A setting for higher order differential equations fields and higher order Lagrange and Finsler spaces

We use the Fr\"olicher-Nijenhuis formalism to reformulate the inverse problem of the calculus of variations for a system of differential equations of order 2k in terms of a semi-basic 1-form of order k. Within this general context, we use the homogeneity proposed by Crampin and Saunders in [14] to formulate and discuss the projective metrizability problem for higher order differential equation fields. We provide necessary and sufficient conditions for higher order projectivpre-e metrizability in terms of homogeneous semi-basic 1-forms. Such a semi-basic 1-form is the Poincar\'e-Cartan 1-form of a higher order Finsler function, while the potential of such semi-basic 1-form is a higher order Finsler function.

The framework for studying higher order differential equation fields, on a configuration manifold M , is the higher order tangent bundle T r M , for some natural number r ≥ 1. In Section 2 we discuss some geometric structures that naturally live on higher order tangent bundles: vertical distributions, Liouville vector fields, tangent structures. We use the Frölicher-Nijenhuis formalism associated to these geometric structures to provide a vertical differential calculus, which is very useful for studying higher order differential equation fields. Motivated by the foliated structure of the higher order tangent bundles, we show that vertical vector fields, as well as their dual, semi-basic 1-forms, play an important role in the vertical differential calculus, which we associate to a higher order differential equations field. We will use the formalism developed in Subsection 2.2 and especially Lemma 2.3 in Sections 3 and 4 to characterize those differential equation fields that may be associated to a variational problem of a Lagrange or a Finsler space of higher order.
The inverse problem of the calculus of variations requires to determine the necessary and sufficient conditions such that a system of ordinary differential equations, of order 2k, may be derived from a variational problem. For k = 1, these conditions can be formulated in terms of a multiplier matrix [23,32,33,34,35], a closed 2-form [2,10], or a semi-basic 1-form [5]. The approach, based on the existence of a closed 2-form, developed by Crampin in [10], was extended by de León and Rodrigues in [26] for k > 1. A deep relationship between variational equations of arbitrary order and closed 2-forms has been found and studied by Krupková in [21,22]. In Section 3 we use the vertical differential calculus, which we develop in Section 2, to provide global formulations for the geometric structures one can associate to higher order Lagrangians and higher order differential equation fields. In Theorem 3.4 we reformulate the inverse problem of the calculus of variations in terms of a semi-basic 1-form of order k. For the variational case, we show that such a semi-basic 1-form is the Poincaré-Cartan 1-form of a Lagrangian of order k. In Proposition 3.6 we prove that some homogeneity properties of a regular Lagrangian transfer to its canonical Euler-Lagrange vector field.
An important case of the inverse problem of the calculus of variations refers to homogeneous systems of ordinary differential equations. For k = 1, this problem contains what is known as the projective metrizability problem or "the Finslerian version of Hilbert's fourth problem", [1,11,12,37,40]. The projective metrizability problem requires to determine if the solutions of a homogeneous system of second order ordinary differential equations coincide with the geodesics of a Finsler metric, up to an orientation preserving reparameterization [7,8,38,41]. For the case k > 1, an attempt to address and study the projective metrizability problem, requires first a good definition of homogeneity for systems of higher order differential equations as well as for higher order Lagrangians. In this work we use the definitions of homogeneity proposed by Crampin and Saunders in [15] to formulate and study the projective metrizability problem in Section 4. In Subsection 4.1 we introduce and discuss higher order Finsler spaces. We show that the regularity condition, which we propose for a higher order Finsler function, is equivalent to the regularity condition proposed by Crampin and Saunders in [15] for parametric Lagrangians and that it reduces, when k = 1, to the classic regularity condition of a Finsler function. We show that the variational problem of a higher order Finsler function uniquely determines a projective class of homogeneous differential equation fields. In Theorem 4.5 we characterize the projective metrizability problem of a homogeneous differential equation field of order 2k in terms of a homogeneous semi-basic 1-form of order k. We prove that, similarly with what happens in the case k = 1, such a semi-basic 1-form is the Poincaré-Cartan 1-form of a Finsler function of order k. Moreover, the potential of such homogeneous semi-basic 1-form is a Finsler function of order k that metricizes the equation field.
In the last section we discuss some examples of higher order differential equation fields and their relations with higher order Lagrange and Finsler spaces. It has been shown in [9] that biharmonic curves, which are solutions of a fourth order differential equations field, are solutions of the Euler-Lagrange equations for a regular Lagrangian L 2 of order 2. See also [4] for a different approach. We use the homogeneity properties of the second order Lagrangian L 2 to obtain some information for the corresponding Euler-Lagrange vector field (biharmonic differential equations field). We provide an example of a second order Finsler functions, which in the Euclidian context reduces to the parametric Lagrangian L, studied by Crampin and Saunders in [15, §6].

Vertical differential calculus on higher order tangent bundles
In this section we discuss first some geometric structures that are naturally defined on higher order tangent spaces: vertical distributions, Liouville vector fields, tangent structures, higher order differential equation fields. We use these geometric structures and the corresponding differential calculus induced by the Frölicher-Nijenhuis formalism to develop a geometric setting, which we will use in Sections 3 and 4 to discuss two important problems associated to a (homogeneous) higher order differential equation field.
2.1. Geometric structures on higher order tangent bundles. In this work M is a real, ndimensional and C ∞ -smooth manifold. We will assume that all objects are smooth where defined. We denote the ring of smooth functions on M by C ∞ (M ), while the Lie algebra of vector fields on M is denoted by X(M ).
The framework to develop a geometric setting for studying systems of higher order ordinary differential equations on a manifold M is the higher order tangent bundle T r M = J r 0 M , for some r ∈ N * , [2,13,25,30,31,42]. This is the jet bundle of order r, of curves c from a neighborhood of 0 in R to M . For a curve c : I → M , c(t) = (x i (t)), consider j r c : I → T r M its jet lift of order r. If (x i ) are local coordinates on M , the induced local coordinates on T r M are denoted by Let y (0)i := x i and denote T 0 M = M . The canonical submersion π r α : T r M → T α M , for each α ∈ {0, 1, ..., r − 1}, induces a natural foliation of T r M . We will consider also the subbundle 0 M for all t ∈ I and some r ∈ N * . The tangent structure (or vertical endomorphism) of order r is the (1, 1)-type tensor field on T r M defined as For each α ≥ 2, we will consider J α , the composition of J, α-times. The following properties are straightforward: J r+1 = 0, Im J α = Ker J r−α+1 , α ∈ {1, ..., r}.
The foliated structure of T r M gives rise to r regular vertical distributions Each distribution V α , for α ∈ {1, ..., r}, is tangent to the fibers of π r α−1 : , for each u ∈ T r M . We will denote by X Vα (T r M ) the Lie subalgebra of vertically valued vector fields.
An important set of vertical vector fields is provided by the Liouville vector fields (or dilation vector fields) C α ∈ X Vα (T r M ), α ∈ {1, ..., r}. These vector fields are locally given by: For the Liouville vector fields, we have the following formulae for their Lie brackets We will make use of the Frölicher-Nijenhuis formalism, [16,17,20], to develop a differential calculus that will be useful to address various problems associated to a differential equation field, [4,5,19,39]. For a vector valued l-form L on T r M consider the derivation of degree l − 1, i L : Λ q (T r M ) → Λ q+l−1 (T r M ) and the derivation of degree l, d L : Λ q (T r M ) → Λ q+l (T r M ). These two derivations are related by the following formula For two vector valued forms K and L on T r M , of degree k and respectively l, consider the Frölicher-Nijenhuis bracket [K, L], which is the vector valued (k + l)-form on T r M , uniquely defined by The Frölicher-Nijenhuis brackets of the Liouville vector fields C α (vector valued 0-forms) and the vertical endomorphisms J β (vector valued 1-forms) are the vector valued 1-forms given by the following formulae For r = 1, semi-basic 1-forms have shown their usefulness to adress various problems associated to second order differential equation fields, [5,7,17]. We will see also that for r > 1, semi-basic 1-forms, of some order, are useful to formulate a geometric setting for higher order differential equation fields. These forms where introduced and discussed in [3,Def 1]. However, in our work a semi-basic 1-form of order α on T r M corresponds to what is called in [3] a semi-basic 1-form of order r + 1 − α.
Definition 2.1. A form on T r M is called semi-basic of order α ∈ {1, ..., r} if it is semi-basic with respect to the submersion π r α−1 . A form θ on T r M is semi-basic of order α if it vanishes whenever one of its argument is a vertical vector field in X Vα (T r M ). Therefore, θ ∈ Λ 1 (T r M ) is semi-basic of order α if and only if i J α θ = 0. Semi-basic 1-forms of order α are the dual equivalent of vertical vector fields in X Vα (T r M ). Hence we have that θ ∈ Λ 1 (T r M ) is semi-basic of order α if and only if there exists η ∈ Λ 1 (T r M ) such that θ = i J r−α+1 η = η • J r−α+1 . Locally, a 1-form θ on T r M is semi-basic of order α if and only if where the α components θ (0)i , ..., θ (α−1)i are smooth functions defined on domains of local charts on T r M .
For a function f ∈ C ∞ (T r M ) and α ∈ {1, ..., r} we have that d J α f is a semi-basic 1-form of order r − α + 1. For a function f ∈ C ∞ (T r M ) we have that df is a semi-basic 1-form of order α ∈ {1, ..., r} if and only if f is constant along the fibers of the submersion π r α−1 and hence one can restrict it to T α−1 M .

2.2.
Higher order differential equation fields. A system of higher order differential equations, whose coefficients do not depend explicitly on time, can be viewed as a special vector field on some higher order tangent bundle. For such systems, we will use the definition for homogeneous differential equation fields of order r, which was proposed by Crampin and Saunders in [15].
As it happens in the case r = 1, Liouville vector fields C α , are important for defining the notion of homogeneity for various geometric structures on T r M . Whenever we want to consider homogeneous structures, which are not necessarily polynomial in the fibre coordinates, we will consider them defined on the subbundle T r 0 M . Definition 2.2. Consider a vector field S on T r M . We say that S is a semispray of order r if it satisfies the condition JS = C 1 .
In induced coordinates for T r M , a semispray of order r is given by for some functions G i defined on domains of induced local charts.
Alternatively, we have that a vector field S on T r M is a semispray of order r if and only if any integral curve of S, γ : I → T r 0 M , is of the form γ = j r (π r 0 • γ). For an integral curve γ : I → T r 0 M of S, we say that curve c = π r 0 • γ is a geodesic of S. Therefore, a regular curve c : I → M is a geodesic of S if and only if S • j r c = (j r c) ′ . Locally, a regular curve c : I → M , c(t) = (x i (t)), is a geodesic of S if and only if it satisfies the system of (r + 1) order ordinary differential equations Therefore semisprays of order r describe systems of higher order differential equations which have regular curves on M as solutions.
We will consider also, d T , the Tulczyjew differential operator on T r M , also called the total derivative operator, which is given by [42] d T = y (1)i ∂ ∂x i + 2y (2)i ∂ ∂y (1)i + · · · + ry (r)i ∂ ∂y (r−1)i . (2.10) Using the Tulczyjew operator, a semispray S of order r can be written as follows The function d T f is basic with respect to the submersion π k α+1 , therefore we can assume that it is defined on T α+1 M and hence d T f ∈ C ∞ (T α+1 M ). In view of formula (2.11), for an arbitrary semispray of order r, S ∈ X(T r M ), and a function The Frölicher-Nijenhuis brackets of an arbitrary semispray S and the vertical endomorphisms J α are useful to fix a (multi) connection on T r M [3,4,13,36]. In this work we will use only the vertical valued components of these vector valued 1-forms. i) The Lie brackets [C α , S] are given by ii) The vertical components of the Frölicher-Nijenhuis brackets [S, J β ] are given by Then the function f can be restricted to T α M and the 1-form θ satisfies the following formulae Proof. For β ∈ {1, ..., r} and for every X ∈ X(T r M ) we have 16) which has been shown in [4, (3.27)].
.., r}, it follows that i J α θ = 0. Using the corresponding commutation rules and formulae (2.6) it follows If we apply i J α−1 to both sides of formula (2.17) and use the commutation rule we obtain It follows that f is constant on the fibres of π r α : T r M → T α M and therefore, we can restrict f to T α M and assume that it is a function defined on T α M .
We will prove now that θ satisfies formulae (2.15). We have seen above that which is formula (2.15) for γ = α − 1. We apply L S to both sides of this formula, use the commutation rule, and obtain We use now formulae (2.17) and (2.14) to obtain which is formula (2.15) for γ = α − 2. We apply again L S to both sides of the above formula, use the commutation rule, and obtain which is formula (2.15) for γ = α − 3. We continue the process and obtain Formula (2.22) represents formula (2.15) for γ = 1. Now, for the last step we use above formula, formula (2.14), for β = 1, as well as formula (2.17): It follows that θ is given by formula which represents formula (2.15) for γ = 0.
Above definition of homogeneity has been proposed in [15,Definition 3.1]. In view of formulae (2.12), a semispray S ∈ X(T r 0 M ) of order r, is homogeneous if and only if for the vertical vector fields U α ∈ X Vr (T r 0 M ), there exist the functions P α ∈ C ∞ (T r 0 M ) such that U α = P α C r , for all α ∈ {1, ..., r}. Therefore, a semispray S of order r is homogeneous if and only if there exists If we write the Jacobi identities for the vector fields S, C 1 , ...., C r , and use the above formulae, we obtain that functions P 1 , ..., P r must satisfy some consistency conditions. Formulae (2.24) and the consistency conditions for functions P 1 , ..., P r were obtained in [15,Prop. 3.2].
For homogeneous higher order differential equation fields, an important concept is that of projective equivalence, which we borrow from [15, Def. 5.1].
Definition 2.5. Consider S 1 and S 2 two homogeneous semisprays of order r. We say that S 1 and Two homogeneous semisprays S 1 and S 2 , locally given by formula (2.8), are projectively equivalent if and only if the semispray coefficients G i 1 and G i 2 are related by G i Above definition was proposed in [15] for generalized sprays and it is motivated by the following arguments. It has been shown in [15,Thm. 5.2] that for two projectively equivalent homogeneous semisprays their geodesics coincide up to an orientation preserving reparameterization. Moreover, according to [15,Thm. 5.2], the projective class of a homogeneous semispray contains a spray, that is a homogenous semispray for which the homogeneity conditions (2.24) hold true with P 1 = P 2 = 0.
3. The inverse problem of the calculus of variations for higher order differential equation fields The inverse problem of the calculus of variations for a semispray (of order 1) was reformulated in [5] in terms of semi-basic 1-forms. In this section we extend these aspects to the higher order case. In Theorem 3.4 we characterize Lagrangian semisprays of order 2k − 1 in terms of semi-basic 1-forms of order k.
3.1. Higher order Lagrangians. In this subsection we discuss some aspects regarding the geometry of a Lagrangian of order k. In Lemma 3.1 we study these geometric aspects in connection with the Poincaré-Cartan 1-form, which is a semi-basic 1-form of order k.
Consider L, a Lagrangian of order k, which is a function defined on T k M . The Poincaré-Cartan where S ∈ X(T 2k−1 M ) is an arbitrary semispray of order 2k − 1. We will see in Lemma 3.1 that The Lagrangian energy function E L ∈ C ∞ (T 2k−1 M ) is given by In the next Lemma we discuss some geometric aspects for a Lagrangian L of order k in terms of its Poincaré-Cartan forms and the Lagrangian energy function. i) The Poincaré-Cartan 1-form θ L is a semi-basic 1-form of order k on T 2k−1 M , which does not depend on the semispray S. ii) The Lagrangian energy function E L ∈ C ∞ (T 2k−1 M ) does not depend on the semispray S and it is related to the Poincaré-Cartan 1-form θ L by the following formula Hessian matrix has maximal rank n on T k M .
Proof. i) Locally, the Poincaré-Cartan 1-form θ L can be expressed as follows Consider d T , the Tulczyjew operator (2.10) on T 2k−1 M . L is a Lagrangian on T k M and ∂L/∂y (α)i are locally defined on T k M , for all α ∈ {1, ..., k}. Therefore, we can view as locally defined functions on T k+β M, for all β ∈ {1, ..., k − 1}. It follows that all components θ (α)i , α ∈ {0, ..., k − 1}, in formula (3.6), do not depend on the semispray S. From formula (3.5) it follows that θ L is a semi-basic 1-form of order k, which does not depend on the semispray S.
ii) Since for all α ∈ {1, ..., k} the functions C α (L) are defined on T k M , it follows that we can view the functions as being defined on T k+α−1 M . Therefore, the right hand side of formula (3.2), and hence the energy E L , is independent of the choice of the semispray S.
If we apply i S to both sides of formula (3.1) it follows In the above formula we did use the commutation rule [17, A.1], as well as the fact that J α S = C α . From formula (3.8) we obtain that (3.3) is true.
iii) Using formula (3.5) and the fact that we can view θ (k−α)i as locally defined functions on T (k+α−1) M , it follows that Since ∂L/∂y (k)i are locally defined functions on T k M , we have Now, from formulae (3.9) and (3.11) it follows that 2kn ≥ rank(dθ L ) ≥ 2k · rank(g ij ). (3.12) We prove the first implication of part iii) of the lemma by contradiction. We assume that ω L = −dθ L is a symplectic structure on T 2k−1 M and also that rank(g ij ) < n. It follows that there are locally defined functions X i such that g ij X j = 0. It follows that the non-zero vector field X = X i ∂/∂y (2k−1)i satisfies i X dθ L = 0, which contradicts the fact that ω L is a symplectic structure. The converse implication of the third item of the lemma follows directly from formula (3.12). If rank(g ij ) = n we obtain that rank(dθ L ) = 2n and hence ω L is a symplectic structure.
The local expression (3.5) -(3.6) for θ L can be written in a more compact form as follows for some Lagrangian L of order k? The equivalence of the two systems (2.9) and (3.14) require that the Hessian matrix (3.4), of the sought after Lagrangian L of order k, has rank n and hence the Lagrangian has to be regular. For a given semispray of order 2k − 1, the Lagrangian to search for can be of order higher then k and the regularity condition can be more general, see [21,23]. In this work, we focus our attention on Lagrangians of minimal-order and hence the regularity condition is given in Definition 3.2 Next theorem provides a characterization for Lagrangian semisprays, in terms of semi-basic 1forms, extending the results obtained in [5]. In [26,Thm. 3.2], Lagrangian semisprays of order 2k − 1 are characterized in terms of a closed 2-form, extending the k = 1 case, which was studied in [10]. The relationship between variational equations of an arbitrary order and closed 2-forms has been investigated in [21,22]. L of order k such that either one, of the following equivalent two conditions, is satisfied ii) S is a Lagrangian semispray if and only if there exists a (locally defined) semi-basic 1-form θ on T 2k−1 M of order k such that rank(dθ) = 2kn and the 1-form L S θ is closed. In this case θ is the Poincaré-Cartan 1-form of some locally defined regular Lagrangian L of order k.
Proof. i) Using the Euler-Lagrange equations (3.14), it follows that the semispray S is Lagrangian if and only if it satisfies the equation for some (locally defined) regular Lagrangian L of order k.
In view of formula (3.3) we obtain that the two equations (3.15) are equivalent. Therefore, we will have to prove that equation (3.16) and first equation (3.15) are equivalent.
Using expression (3.13) for the Poincaré-Cartan 1-form θ L and the fact that L S dy (α−1)i = αdy (α)i it follows If we use the above expression for L S θ L it follows that ii) For the direct implication of this part, we assume that S is a Lagrangian semispray. Therefore, semispray S satisfies first equation (3.15), for some regular Lagrangian L of order k. We consider θ = θ L ∈ Λ 1 (T 2k−1 M ), its Poincaré-Cartan 1-form, which is a semi-basic 1-form of order k and satisfies first equation (3.15). By Definition 3.2 we have that rank(dθ) = 2kn.
For the converse, let us consider θ ∈ Λ 1 (T 2k−1 M ), a semi-basic 1-form of order k such that L S θ is a closed 1-form. Therefore L S θ is locally exact and hence there exists L, a locally defined function on T 2k−1 M , such that We want to prove now that L is constant on the fibres π 2k−1 k and hence we can view it as a function defined on some open domain of T k M . Moreover, we will prove that θ is the Poincaré-Cartan 1form θ L of L. For these, as we have seen in the last part of Lemma 2.3, we need a condition weaker then (3.18), namely we will use the fact that L S θ − dL is a semi-basic 1-form of order 1. This means that According to part v) of Lemma 2.3 it follows that one can restrict the function L to some open domain of T k M and the semi-basic 1-form θ is given by formula (2.23), where f = L and α = k. It follows that θ is given by formula (3.1) and hence it is the Poincaré-Cartan 1-form of the function L, which means that θ = θ L . Using the assumption rank(dθ) = 2kn it follows that the Poincaré-Cartan 2-form of L, ω L = −dθ L = −dθ, is a symplectic structure. Hence L is a (locally defined) regular Lagrangian of order k. If we replace θ = θ L in formula (3.18) it follows that the semispray S satisfies first formula (3.15) for the Lagrangian L. In view of the first part of the theorem it follows that the semispray S is Lagrangian. According to Definition 3.2, we have that for a regular Lagrangian L of order k, second equation (3.15) has a unique solution. This way, to each regular Lagrangian L on T k M it corresponds a unique Lagrangian semispray S ∈ X(T 2k−1 M ). We will refer to this semispray as to the canonical semispray (or the Euler-Lagrange vector field) associated to the Lagrangian L of order k. Using the terminology introduced by Krupková in [23, Ch. 4] we can say that for a regular Lagrangian its Euler-Lagrange distribution has a constant rank equal to one and it is spanned by the semispray S.
If we want to determine the local coefficients G i of a Lagrangian semispray S of order 2k − 1, we use formula (3.10) and write equations (3.16) in the following equivalent form It follows that for a regular Lagrangian, the Hessian matrix g ij is invertible and hence equations (3.19) uniquely determine the semispray coefficients G i .
For a Lagrangian semispray S, its geodesics are solutions of the Euler-Lagrange equations (3.14). Moreover, the geodesic equations (2.9), with r = 2k − 1, for the Lagrangian semispray S and the Euler-Lagrange equations (3.14) are related by where g ij is the Hessian matrix (3.4). The two systems of equations (2.9) and (3.14) coincide if the Lagrangian is regular.
Next lemma presents some compatibility conditions between the geometric structures associated to a Lagrangian and the Liouville vector fields. These properties will be useful in the next section to see how the homogeneity properties of a Finsler function transfer to the induced geometric structures.
Lemma 3.5. Consider L a Lagrangian on T k M and θ L ∈ Λ 1 (T 2k−1 M ) its Poincaré-Cartan 1-form. The following formulae are true: Proof. For the Lagrangian function L consider S a semispray, solution to one of the two equivalent equations (3.15), which means L S θ L = dL. If we apply L C1 to both sides of this formula and use the commutation rule we obtain Using the local expression (3.5) of the Poincaré-Cartan 1-form θ L and the fact that its only component that depends on y (2k−1)i is θ (0)i , which is given in formula (3.6), it follows that Therefore L U1 θ L is a semi-basic 1-form of order 1. Using formula (3.23) it follows According to part iv) of Lemma 2.3 it follows that L C1 θ L is a semi-basic 1-form of order k. We use now part v) of Lemma 2.3 to conclude, from formula (3.24), that the semi-basic 1-form of order k, L C1 θ L , satisfies formula (2.23) for α = k and f = C 1 (L) − L. In view of formula (3.1), this means that L C1 θ L is the Poincaré-Cartan 1-form of the function C 1 (L) − L, which is first formula (3.21). Now, we use formula L S θ L = dL and compose both sides with i J , which means that i J L S θ L = d J L. Using this formula and part v) of Lemma 2.3 it follows that the semi-basic 1-form of order k, θ L satisfies formulae (2.15) for α = k, γ ∈ {0, ..., k − 1} and f = L, which can be written as follows We note that both sides in above formulae do not depend on the chosen semispray S. If we compose with i S in both sides of formulae (3.25), we obtain formulae (3.21) for α ∈ {1, ..., k − 1}.
Since θ L is a semi-basic 1-form of order k, it follows that there exists η ∈ Λ 1 (T 2k−1 M ) such that θ L = i J k η. For α ∈ {k, ..., 2k − 1}, we have that J k (C α ) = 0. Therefore, i Cα θ L = i J k (Cα) η = 0 and hence we proved all formulae (3.21) We prove in the next proposition that some homogeneity properties of a regular Lagrangian are inherited by its canonical semispray. Proposition 3.6. Consider L a regular Lagrangian of order k such that C 1 (L) = aL, for a = 1, and let S be its canonical semispray of order 2k − 1. It follows that [C 1 , S] = S.
Proof. Since L is a regular Lagrangian of order k it follows that the semispray S ∈ X(T 2k−1 0 M ) is the unique solution of the second equation (3.15). Using the fact that L S ω L = 0, it follows that If we use first formula (3.21) and the homogeneity condition C 1 (L) = aL we obtain L C1 θ L = (a − 1)θ L . We replace this and first formula (3.15) in (3.26). It follows Using second formula (3.21), for α = 1, we obtain the following expression for the energy Lagrangian function E L , which is given by formula (3.2) We replace the expression for L S i C1 θ L from above formula in (3.27) and obtain Since ω L is a symplectic structure it follows that [C 1 , S] = S.
For the case k = 1, above formulae show that the homogeneity of a regular Lagrangian transfers to the canonical Euler-Lagrange vector field, which makes it into a spray.

Projective metrizability for homogeneous higher order differential equation fields
A particular aspect of the inverse problem of the calculus of variations deals with homogeneous systems of differential equations. For k = 1, this problem is known as the projective metrizability problem, or as the Finslerian version of Hilbert's fourth problem [1,11,12,37,40]. The most important aspect that is needed to formulate and address the projective metrizability problem for k > 1 relies on a correct definition of homogeneity for systems of higher order differential equations and corresponding Lagrangians. We believe that such definition of homogeneity is that proposed by Crampin and Saunders in [15], which we use in this paper. In this section we formulate and discuss some aspects regarding the projective metrizability problem for the case k > 1, extending some results obtained in [5,7] for k = 1.

4.1.
Higher order Finsler spaces. For k = 1, a Finsler function is characterized by the following important aspect: its variational problem uniquely determines a class of projectively related systems of second order ordinary differential equations. This property is due to the fact that a Finsler function satisfies some homogeneity condition and a regularity condition. Inspired by the work of Crampin and Saunders [15], we propose the following definition for a Finsler function of order k > 1. ii) the tensor with components has rank n − 1 on T k 0 M . A Lagrangian L on T k 0 M that satisfies the Zermelo conditions (4.1) in Definition 4.1 is called parametric Lagrangian in [15, §4] since the solutions of the corresponding variational problem are invariant under orientation preserving reparameterization. The Zermelo conditions and the invariance under reparameterizations for the integral curves of some higher order differential equations, as well as their relation with the variational equations related to Finsler geometry, has been studied very recently by Urban and Krupka in [43].
Spaces with functions that satisfy the Zermelo conditions (4.1) as well as the regularity condition ii) of Definition 4.1 where studied by Kawaguchi, [18], and also referred to as Kawaguchi spaces.
Definition 4.1 reduces to the classic definition of a Finsler space when k = 1, and the tensor (4.2) becomes the angular metric tensor [28, §16]. Indeed, if k = 1, we have that the tensor (4.2) satisfies It is well known that rank(h ij ) = n − 1 if and only if rank(∂ 2 F 2 /∂y i ∂y j ) = n, [28, §16]. Due to a recent result by Lovas [27], the regularity condition rank(∂ 2 F 2 /∂y i ∂y j ) = n and the positivity of the Finsler function F is equivalent to the fact that Hessian matrix of F 2 , g ij = ∂ 2 F 2 /∂y i ∂y j is positive definite. Using [12,Section 3] or [37, Section 3] the Hessian matrix of F 2 is positive definite if and only if the Hessian matrix of F is positive quasi-definite.
Due to the homogeneity conditions of a Finsler function of order k, the energy function E F and the Poincaré-Cartan forms θ F and ω F = −dθ F have special properties. These properties are presented in the next lemma.
Last part of the next lemma also shows that the regularity condition ii) in Definition 4.1 is equivalent to rank(dθ F ) = 2k(n − 1), which is the regularity condition for parametric Lagrangians considered by Crampin and Saunders in [15]. i) The Poincaré-Cartan 1-form θ F satisfies the homogeneity conditions (4.3).
ii) The following formulae are true iii) F is a Finsler function of order k if and only if rank(dθ F ) = 2k(n − 1).
First formula (3.21) shows that the Zermelo condition is a semispray and satisfies the equation L S θ F = dF . For α ≥ 2, we apply L Cα to both sides of this equation. It follows We replace this in formula (4.6) and obtain Using a similar argument that we have used in the proof of Lemma 3.5 it follows that L Uα θ F are semi-basic 1-forms of order 1, for all α ∈ {2, ..., 2k − 1}. For α = 2, in formula (4.7), it follows that L S L C2 θ F is a semi-basic 1-form of order 1. Item v) of Lemma 2.3 implies L C2 θ F = 0. We continue with α ∈ {3, ..., 2k − 1} in formula (4.7), use a similar argument as above, and obtain L Cα θ F = 0.
For θ F , the Poincaré-Cartan 1-form of a Finsler function F , given by formula (3.1), we use formula (3.8), as well as the Zermelo conditions (4.1), to obtain i S θ F = C 1 (F ) = F , which is first formula (4.4). These considerations and formula (3.3) imply that second formula (4.4) is true.
The Poincaré-Cartan 1-form is homogeneous, which means that it satisfies formulae (4.3). The two formulae (4.3) imply that formulae (4.5) are true as well.
iii) We have seen already that the {S, C 1 , ..., C 2k−1 } ⊂ Ker ω F . Based on this aspect and using a similar argument we did use for the proof of third item in Lemma 3.1, formula (3.9) has the following correspondent 2k(n − 1) ≥ rank(ω F ) ≥ 2k · rank(h ij ). (4.8) We assume now that F is a Finsler function of order k, which means that it satisfies the regularity condition ii) of Definition 4.1. From formula (4.8) it follows that rank(ω F ) = 2k(n − 1), which is the regularity condition for parametric Lagrangians in [15].
We prove the other implication by contradiction. We assume that rank(ω F ) = 2k(n − 1) and that rank(h ij ) < (n − 1). From the Zermelo condition C k (F ) = 0 we obtain that h ij y (1)j = 0. Therefore, in view of our assumption, there exist the functions X j = P y (1)j that satisfy h ij X j = 0. It follows that the non-zero vector field X = X j ∂/∂y (2k−1)j satisfies i X ω F , which contradicts the assumption that rank(ω F ) = 2k(n − 1).
The homogeneity properties of the Poincaré-Cartan forms θ F and ω F were proven in a different context in Proposition 6.1 and Theorem 6.4 of [14].

4.2.
Higher order projective metrizability. In this subsection we formulate and discuss the projective metrizability problem for homogeneous higher order systems. We show first that the variational problem of a Finsler function of order k uniquely determines a projective class of homogeneous higher order systems. Then, we characterize the metrizability of a homogeneous higher order systems in terms of some homogeneous semi-basic 1-forms. The variational problem for a regular Lagrangian on T k M uniquely determines a Lagrangian semispray of order 2k − 1. In Theorem 3.4 we gave characterizations for a semispray S to be a Lagrangian semispray.
We will see now that in the case of a Finsler function of order k, the variational problem uniquely determine a projective class of sprays. In the next theorem, which represents the homogeneous version on Theorem 3.4, we provide characterizations of projectively metrizable homogeneous semisprays in terms of homogeneous semi-basic 1-forms, extending the case k = 1 studied in [5, §4.3].
for some (locally defined) Finsler functions F of order k. ii) S is projectively metrizable if and only if there exists a (locally defined) homogeneous semibasic 1-form θ on T 2k−1 M of order k, such rank(dθ) = 2k(n − 1) and the 1-form L S θ is closed.
Proof. i) In view of the two formulae (4.4) we have that the two equations (4.9) are equivalent. For the direct implication, we assume that S is projectively metrizable. Then, the semispray S satisfies the equation for some Finsler function F on T k M . Using similar arguments as we did use in the proof of Theorem 3.4, it follows that equation (4.10) is equivalent to first equation (4.9).
For the converse implication, consider F a Finsler function of order k. We assume that the semispray S is a solution of the second equation (4.9). Locally, first equation (4.9) is equivalent to It follows that two homogeneous semisprays S 1 and S 2 are solutions of either one of the two equations (4.9) if and only if the semispray coefficients G i The regularity condition for the Finsler function F implies that the only solutions of equation (4.12) are given by G i 1 − G i 2 = P y (1)i , for some function P ∈ C ∞ (T 2k−1 0 M ), and hence the two homogeneous semisprays S 1 and S 2 are projectively equivalent. Therefore, equations (4.9) uniquely determine the projective class of a homogeneous semispray S, and this homogeneous semispray is projectively metrizable.
ii) For the first implication we assume that the homogeneous semispray S is projectively metrizable. Therefore, it satisfies first equation (4.9), for some (locally defined) Finsler function F of order k. We consider θ = θ F , the Poincaré-Cartan 1-form of F . We have that θ ∈ Λ 1 (T 2k−1 0 M ) is a homogeneous, semi-basic 1-form of order k, the 1-form L S θ is closed, and according to iii) of Lemma 4.3 we have rank(dθ) = 2k(n − 1).
We will prove first that the condition L S θ is closed implies L S θ = di S θ. Since S is a homogeneous semispray of order 2k − 1 it follows [J, S]S = S + P 1 C 2k−1 , for some function P 1 ∈ C ∞ (T 2k−1 0 M ). Due to the homogeneity of the semi-basic 1-form θ it follows that L P1C 2k−1 θ = 0. Using the commutation rule [17, 1a, p.205] it follows Since θ is a semi-basic 1-form of order k it follows that for k ≥ 2 the 1-form L S θ is semi-basic of order k + 1 ≤ 2k − 1. Using formula (2.14) for β = 1 and the commutation rule [17, 1c, p.205] it follows Formulae (2.4), (2.14) for β = 1 and the condition L S dθ = 0 imply: In the above formula we compose with i S , use the homogeneity condition i C1 dθ = 0 and obtain If we replace now formulae (4.14) and (4.15) and the fact that d [J,S] Consider the function F = i S θ. Above formula shows that the function F satisfies formula (3.18), for L = F , which means L S θ = dF . Since L S θ ∈ Λ 1 (T 2k−1 0 M ) is semi-basic of order (k + 1), it follows that the function F is constant along the fibres of the projection π 2k−1 k : T 2k−1 0 M → T k 0 M and hence we can assume that F ∈ C ∞ (T k 0 M ). Using part v) of Lemma 2.3 we obtain that θ = θ F . We have to prove now that the function F is a Finsler function. From first formulae (3.21) it follows that i Cα θ F = 0 if and only if C α+1 (F ) = 0 for all α ∈ {1, ...., k −1}. Since θ is homogeneous, we obtain C 2 (F ) = · · · = C k (F ) = 0. These arguments, the definition of function F and formula (3.8) imply F = i S θ F = C 1 (F ). It follows that Zermelo conditions (4.1) are satisfied. Finally, we have that rank(dθ) = 2k(n − 1), and using part iii) of Lemma 4.3 implies that F is a Finsler function of order k. Now the condition L S θ F = dF says that S is projectively metrizable.
For k = 1, second equation (4.9) reduces to Rapcsák equation [40,Rap 1]. We note that for the converse implication of the first part of Theorem 4.5 we do not need the requirement that the semispray S is homogeneous. The argument is as follows, and it is due to Crampin and Saunders [15,Thm. 4.4]. For a semispray S, solution of second equation (4.9), using formulae (4.5) it follows that D = span{S, C 1 , ...., C 2k−1 } = Ker ω F . Since the Poincaré-Cartan 2-form ω F = −dθ F is closed it follows that its characteristic distribution D is involutive and hence S is a homogeneous semispray.

Examples
For a Finsler function F , of order k ≥ 1, its variational problem uniquely determines a projective class of homogeneous semisprays, solutions of either one of the two equivalent equations (4.9).
For k = 1, in this projective class of homogeneous semisprays, we can single out one spray, which is called the geodesic spray. The geodesic spray is the only semispray determined by the variational problem of the regular Lagrangian L = F 2 . Moreover, the geodesic spray is the only spray, in the projective class determined by the Finsler function F , whose geodesics are parameterized by arclength.
When k > 1, we do not know yet if it is possible, and eventually how, to associate to a Finsler function F of order k, a regular Lagrangian of order k. Therefore, the only option to fix a homogeneous semispray, which was suggested to me by David Saunders, in the projective class determined by the variational problem of F , is to use the arc-length induced by F .
Next we use some examples to discuss the above considerations as well as the results obtained in the previous sections.

(5.3)
We have that rank(h (1) ij ) = n − 1 and the variational problem for F 1 uniquely determines the projective class of the geodesic spray S 1 . Within this projective class, S 1 is the only spray whose geodesics are parameterized by the arc-length of the given riemannian metric g.
It follows that z (2)i behave as the components of a vector field on M . These components were interpreted as the covariant form of acceleration in [6, (6.5)], as half of the components of the tension field in [9]. It follows that the function L 2 : T 2 M → R, given by L 2 (x, y (1) , y (2) ) = 1 2 g ij (x)z (2)i (x, y (1) , y (2) )z (2)j (x, y (1) , y (2) ) = 1 2 z (2) 2 g , is a second order regular Lagrangian. The Hessian matrix of L 2 , given by formula (3.4), is the Riemannian metric g ij . The variational problem for L 2 uniquely determines a semispray of order 3, S 3 ∈ X(T 3 M ), whose geodesics are biharmonic curves [9]. We call S 3 the biharmonic semispray and it is uniquely determined by either one of the two equivalent equations (3.15). The local coefficients of the biharmonic semispray can be determined as in [4, (4.6)], while the biharmonic equations can be written as in [4, (4.8)].
Consider the function F 2 : T 2 0 M → R, The numerator of the right hand side of the above formula is z (2) 2 g y (1) 2 g − g ij y (1)i z (2)j 2 ≥ 0, and hence F 2 ≥ 0 on T 2 0 M . Using the homogeneity properties (5.4) of the Lagrangian L 2 , we obtain that F 2 satisfies the Zermelo conditions (4.1), for k = 2. Moreover, the tensor (4.2) that corresponds to F 2 is given by It follows that rank(h (2) ij ) = rank(h (1) ij ) = n − 1 and therefore F 2 is a Finsler function of order 2. Using formulae (5.3), (5.5), and (5.6), it follows that one can recover the Finsler function of order 2, F 2 , from either one of the angular metrics h (1) ij or h (2) ij as follows ij (x, y (1) , y (2) )z (2)i z (2)j . (5.7) In the Euclidean context, F 2 reduces to the parametric Lagrangian considered by Crampin and Saunders in [15]. The function F 2 /F 1 ∈ C ∞ (T 2 0 M ), which connects the angular metrics of the two Finsler functions, is related to the first curvature κ of a curve. Indeed we have F 2 /F 1 = κ 2 . See also formula (39) in [29] for A = 0.
The variational problem for F 2 uniquely determines a system of fourth order differential equations, which is invariant under orientation preserving reparameterizations. By fixing the parameter to be the arc-length, the system reduces to the dynamical equation of motion (38) studied by Matsyuk [29]. In the Euclidean context, a homogeneous semispray, in the projective class determined by the variational problem of F 2 was obtained in [15].