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The concept of homogeneity, which picks out sprays from the general run of systems of second-order ordinary differential equations in the geometrical theory of such equations, is generalized so as to apply to equations of higher order. Certain properties of the geometric concomitants of a spray are shown to continue to hold for higher-order systems. Third-order equations play a special role, because a strong form of homogeneity may apply to them. The key example of a single third-order equation which is strongly homogeneous in this sense states that the Schwarzian derivative of the dependent variable vanishes. This equation is of impor-tance in the theory of the association between third-order equations and pseudo-Riemannian manifolds due to Newman and his co-workers.

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... 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. ...

... , 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 [∆ 1 , Γ] = Γ, [∆ 2 , Γ] = 2∆ 1 are consistent. ...

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.

A Lie system is a system of first-order differential equations admitting a
superposition rule, i.e., a map that expresses its general solution in terms of
a generic family of particular solutions and certain constants. In this work,
we use the geometric theory of Lie systems to prove that the explicit
integration of second- and third-order Kummer--Schwarz equations is equivalent
to obtaining a particular solution of a Lie system on SL(2,R). This same result
can be extended to Riccati, Milne--Pinney and other related equations. We
demonstrate that all the above-mentioned equations associated with exactly the
same Lie system on SL(2,R) can be integrated simultaneously. This retrieves and
generalizes in a unified and simpler manner previous results appearing in the
literature. As a byproduct, we recover various properties of the Schwarzian
derivative.

The study of higher-order mechanics, by various geometrical methods, in the framework of the theory of higher-order tangent bundles or jet spaces, has been undertaken by a number of authors recently: for example, Tulczyjew [16, 17], Rodrigues [14, 15] de León [8], Krupka and Musilova [11, and references therein]. In this article we wish to complement these studies by approaching the subject from a new point of view, one which we developed for second-order differential equation fields and first-order Lagrangian mechanics in [19]. In particular, our aim is to show that many of the results we obtained there may be extended to the higher-order case.(Received May 21 1985)(Revised September 30 1985)

Associated with any horizontal distribution on the tangent bundle of a differentiable manifold is a certain linear connection which can be regarded as a linearization of the corresponding non-linear connection. I call this linear connection a connection of Berwald type, and show how its covariant derivative operator can be specified in terms of the projections of the horizontal distribution. I explain how the Berwald connection of Finsler geometry can be regarded as a special case of this general construction, and describe the relation between the Berwald connection and the other standard Finsler connections from this point of view.

In this book we study sprays and Finsler metrics. Roughly speaking, a spray on a manifold consists of compatible systems of second-order ordinary differential equations. A Finsler metric on a manifold is a family of norms in tangent spaces, which vary smoothly with the base point. Every Finsler metric determines a spray by its systems of geodesic equations. Thus, Finsler spaces can be viewed as special spray spaces. On the other hand, every Finsler metric defines a distance function by the length of minimial curves. Thus Finsler spaces can be viewed as regular metric spaces. Riemannian spaces are special regular metric spaces.

We discuss the infinitesimal affine transformations of the Berwald connection of a spray, and the relation between the projective transformations of a spray and the affine transformations of its Berwald–Thomas–Whitehead connection.

Given a regular Lagrangian L of order k on M we show that there exists a canonical connection
L
on T
2k–1
M whose paths are the solutions of the Euler-Lagrange equations for L. If L is the kinetic energy defined by a Riemannian metric then
L
is the Riemannian connection.

A paper of Anderson and Thompson demonstrates that the inverse problem in the calculus of variations for systems of fourth-order ordinary differential equations gives rise to a system of PDEs in Frobenius form for the multiplier functions, and therefore has a structure much simpler than that of the corresponding problem for second-order equations. We show that a similar simplification holds for systems of equations of order 2k (k>2).

We construct a family of split signature Einstein metrics in four dimensions, corresponding to particular classes of third order ODEs considered modulo fiber preserving transformations of variables.

A geometric setting for systems of ordinary differential equations

- I Bucataru
- O Constantinescu
- M F Dahl

I. Bucataru, O. Constantinescu, M.F. Dahl: A geometric setting for systems of ordinary
differential equations. preprint: arXiv:1011.5799 [math.DG].

Differential geometry from differential equations

- S Fritelli
- C Kozameh
- E T Newman

S. Fritelli, C. Kozameh, E.T. Newman: Differential geometry from differential equations.
Comm. Math. Phys. 223 (2001) 383-408.

- M Godliński
- P Nurowski

M. Godliński, P. Nurowski: Geometry of third-order ODEs. preprint: arXiv:0902.4129v1
[math.DG].