# Equivalence problem for systems of second-order ordinary differential equations

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Let , , and be coordinates in an open connected subset of the Euclidean -dimensional space . Suppose that there is given a second-order system (a1)

for which each is in a neighbourhood of initial conditions .

Following D. Kosambi [a8], one wishes to solve the problem of finding the intrinsic geometric properties (i.e., the basic differential invariants) of (a1) under non-singular coordinate transformations of the type A similar problem was solved by E. Cartan and S.S. Chern [a4], [a5], but in the real-analytic case with transformations replaced by Below, only is considered; see the references for .

Define the KCC-covariant differential of a contravariant vector field on by (a2)

where the semi-colon indicates partial differentiation with respect to . Note that the Einstein summation convention (cf. Einstein rule) on repeated upper and lower indices is used throughout. Using (a2), equation (a1) can be re-expressed as (a3)

The quantity is a contravariant vector field on and constitutes the first KCC-invariant of (a1). It represents an "external force" .

If the trajectories of (a1) are varied into nearby ones according to (a4)

where denotes a constant with small and the are the components of some contravariant vector defined along , substitution of (a4) into (a1) and taking the limit as results in the variational equations (a5)

where the comma indicates partial differentiation with respect to . Using the KCC-covariant differential (a2), this can be re-expressed as (a6)

where (a7) The tensor is the second KCC-invariant of (a1). The third, fourth and fifth invariants are: (a8)

The main result of KCC-theory is the following assertion: Two systems of the form (a1) on are equivalent relative to if and only if the five KCC-invariant tensors , , , , and are equivalent. In particular, there exist coordinates for which the all vanish if and only if all KCC-invariants are zero.

### Remarks. if and only if are positively homogeneous of degree two in the variable . In this case, the structure of must accommodate possible non-differentiability in . This happens in Finsler geometry, but not in affine and Riemannian geometries, where (a1) are geodesics or autoparallels of a linear connection whose coefficients are . The latter are known as the coefficients of the Berwald connection in Finsler geometry, and of the Levi-Civita connection for Riemannian theory [a3], [a9], [a10]. Furthermore, in the Finsler case, , are the Berwald torsion and curvature tensors. Also, is the Douglas tensor, whose vanishing is necessary and sufficient for all to be quadratic in the variables . The latter is always zero in Riemannian and affine geometries, and also for Berwald spaces in Finsler theory [a6], [a3].

Finally, the KCC-invariants can be readily computed in each of the two following cases:

1) where , are functions of only. In this case and The trajectories of this equation are Lyapunov stable if , and unstable if ;

2) where are the coefficients of the Levi-Civita connection of a two-dimensional Riemannian metric, are fixed constants and where the bracket on the right-hand side indicates no summation, [a3]. The KCC-invariants in the case where are close to Riemannian, but has a significant effect on Lyapunov stability.

Further applications of KCC-theory can be found in [a2], [a1]. The equivalence problem can be found in a more general context in [a7], [a11].

How to Cite This Entry:
Equivalence problem for systems of second-order ordinary differential equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equivalence_problem_for_systems_of_second-order_ordinary_differential_equations&oldid=16836
This article was adapted from an original article by P.L. Antonelli (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article