemann volume element.

The principal Lepage equivalent of $\lambda$ (17) (cf. Formula (6)) reads

$${\Theta }_{\lambda }=\mathcal{R}{\omega }_0+\left(\left(\frac{\partial \mathcal{R}}{\partial y_j^{\sigma }}-d_p\frac{\partial \mathcal{R}}{\partial y_{pj}^{\sigma }}\right){\omega }^{\sigma }+\frac{\partial \mathcal{R}}{\partial y_{ij}^{\sigma }}{\omega }_i^{\sigma }\right)\wedge {\omega }_j,$$

(18)

and it is a globally defined n-form on $J^1\mathrm{Met}X$. (18) was used for a study of the Einstein equations as a system of first-order partial differential equations (see [23]). Another Lepage equivalent for a second-order Lagrangian in field theory which could be studied in this context is given by (10),

$$\begin{array}{cc}\hfill & {\Lambda }_{\lambda }=\frac{1}{{\mathcal{R}}^{n-1}}\underset{k=1}{\overset{n}{\bigwedge }}\left(\mathcal{R}d{x}^k+\left(\frac{\partial \mathcal{R}}{\partial g_{ij,k}}-d_l\frac{\partial \mathcal{R}}{\partial g_{ij,kl}}\right){\omega }_{ij}+\frac{\partial \mathcal{R}}{\partial g_{ij,kl}}{\omega }_{ij,l}\right),\hfill \end{array}$$

where ${\omega }_{ij}=d{g}_{ij}-g_{ij,s}d{x}^s$, ${\omega }_{ij,l}=d{g}_{ij,l}-g_{ij,ls}d{x}^s$. Using a chart expression of the scalar curvature, we obtain

$$\begin{array}{cc}\hfill & {\Lambda }_{\lambda }=\frac{1}{{\mathcal{R}}^{n-1}}\underset{k=1}{\overset{n}{\bigwedge }}\left(\mathcal{R}d{x}^k+\frac{1}{2}\sqrt{g}\left(g^{qp}{g}^{si}{g}^{jk}-2g^{sq}{g}^{pi}{g}^{jk}+g^{pi}{g}^{qj}{g}^{sk}\right)g_{pq,s}{\omega }_{ij}\right.\hfill \\ \hfill & +\sqrt{g}\left(g^{il}{g}^{kj}-g^{kl}{g}^{ji}\right){\omega }_{ij,l}).\hfill \end{array}$$

This is the Carathéodory equivalent of the Hilbert Lagrangian.

6. Discussion

The Lepage equivalents of Lagrangians belong to the basic themes in the global variational theory (in both fibered spaces and the theory for submanifolds) and its applications in differential geometry and mathematical physics. These objects completely describe a global variational problem by means of intrinsic geometric operations. Since the pioneering works by García [6] and Krupka [7,11,12], there have been several attempts at finding new Lepage equivalents in higher-order field theory (cf. [3,20,21,24]).

In this work, we contribute to this area namely by establishing a new relationship between the Poincaré–Cartan and Carathéodory forms and, on this basis, by generalizing the Carathéodory form for higher-order Lagrangians in field theory. For the order $r=2$, the result is general, whereas, for $r\ge 3$, the Carathéodory form in higher-order theory is obtained under specific assumptions on global properties, defined by the initial Lagrangian and the underlying topological space under consideration. In the forthcoming works, these results will be further studied in the sense of the mentioned topological obstructions and applied in physical field theories.