On the Carath\'eodory form in higher-order variational field theory

The Carath\'eodory 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 intrisic geometric operations applied to the well-known Poincar\'e--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 Carath\'eodory equivalent of the Hilbert Lagrangian in general relativity is discussed.


Introduction
In this note, we describe a generalization of the Carathéodory form of the calculus of variations in second-order and, for specific Lagrangians, in higherorder field theory. Our approach 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, (1.1) Θ λ = L ω 0 + r−1 k=0 r−1−k l=0 (−1) l d p1 . . . d p l ∂L ∂y σ j1...j k p1...p l i ω σ j1...j k ∧ ω i , which generalize the well-known Poincaré-Cartan form of the calculus of variations, define, in general, differential form Θ λ globally for Lagrangians λ = L ω 0 of order r = 1 and r = 2 only; see Krupka [7] (Θ λ is known as the principal component of a Lepage equivalent of Lagrangian λ), and Horák and Kolář [5] (for higher-order Poincaré-Cartan morphisms). We show that if Θ λ is globally defined differential form for a class of Lagrangians of order r ≥ 3, then a higher-order Carathéodory equivalent for Lagrangians belonging to this class naturally arises by means of geometric operations acting on Θ λ . To this purpose, for order r = 3 we analyze conditions, which describe the obstructions for globally defined principal components of Lepage equivalents (1.1) (or, higher-order Poincaré-Cartan forms). The above-mentioned differential forms are examples of Lepage forms; for a comprehensive exposition and original references see Krupka [9,10]. Similarly as the well-known Cartan form describes analytical mechanics in a coordinate-independent way, in variational field theory (or, calculus of variations for multiple-integral problems) this role is played by Lepage forms, in general. 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 integrandsthe Lepage equivalents of a Lagrangian.
A concrete application of our result in second-order field theory includes the Carathéodory equivalent of the Hilbert Lagrangian in general relativity, which we determine and it will be further studied in future works.
Basic underlying structures, well adapted to this paper, can be found in Volná and Urban [14]. If (U, ϕ), ϕ = (x i ), is a chart on smooth manifold X, we set where ε i1i2...in is the Levi-Civita permutation symbol. If π : Y → X is a fibered manifold and W an open subset of Y , then there exists a unique morphism h : Ω r W → Ω r+1 W of exterior algebras of differential forms such that for any fibered chart (V, ψ), ψ = (x i , y σ ), where V ⊂ W , and any differentiable function f : W r → R, where W r = (π r,0 ) −1 (W ) and π r,s : J r Y → J s Y the jet bundle projection, is the i-th formal derivative operator associated with (V, ψ). A differential form q-form ρ ∈ Ω r q W satisfying hρ = 0 is called contact, and ρ is generated by contact 1-forms ω σ j1...j k = dy σ j1...j k − y σ j1...j k s dx s , 0 ≤ k ≤ r − 1. Throughout, we use the standard geometric concepts: the exterior derivative d, the contraction i Ξ ρ and the Lie derivative ∂ Ξ ρ of a differential form ρ with respect to a vector field Ξ, and the pull-back operation * acting on differential forms.

Lepage equivalents in first-and second-order field theory
By a Lagrangian λ for a fibered manifold π : Y → X of order r we mean an element of the submodule Ω r n,X W of π r -horizontal n-forms in the module of nforms Ω r n W , defined on an open subset W r of the r-th jet prolongation J r Y . In a fibered chart (V, ψ), ψ = (x i , y σ ), where V ⊂ W , Lagrangian λ ∈ Ω r n,X W has an expression where ω 0 = dx 1 ∧ dx 2 ∧ . . . ∧ dx n is the (local) volume element, and L : V r → R is the Lagrange function associated to λ and (V, ψ). An n-form ρ ∈ Ω s n W is called a Lepage equivalent of λ ∈ Ω r n,X W , if the following two conditions are satisfied: (i) (π q,s+1 ) * hρ = (π q,r ) * λ (i.e. ρ is equivalent with λ), and (ii) hi ξ dρ = 0 for arbitrary π s,0 -vertical vector field ξ on W s (i.e. ρ is a Lepage form).
The following theorem describe the structure of the Lepage equivalent of a Lagrangian (see [7,9]). Theorem 1. Let λ ∈ Ω r n,X W be a Lagrangian of order r for Y , locally expressed by (2.1) with respect to a fibered chart (V, ψ), ψ = (x i , y σ ). An n-form ρ ∈ Ω s n W is a Lepage equivalent of λ if and only if where n-form Θ λ is defined on V 2r−1 by (1.1), µ is a contact (n − 1)-form, and an n-form η has the order of contactness ≥ 2.
Θ λ is called the principal component of the Lepage form ρ with respect to fibered chart (V, ψ). In general, decomposition (2.2) is not uniquely determined with respect to contact forms µ, η, and the principal component Θ λ need not define a global form on W 2r−1 . Nevertheless, the Lepage equivalent ρ satisfying (2.2) is globally defined on W s ; moreover E λ = p 1 dρ is a globally defined (n + 1)-form on W 2r , called the Euler-Lagrange form associated to λ.
We recall the known examples of Lepage equivalents of first-and second-order Lagrangians, determined by means of additional requirements.

Lemma 2. (Principal Lepage form) (a)
For every Lagrangian λ of order r = 1, there exists a unique Lepage equivalent Θ λ of λ on W 1 , which is π 1,0 -horizontal and has the order of contactness ≤ 1. In a fibered chart (V, ψ), Θ λ has an expression (b) For every Lagrangian λ of order r = 2, there exists a unique Lepage equivalent Θ λ of λ on W 3 , which is π 3,1 -horizontal and has the order of contactness ≤ 1. In a fibered chart (V, ψ), Θ λ has an expression For r = 1 and r = 2, the principal component Θ λ (1.1) is a globally defined Lepage equivalent of λ. We point out that for r ≥ 3 this is not true (see [5,7]). (2.3) is the well-known Poincaré-Cartan form (cf. García [4]), and it is generealized for second-order Lagrangians by globally defined principal Lepage equivalent n,X W be a Lagrangian of order 1 for Y , locally expressed by (2.1). There exists a unique Lepage equivalent Z λ ∈ Ω 1 n W of λ, which satisfies Z hρ = (π 1,0 ) * ρ for any n-form ρ ∈ Ω 0 n W on W such that hρ = λ. With respect to a fibered chart (V, ψ), Z λ has an expression Z λ (2.5) is known as the fundamental Lepage form [6], [1]), and it is characterized by the equivalence: Z λ is closed if and only if λ is trivial (i.e. the Euler-Lagrange expressions associated with λ vanish identically). Recently, the form (2.5) was studied for variational problems for submanifolds in [12], and applied for studying symmetries and conservation laws in [11].
n,X W be a non-vanishing Lagrangian of order 1 for Y (2.1). Then a differential n-form Λ λ ∈ Ω 1 n W , locally expressed as is a Lepage equivalent of λ.

The Carathéodory form: second-order generalization
Let λ ∈ Ω 1 n,X W be a non-vanishing, first-order Lagrangian on W 1 ⊂ J 1 Y . In the next lemma, we describe a new observation, showing that the Carathéodory form Λ λ (2.6) arises from the Poincaré-Cartan form Θ λ (2.3) by means of contraction operations on differential forms with respect to the formal derivative vector fields d i (1.2).
Lemma 5. The Carathéodory form Λ λ (2.6) and the Poincaré-Cartan form Θ λ (2.3) satisfy Proof. From the decomposable structure of Λ λ , we see that what is needed to show is the formula Applying the contraction operations to Θ λ , we obtain by means of a straightforward computation for every j, Following the inductive structure of the preceding expressions, we get after the next n − j − 1 steps, An intrinsic nature of Lemma 5 indicates a possible extension of the Carathéodory form (2.6) for higher-order variational problems. We put where Θ λ in (3.1) denotes the principal Lepage equivalent (2.4) of a second-order Lagrangian λ, and verify that formula (3.1) defines a global form.
2. Analogously to the proof of Lemma 5, we find a chart expression of 1-form where Θ λ is the principal Lepage equivalent (2.4). Using dx k ∧ ω j = δ k j ω 0 , we have After another n − j − 1 steps we obtain 3. From (3.2) it is evident that Λ λ (3.1) is decomposable, π 3,1 -horizontal, and obeys hΛ λ = λ. It is sufficient to verify that Λ λ is a Lepage form, that is hi ξ dΛ λ = 0 for arbitrary π 3,0 -vertical vector field ξ on W 3 ⊂ J 3 Y . This follows, however, by means of a straightforward computation using chart expression (3.2). Indeed, we have , and the contraction of dΛ λ with respect to π 3,0 -vertical vector field ξ reads Hence the horizontal part of i ξ dΛ λ satisfies where the identity dx k ∧ ω l = δ k l ω 0 is applied. Lepage equivalent Λ λ (3.1) is said to be the Carathéodory form associated to λ ∈ Ω 2 n,X W .

The Carathéodory form and principal Lepage equivalents in higher-order theory
We point out that in the proof of Theorem 6, (a), the chart independence of formula (3.1) is based on principal Lepage equivalent Θ λ (2.4) of a second-order Lagrangian, which is defined globally. Since for a Lagrangian of order r ≥ 3, principal components of Lepage equivalents are, in general, local expressions (see the Introduction), we are allowed to apply the definition (3.1) for such class of Lagrangians of order r over a fibered manifold which assure invariance of local expressions Θ λ (1.1).
Consider now a third-order Lagrangian λ ∈ Ω 3 n,X W . Then the principal component Θ λ of a Lepage equivalent of λ reads In the following lemma we describe conditions for invariance of (4.1).
Lemma 7. The following two conditions are equivalent: (b) For arbitrary two overlapping fibered charts on Y , (V, ψ), ψ = (x i , y σ ), and (V ,ψ),ψ = (x i ,ȳ σ ), Proof. Equivalence conditions (a) and (b) follows from the chart transformation applied to Θ λ , where Lagrange function L is transformed by (3.3) and the following identities are employed, Theorem 8. Suppose that a fibered manifold π : Y → X and a non-vanishing third-order Lagrangian λ ∈ Ω 3 n,X W satisfy condition (4.2). Then Λ λ (3.1), where Θ λ is given by (4.1), defines a differential n-form on W 5 ⊂ J 5 Y , which is a Lepage equivalent of λ, decomposable and π 5,2 -horizontal. In a fibered chart (V, ψ), ψ = (x i , y σ ), on W, Λ λ has an expression Proof. This is an immediate consequence of Lemma 7 and the procedure given by Lemma 5 and Theorem 6.
Remark 9. Note that according to Lemma 7, characterizing obstructions for the principal component Θ λ (4.1) to be a global form, the Carathéodory form (4.3) is well-defined for third-order Lagrangians on fibered manifolds, which satisfy condition (4.2). Trivial cases when (4.2) holds identically include namely (i) Lagrangians independent of variables y σ ijk , and (ii) fibered manifolds with bases endowed by smooth structure with linear chart transformations. An example of (ii) are fibered manifolds over two-dimensional open Möbius strip (for details see [13]).
Suppose that a pair (λ, π), where π : Y → X is a fibered manifold and λ ∈ Ω r n,X W is a Lagrangian on W r ⊂ J r Y , induces invariant principal component Θ λ (1.1) of a Lepage equivalent of λ, with respect to fibered chart transformations on W . We call n-form Λ λ on W 2r−1 , where Θ λ is given by (1.1), the Carathéodory form associated to Lagrangian λ ∈ Ω r n,X W . In a fibered chart (V, ψ), ψ = (x i , y σ ), on W, Λ λ has an expression Consider a fibered manifold MetX of metric fields over n-dimensional manifold X (see [14] for geometry of MetX). In a chart (U, ϕ), ϕ = (x i ), on X, section g : X ⊃ U → MetX is expressed by g = g ij dx i ⊗ dx j , where g ij is symmetric and regular at every point x ∈ U . An induced fibered chart on second jet prolongation J 2 MetX reads (V, ψ), ψ = (x i , g jk , g jk,l , g jk,lm ).
The Hilbert Lagrangian is an odd-base n-form defined on J 2 MetX by where R = R | det(g ij )|, R = R(g ij , g ij,k , g ij,kl ) is the scalar curvature on J 2 MetX, and µ = | det(g ij )|ω 0 is the Riemann volume element.
The principal Lepage equivalent of λ (5.1) (cf. formula (2.4)), reads and it is a globally defined n-form on J 1 MetX. (5.2) was used for analysis of structure of Einstein equations as a system of first-order partial differential equations (see [8]). Another Lepage equivalent for a second-order Lagrangian in field theory which could be studied in this context is given by (3.2), where ω ij = dg ij − g ij,s dx s , ω ij,l = dg ij,l − g ij,ls dx s . Using a chart expression of the scalar curvature, we obtain Λ λ = 1 R n−1 n k=1 Rdx k + 1 2 √ g g qp g si g jk − 2g sq g pi g jk + g pi g qj g sk g pq,s ω ij + √ g g il g kj − g kl g ji ω ij,l .
This is the Carathéodory equivalent of the Hilbert Lagrangian.