Homogeneity and projective equivalence of differential equation fields

We propose definitions of homogeneity and projective equivalence for systems of ordinary differential equations of order greater than two, which allow us to generalize the concept of a spray (for systems of order two). We show that the Euler-Lagrange fields of parametric Lagrangians of order greater than one which are regular (in a natural sense that we define) form a projective equivalence class of homogeneous systems. We show further that the geodesics, or base integral curves, of projectively equivalent homogeneous differential equation fields are the same apart from orientation-preserving reparametrization; that is, homogeneous differential equation fields determine systems of paths.


Introduction
This paper is concerned with systems of ordinary differential equations y i n+1 = Γ i (y i , y i 1 , . . . , y i n ); y i r = d r y i dx r of order n + 1, n ≥ 1, in m dependent variables y i , i = 1, 2, . . . , m and one independent variable x. Second-order systems of this kind (n = 1) have been studied intensively using methods of differential geometry, by taking advantage of the fact that the equations are those satisfied by the integral curves of a vector field of a certain type, a so-called second-order differential equation field (often abbreviated to SODE) on the tangent bundle T M of an m-dimensional differential manifold M , or some open submanifold of it. Higher-order systems may be studied by analoguous methods, where now the differential equation vector field in question lives (in the case of equations of order n + 1) on the bundle T n M of n-velocities, in other words n-jets at zero of curves in M (whose domains contain zero), or some open submanifold of it. This is the approach adopted here. (For a recent account of the geometric theory of differential equations, covering both second-and higher-order systems, and containing many references, see [1]. However, this reference is mainly concerned with aspects of the theory not covered here, namely generalized connections and related matters.) In the theory of second-order differential equation fields an important role is played by a class of fields which are homogeneous in a certain sense. Let T • M be the slit tangent bundle of M (T M with the zero section deleted). Let ∆ be the Liouville vector field, that is, the infinitesimal generator of dilations of the fibres of T • M . A second-order differential equation field Γ on T • M is called a spray if it satisfies [∆, Γ] = Γ. The canonical spray of a Finsler space, whose integral curves project onto geodesics with constant speed parametrization, is a helpful example. If Γ is a spray then the distribution D + spanned by Γ and ∆ is involutive. Any second-order differential equation field Γ ′ contained in D + is said to be projectively equivalent to Γ, and its base integral curves are obtained from those of Γ by reparametrization. Slightly more generally, for the purposes of this paper we shall say that a second-order differential equation field Γ is homogeneous if the distribution D + spanned by Γ and ∆ is involutive. It is not difficult to show that if Γ is homogeneous in this sense then there is a projectively equivalent second-order differential equation field which is a spray. In Finsler geometry the canonical spray is the Euler-Lagrange field of the energy, which is a regular Lagrangian. But instead of the energy one may consider the Finsler function itself as a Lagrangian. Because of its assumed homogeneity it is parametric: that is to say, the corresponding action integral is invariant under reparametrizations. Such a parametric Lagrangian determines, not a single Euler-Lagrange field, but a projective equivalence class of second-order differential equation fields, which in the Finslerian case is the projective class of the canonical spray.
Our purpose in the present paper is to generalize the concept of homogeneity so that it applies to differential equation fields of higher order. For this purpose we introduce the jet group of order n, L n , which is defined as follows. Consider the local diffeomorphisms φ of R, defined in a neighbourhood of zero and satisfying φ(0) = 0. Then L n is the set of n-jets at zero of such local diffeomorphisms; it is a group under composition, and even a Lie group. Moreover, it acts to the right on T n M . Let D be the distribution spanned by the generators of the action. We say that a differential equation field Γ is homogeneous if the distribution D + spanned by Γ and D is involutive. This generalizes the concept described above for second-order differential equation fields because when n = 1, D is just the one-dimensional distribution spanned by ∆. (A preliminary attempt to examine homogeneity of higher-order differential equation fields is to be found in [2]; however, this paper is again concerend mainly with connection theory.) Section 3 below is devoted to the definition of homogeneity and its immediate consequences.
With this definition we show that the other properties of homgeneous second-order differential equation fields discussed above also generalize. The extremals of a Lagrangian of order n satisfy differential equations of order 2n. If the Lagrangian is parametric, so that it satisfies the so-called Zermelo conditions, then its extremals are determined only up to reparametrization. We show in Section 4 that, provided it is sufficiently regular, a parametric Lagrangian determines a projective equivalence class of homogeneous differential equation fields. This result gives us the reassurence that our definition is not vacuous.
In Section 5 we show that the base integral curves of projectively equivalent homogeneous differential equation fields differ only in parametrization. Section 6 of the paper contains some examples, while Section 2 is devoted to the requisite geometry of jet manifolds and jet groups.
We shall use the Einstein summation convention for coordinate indices such as i; other sums will be indicated explicitly.

Geometrical background
We consider differentiable manifolds of class C ∞ which are Hausdorff, secondcountable and (unless otherwise specified) connected.
Let M be such a manifold with dim M = m and with local coordinates (y i ). Consider curves γ : (a, b) → M , where 0 ∈ (a, b), and let T n M be the set of n-jets at zero {j n 0 γ}, with coordinates (y i r ) for 0 ≤ r ≤ n given by It is a standard result that T n M is a manifold with dim T n M = (n + 1)m, and that the maps τ n : T n M → M and τ n,n ′ : The manifold T n+1 M has a natural identification with a submanifold of T T n M , obtained because the map t → j n t γ defines a curvẽ We therefore obtain the inclusion given in coordinates by y i r+1 →ẏ i r . We may think of this inclusion as a section of the pull-back vector bundle τ * n+1,n T T n M → T n+1 M , in other words as a vector field along the projection τ n+1,n ; with such an interpretation it is the total derivative The projection τ n+1,n : T n+1 M → T n M is in fact an affine bundle, modelled on the vector bundle V τ n,n−1 → T n M of very vertical tangent vectors on T n M . The affine action is just addition in the fibres of T T n M → T n M : gives rise, using the identificationẏ i r = y i r+1 on T n+1 M , to We shall make use later of the vertical endomorphism on T n M . This is the type (1, 1) tensor field S given in coordinates by it is canonically defined, and generalizes the well-known vertical endomorphism or tangent structure on T M . It is not difficult to show that for any 1-form α on T n M , ,n α (where in the first instance S is the vertical endomorphism on T n+1 M , in the second the one on T n M ). This may conveniently be written Sd T − d T S = 1, with the pull-back map understood (note that we assume action on 1-forms; there is a more complicated formula for action on forms of higher degree).
A sectionΓ : T n M → T n+1 M of the affine bundle τ n+1,n is called a differential equation field; in coordinates it is indeed a differential equation every integral curveγ : (a, b) → T n M of Γ is of the formγ = n γ for some geodesic γ ofΓ; that is to say, the geodesics ofΓ are the base integral curves of Γ. We use the term 'differential equation field' to refer to Γ as well asΓ. Now consider local diffeomorphisms φ of R defined in a neighbourhood of zero and satisfying φ(0) = 0. Let L n be the set of n-jets at zero {j n 0 φ} of these local diffeomorphisms; this is a group under composition, and is a Lie group. As a manifold it has two connected components. The component of the identity is a subgroup L n+ of index 2 containing jets of local diffeomorphisms satisfying φ ′ (0) > 0. The map is a global coordinate system on L n (and L n+ ). We can obtain an explicit formula for the product in these coordinates by using an expression for the nth derivative of a composition of functions. This is Faà di Bruno's formula [4], which we take in the form involving the Bell polynomials: here ξ and η are functions of some variable t with (in our case) ξ(0) = η(0) = 0, and B r n (η 1 , η 2 , . . . , η n+1−r ) is a polynomial in the n + 1 − r variables η p , for which the following explicit formula is known: where the sum is over all non-negative integers q 1 , q 2 , . . . , q n+1−r such that q 1 + q 2 + . . . + q n+1−r = r and q 1 + 2q 2 + . . . + (n + 1 − r)q n+1−r = n. (This formula may easily be derived by taking ξ(x) = x r and expressing η(x) as a formal power series in x: then (r!/n!)B r n (η 1 , η 2 , . . . , η n+1−r ) is the coefficient of x n in the formal power series for (η(x)) r . In principle B r n could depend on η p with p > n + 1− r, but in practice, since η(x) contains no constant term there can be no contribution to x n in (η(x)) r coming from the term η p x p /p! if p > n + 1 − r.) To express multiplication in L n in terms of the global coordinate system introduced above, a vectorial notation is convenient. We denote by ξ, η elements of R n , considered as row vectors, and by B(η) the (upper triangular) matrix whose p, q entry is B p q (η 1 , η 2 , . . .). Then the product in L n is given by we must of course assume that ξ 1 and η 1 are non-zero. Note that the matrix which determines multiplication in L n ′ with n ′ < n is just the n ′ × n ′ submatrix of B(η) in the upper left corner. For instance, taking n = 4 gives There are obvious projections λ n,1 : L n → L 1 and λ + n,1 : L n+ → L 1+ given by j n 0 φ → j 1 0 φ; they are homomorphisms because The kernel of λ n,1 is therefore a normal subgroup K n ⊳ L n . By inspection of the matrix B above in the case n = 4 we see that K 2 and K 3 are abelian but K 4 is not; indeed K n is non-abelian whenever n ≥ 4. We may, in addition, identify L 1 with a subgroup of L n by mapping Thus L n may be regarded as a semidirect product L 1 ⋊ K n , using the action of L 1 on K n by conjugation.
Furthermore, we may write L n = L 2 ⋊ J n , where J n is the kernel of the homomorphism λ n,2 : L n → L 2 given by j n 0 φ → j 2 0 φ. For the inclusion of L 2 in L n we take, as a representative of j 2 0 φ, not the quadratic polynomial φ(x) = ax + 1 2 bx 2 but the Möbius transformation because the composition of two of these transformations is another map of the same form. By calculating the derivatives of φ one finds a polynomial representative of j n 0 φ, n r=1 b r−1 x r a r−2 (see [5]). Note that the kernels of the homomorphisms λ n,r do not give semidirect product decompositions when r > 2. We next obtain a basis for the Lie algebra l n of the jet group L n in terms of the coordinates defined above. For this purpose we need the following lemma.
Lemma 1. For given n, p = 1, 2, . . . , n, and r = 1, 2, . . . , n + 1 − p, Proof. We consider the cases r = 1, r ≥ 2 separately. We have where the sum is over all non-negative integers q 1 , q 2 , . . . , q n+1−r with q 1 > 0 such that q 1 + q 2 + . . . + q n+1−r = p and q 1 + 2q 2 + . . .+ (n + 1− r)q n+1−r = n. With η 1 = 1, η r = 0 for r ≥ 2 we get a non-zero contribution to the sum only if q r = 0 for r ≥ 2, when we must have q 1 = p and q 1 = n. That is, there is no non-zero term in the sum unless p = n, and so For r ≥ 2 on the other hand where the sum is now over all non-negative integers q 1 , q 2 , . . . , q n+1−r with q r > 0 such that q 1 +q 2 +. . .+q n+1−r = p and q 1 +2q 2 +. . .+(n+1−r)q n+1−r = n. With η 1 = 1, η s = 0 for s ≥ 2 we get a non-zero contribution to the sum only if q r = 1, q s = 0 for s ≥ 2, s = r. We must therefore have q 1 + 1 = p and q 1 + r = n, that is, p = n + 1 − r and q 1 = n − r. Then We denote by δ r , r = 1, 2, . . . , n, the left-invariant vector field on L n which takes the value r! ∂ ∂y r at the identity. Here (y 1 , y 2 , . . . , y n ) are the coordinates on L n . The factor r! is included for later convenience. We now obtain an explicit expression for δ r in terms of coordinates.
Proof. Let y = (y 1 , y 2 , . . . , y n ) be a generic point of L n . The identity e has coordinates (1, 0, . . . , 0). We have where L y is left multiplication by y: L y η = yB(η). Then One reason for introducing the factor r! in the definition of δ r is to simplify the expression for the bracket.
We can now make some observations about the Lie algebra l n which correspond to properties of the Lie group L n mentioned earlier. In the first place, {δ 1 , δ 2 } span a subalgebra. Furthermore, for any p = 1, 2, . . . , n, δ r : r ≥ p is an ideal in l n (since for any r ≥ p and s ≥ 1, r + s − 1 ≥ p). Let l n p ⊂ l n be the ideal just defined (so that in particular l n = l n 1 ). Then for p > 1, l n /l n p ∼ l p−1 . The ideals l n p for p > 1 are nilpotent. It is known that for a nilpotent Lie algebra, exponentiation is a surjective map onto the simply connected Lie group of which it is the algebra; it follows in particular that exponentiation maps l n 2 onto K n (see [5]). But it is possible to prove this directly, as we now show. The Lie algebra l n consists of linear vector fields on R n , and so exponentiation in the group L n coincides with matrix exponentiation. In the case of an element κ = n r=2 k r δ r of l n 2 the matrix K n to be exponentiated is strictly lower triangular: its elements are given by Notice that the (n − 1) × (n − 1) submatrix in the upper left corner is just the matrix K n−1 corresponding to the element n−1 r=2 k r δ r of l n−1
Proof. As a column vector in R n , exp κ ∈ L n is just (exp K n )e, where e = (1, 0, . . . , 0) T ; that is to say, exp κ is the left column of exp K n . It is easy to see that the first n − 1 entries in the left column of exp K n comprise the left column of the (n − 1) × (n − 1) matrix exp K n−1 , while the lower left corner of exp K n is of the form n!k n + p(k 2 , k 3 , . . . , k n−1 ) where p is a polynomial in the indicated variables. Since K n is strictly lower triangular, exp K n has 1s down the diagonal. We have to show that for any y 2 , y 3 , . . . , y n we can choose k 2 , k 3 , . . . , k n such that the left column of exp K n is (1, y 2 , y 3 , . . . , y n ) T . We proceed by induction. Thus we assume that we can find k 2 , k 3 , . . . , k n−1 such that the left column of exp K n−1 is (1, y 2 , y 3 , . . . , y n−1 ) T . With these values of k 2 , k 3 , . . . , k n−1 we set k n = (y n − p(k 2 , k 3 , . . . , k n−1 ))/n!: then the left column of exp K n is (1, y 2 , y 3 , . . . , y n ) T as required. We have so the first step certainly works.
We next turn to the relationship between l n and certain vector fields on R. An element δ of l n determines a one-parameter subgroup of L n , namely t → exp(tδ): observe first that the integral curve of δ through y ∈ L n is t → y · exp(tδ). Now let φ t be a one-parameter group of diffeomorphisms of R such that φ t (0) = 0. Thenφ t = j n 0 φ t is evidently a one-parameter subgroup of L n whose infinitesimal generator, a vector field χ on L n , is left invariant and therefore an element of l n . For any y ∈ L n , χ y is the tangent vector at t = 0 to the curve t → y ·φ t . But the sth component of y ·φ t is given by where X(x)∂/∂x is the infinitesimal generator of φ t (a vector field on R). On differentiating the formula for (y ·φ t ) s with respect to t at t = 0 we obtain In fact is a linear map from the space of vector fields on R vanishing at the origin onto l n , such that Φ(X∂/∂x) depends only on j n 0 X. In particular, Φ(x r ∂/∂x) = δ r for r = 1, 2, . . . , n while Φ(x r ∂/∂x) = 0 for r > n. Now so Φ is an anti-homomorphism. Let p be the Lie algebra of vector fields on R whose coefficients are formal power series in x, and let p n be the subalgebra of those vector fields which vanish to order n at 0, that is, whose coefficient begins with x n+1 . Then p 0 is the subalgebra of formal power series vector fields which vanish at 0; and for n > 0, p n is an ideal in p 0 . If we think of Φ as a map p 0 → l n it is a surjective anti-homomorphism with kernel p n , and therefore defines an anti-isomorphism p 0 /p n → l n . To put things another way, we can realise l n as the space of vector fields on R whose coefficients are polynomials of order n which vanish at the origin, with bracket the negative of the ordinary bracket of vector fields followed by truncation at order n. This alternative realisation is found elsewhere in the literature. The fact that the bracket is related to the negative of the ordinary bracket is not too surprising when one recalls that if the diffeomorphism group of R is regarded as an infinite dimensional Lie group, its Lie algebra is the space of vector fields on R with compact support, but with bracket the negative of the ordinary vector field bracket.
The group L n has a right action α n on T n M given by composition of jets, ; the action is fibred over the identity on M . It restricts to the 'regular' submanifold T n • M (that is, of n-jets j n 0 γ where γ is a curve with γ ′ (0) = 0, so that γ is an immersion near zero) because composing a diffeomorphism with an immersion gives another immersion. We may use Faá di Bruno's formula recursively to see that the restricted action is free, using the coordinates (y i r ), 1 ≤ i ≤ n, defined on complete fibres over M : by regularity at least one coordinate y i 1 must be non-zero at any given point, so if α n (j n 0 φ, j n 0 γ) = j n 0 γ we see successively that φ ′ (0) = 1 and then that φ ′′ (0) = 0, φ ′′′ (0) = 0, . . . , so that j n 0 φ = 1 L n . The orbit space of T n • M under the action α n has a manifold structure (and is, indeed, a Hausdorff manifold); we shall denote it by P T n M . The subgroup L n+ acts in the same way, and its orbit space will be denoted by P + T n M ; this is a double cover of P T n M . Let ρ n : T n • M → P T n M and ρ + n : T n • M → P + T n M be the projections. In the present work we shall in effect be interested in the circumstances when a differential equation field on T n M (more accurately, on T n • M ) 'passes to the quotient' to determine a line-element field on P T n M or an oriented line-element field on P + T n M . In view of the identification of T n+1 M with an affine sub-bundle of T T n M over T n M , it is of some interest to consider the identification of P T n+1 with a submanifold of P T (P T n M ). One may do this, for instance, by using the action of the tangent group T L n on T T n • M to give T (P T n M ), and then the action of L 1 on the open submanifold T • (P T n M ) to give P T (P T n M ). In fact it is known that P T n+1 is an affine bundle over P T n M , even though P T (P T n M ) → P T n M certainly does not have an affine structure, as its fibres are compact (they are projective spaces): see, for instance, the discussion in [6], which uses the fact that the kernel of the homomorphism L n+1 → L n is abelian. Similar arguments hold in the oriented case.
We shall need to know the fundamental vector fields on T n M of the action of L n . We denote by ∆ r the fundamental vector field corresponding to δ r ∈ l n . Then ∆ r is the infinitesimal generator of the one-parameter group R exp(tδ r ) (where R denotes the right action). In terms of the coordinates (y i r ) on T n M , the action of η ∈ L n is given by By a by now familiar type of argument invoking Lemma 1 we obtain Proposition 3.
We shall not use the results of Corollary 3 directly, but we mention them because they provide an alternative starting point for the geometrical analysis of differential equation fields which may be found elsewhere in the literature.
We shall however use in Section 4 a result which is related to item 1 of Corollary 3, which we now explain. We denote by i the operator of interior product of a vector field with a form, so that for example for a 1-form α, i X α = α(X). We extend this notation to apply to the total derivative: if α is a 1-form on T n • M then i T α is a function on T n+1 • M . Then i T (Sα) = i ∆ 1 (τ * n+1,n α), which may conveniently be written i T S = i ∆ 1 , with the pull-back map understood. Note that we assert only that this formula holds for action on 1-forms (there is a more complicated formula for action on forms of higher degree). Similarly, in relation to item 2, we have (again for action on 1-forms) i T S r = i ∆ r .

Homogeneous higher-order systems
We now consider differential equation (vector) fields on T n • M .
Definition 1. A differential equation field Γ on T n • M is homogeneous if the distribution D + spanned by Γ and the ∆ r is involutive.
When they exist, such functions must satisfy the following consistency conditions, where 1 < r, s ≤ n, r = s: If Γ is a differential equation field such that D + is involutive, so is Γ+µ∆ n for any function µ on T n • M , and of course it belongs to the same involutive distribution.
One cannot, in general, demand that for n ≥ 3 a differential equation field Γ satisfies the strong conditions [∆ r , Γ] = r∆ r−1 , r = 2, 3, . . . , n; the Jacobi identities would be inconsistent with the bracket relations for the ∆ r . (This point is discussed more fully in [2], where examples of equations which do satisfy the conditions for n = 2 are given.) Recall that {∆ 1 , ∆ 2 } span a subalgebra of D; it is easily verified that the conditions are consistent. We shall now show that for any homogeneousΓ it is possible to find µ such that if Γ =Γ + µ∆ n then Γ satisfies the conditions above (as well as [∆ r , Γ] = r∆ r−1 + λ r ∆ n for r > 2); in other words, if Γ is homogeneous, without loss of generality we may assume that λ 1 = λ 2 = 0. We need a couple of lemmas.

Lemma 2.
There is a submanifold S of T n • M of codimension 1 such that ∆ 1 is transverse to S and ∆ 2 is tangent to it. There is a submanifold S ′ of S of codimension 1 such that ∆ 2 | S is transverse to S ′ .
Proof. Let g be any Riemannian metric on M . Then g ij y i 1 y j 1 is a welldefined function on T n M (because the y i 1 transform as the components of tangent vectors), and g ij y i 1 y j 1 > 0 on T n • M . Let ϕ(y) = g ij y i 1 y j 1 , and S = {y ∈ T n • M : ϕ(y) = 1}. Then ∆ 1 (ϕ) = ϕ and ∆ 2 (ϕ) = 0, so ∆ 1 and ∆ 2 are respectively transverse and tangent to S. Next, let ϕ ′ = 1 2 Γ(ϕ) for any differential equation field Γ on T n • M . We have where the Γ k ij are the connection coefficients of the Levi-Civita connection of the metric g, from which it is clear that S ′ = {y ∈ S : ϕ ′ (y) = 0} is a codimension 1 submanifold of S, and ∆ 2 | S is transverse to it. Proof. One can integrate the differential equation where x is any fixed point of S, by the integrating factor method.
(The restriction that k should be constant covers the situation encountered below, but clearly more general equations could be considered.) Theorem 1. LetΓ be a homogeneous higher-order differential equation field on T n • M , so that D + is involutive. Then there is a higher-order differential equation field Γ =Γ + µ∆ n ∈ D + such that Proof. Let S be a codimension 1 submanifold of T n • M such that ∆ 1 is transverse to S and ∆ 2 is tangent to it, and S ′ a codimension 1 submanifold of S such that ∆ 2 | S is transverse to S ′ .
We may write Then with Γ =Γ + µ∆ n , By Lemma 3 we can solve the equation ∆ 1 (µ) + (2 − n)µ = −λ 1 , assigning the value of µ on S arbitrarily. Also, Now ∆ 2 is tangent to S, and µ is undetermined on that submanifold, so we may solve the equation ∆ 2 (µ) = −λ 2 there, assigning the value of µ on S ′ arbitrarily. With µ satisfying these two equations, for Γ we have λ 1 = 0 everywhere on T n • M and λ 2 = 0 on S. But from the consistency condition for λ 1 and λ 2 applied to Γ we have ∆ 1 (λ 2 ) = (n − 1)λ 2 on T n • M . It follows from Lemma 3 and the fact that λ 2 = 0 on S 2 that λ 2 = 0 everywhere.
Finally in this section we observe that a homogeneous differential equation field on T n • M does indeed pass to the quotient under the action of L n+ .
Theorem 2. Let Γ be a homogeneous differential equation field on T n • M . Then the involutive distribution D + defines an oriented line-element field on P + T n M .
Proof. The distribution D + evidently projects to a line-element field on P + T n M . To orient it we take, at any point ρ + n (j n 0 γ), the positive multiples of ρ + n * Γ j n 0 γ for any representative point j n 0 γ ∈ T n • M .

Parametric Lagrangians
To show that the theory described above is not vacuous, we devote this section to an important source of examples, the higher-order differential equation fields obtained from parametric variational problems. (Related results, but for second-order variational problems only, may be found in [8]. Multiple-integral parametric variational problems are discussed at length in [3], and indeed the methods used here are related to those used in that paper, but of course specialized to the case of a single integral.) Let L be a Lagrangian function, defined on T n • M . The Hilbert form of L is the 1-form M . This may be used to construct the Euler-Lagrange form of the Lagrangian, We may see that ε is horizontal over M , because Sd T = d T S + 1 (modulo pullback), so that Sd p T = d p T S + pd p−1 T and therefore Thus Sε is a collapsing sum, and indeed as L has order n. Proof. An extremal of such a problem is a curve γ in M such that b a ( n γ) * L X n L dt = 0 for any vector field X on M vanishing at the (fixed) image of the endpoints of [a, b], where X n denotes the prolongation of X to T n as X vanishes at the endpoints, so that because vector field prolongations commute with total derivatives, and the pull-back of d T to R is d.
Now suppose that L is positively homogeneous; that is, that it satisfies the Zermelo conditions Such a Lagrangian is called a parametric Lagrangian, and the geodesics of the corresponding variational problem are invariant under orientationpreserving reparametrizations [3]; in other words, they are paths. We shall show that (subject to suitable regularity conditions) they are also the geodesics of homogeneous differential equation fields. In order to investigate this, we first establish a technical lemma.
Lemma 5. The Hilbert form ϑ and the Euler-Lagrange form ε satisfy (omitting the pull-back maps).
In addition, so that, from the definition of ε, Proposition 5. The characteristic distribution of the 2-form dϑ satisfies whenever Γ is a differential equation field on T n • M whose geodesics are extremals of the variational problem with fixed endpoints defined by L.
Proof. We remark first that ϑ satisfies i ∆ r ϑ = 0 and L ∆ r ϑ = 0 for 1 ≤ r ≤ 2n − 1 ([3], Proposition 6.1 and Theorem 6.4). It follows that because γ is an extremal of L so that ε j 2n 0 γ = 0. This calculation also uses the fact that the contraction i T is just the inclusion map i : T 2n We say that the Lagrangian is regular if the characteristic distribution of dϑ has dimension 2n. Any differential equation field in this distribution is an Euler-Lagrange field of the variational problem. Since the characteristic distribution of a closed 2-form is involutive we have the following theorem.
Theorem 3. Any Euler-Lagrange field of a regular parametric Lagrangian is homogeneous.

Projective equivalence and reparametrization
We now return to the study of general higher-order differential equation fields.
Projective equivalence is an equivalence relation. The projective equivalence class of a homogeneous differential equation field consists of all the differential equation fields in the involutive distribution D + . We may summarize the result of Theorem 1 by saying that every projective equivalence class of a homogeneous differential equation field contains a generalized spray.
We have seen that the Euler-Lagrange fields of a regular parametric Lagrangian consist of a projective equivalence class of homogeneous differential equation fields. So we should expect in this case that the geodesics of projectively equivalent homogeneous differential equation fields should define the same paths, that is, differ only in parametrization. In this section we shall show that this is true for any projective equivalence class of homogeneous differential equation fields, whether or not they come from a parametric Lagrangian.
In fact we shall prove two results about projective equivalence in homogeneous systems: firstly, if Γ ′ is projectively equivalent to Γ then a geodesic of Γ ′ is a reparametrization of a geodesic of Γ; secondly, the jet group acts on differential equation fields in such a way as to map any homogeneous field to one which is projectively equivalent to it. Theorem 4. Let Γ be a homogeneous differential equation field, and Γ ′ another which is projectively equivalent to it. Let γ ′ be a geodesic of Γ ′ . Then there is a geodesic γ of Γ such that γ ′ is obtained from γ by an orientationpreserving reparametrization.
Proof. By assumption the distribution D + is involutive, and thus integrable. Let L be a leaf of D + . Then Γ ′ is tangent to L, and any integral curve of Γ ′ which meets L lies in it. If γ ′ is a geodesic of Γ ′ then n γ ′ is an integral curve of Γ ′ : let L be in fact the leaf of D + containing (the image of) n γ ′ . Since D ⊂ D + , L is invariant under the action of L n+ , the identity component of the jet group L n , and consists of the orbits of points on n γ ′ under the action of L n+ . Under projection τ n : T n • M → M , we have of course τ n • n γ ′ = γ ′ , and for any g ∈ L n+ , τ n • g = τ n . Thus τ n (L) is the 1-dimensional submanifold of M which is the oriented path of γ ′ . Now Γ also belongs to D + , and so likewise any integral curve of Γ which meets L lies in it. Take any point of L, and the integral curve of Γ through that point: it is of the form n γ where γ is a geodesic of Γ. Then γ = τ n • n γ lies in τ n (L), that is to say, γ ′ and γ have the same oriented path. Since we are restricted to (y i 1 ) = 0, we must assume that γ and γ ′ have nowhere vanishing tangent vectors. It follows that one is a reparametrization of the other, by a reparametrization that preserves orientation.
Corollary 4. Let Γ be a homogeneous differential equation field of order n + 1. Let γ be the geodesic of Γ with initial conditions (y i , y i 1 , . . . , y i n ). Let (z i 1 , . . . , z i n ) be the image of (y i 1 , . . . , y i n ) under some element of L n+ (so that (z i 1 ) is a positive scalar multiple of (y i 1 )). Then the geodesic of Γ with initial conditions (y i , z i 1 , . . . , z i n ) is obtained from γ by an orientation-preserving reparametrization.
The second result is rather more complicated to prove, and requires some preliminaries.
We shall need to distinguish notationally between the fundamental vector fields of the actions of L n on T n M and L n+1 on T n+1 M . Let us denote by ∆ r n the vector field operating on T n M : that is to say With a bit of licence we can write, for r = 1, 2, . . . , n, In particular, Now the homogeneity conditions for an (n + 1)th order differential equation field Γ, when expressed in terms of the Γ i , are ∆ 1 n (Γ i ) = (n + 1)Γ i + n!λ 1 y i 1 , ∆ r n (Γ i ) = (n + 1)! (n + 1 − r)! y i n+2−r + n!λ r y i 1 .
Comparison with the formulae for ∆ r n+1 (y i n+1 ) is suggestive.
That is to say, when Γ is homogeneous there is a representation of the algebra l n on imΓ.
Note that when this holds Γ φ = Γ + ν φ ∆ n n , so Γ φ is projectively equivalent to Γ. From the point of view of representation by sections one might say thatΓ φ is obtained by slidingΓ along the integral curves of ∆ n+1 n+1 . To be more precise, recalling that T n+1 • M is an affine bundle over T n • M modelled on the bundle of very vertical tangent vectors on T n • M ,Γ φ is the translate ofΓ by the very vertical vector field ν φ ∆ n+1 n+1 .
Theorem 5. Let Γ be a homogeneous differential equation field and let φ be an orientation-preserving local diffeomorphism of R leaving 0 fixed. Then Γ φ is projectively equivalent to Γ.
Proof. Note first that Γ φ 1 •φ 2 = (Γ φ 1 ) φ 2 . So if for some φ 1 , φ 2 and every homogeneous Γ, it is the case that Γ φ 1 is projectively equivalent to Γ and Γ φ 2 is projectively equivalent to Γ then Γ φ 1 •φ 2 is projectively equivalent to Γ. We pointed out in Section 2 that the finite order jet group L m is the semi-direct product of L 1 and K m , the subgroup consisting of the jets of diffeomorphisms with derivative at 0 equal to the identity, and further that every element of the latter is the exponential of some element of its Lie algebra. So the result about Γ φ , or equivalentlyΓ φ , will be proved if it can be proved when φ(x) = kx (k a positive constant) and when φ n+1 is the exponential of an element of ∆ r n+1 for r = 2, . . . , n. Consider first the case where φ(x) = kx. Then φ n (y i r ) = (k r y i r ). We have ∆ 1 n (Γ i ) = (n + 1)Γ i + n!λ 1 y i 1 , which integrates to give e −(n+1)t Γ i (e rt y j r ) − Γ i (y j r ) = n! t 0 e −ns λ 1 (e rs y j r )y i 1 ds.