Equivalence problem for systems of second-order ordinary differential equations
Let , , and be coordinates in an open connected subset of the Euclidean -dimensional space . Suppose that there is given a second-order system
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
Below, only is considered; see the references for .
Define the KCC-covariant differential of a contravariant vector field on by
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
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
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
where the comma indicates partial differentiation with respect to . Using the KCC-covariant differential (a2), this can be re-expressed as
The tensor is the second KCC-invariant of (a1). The third, fourth and fifth invariants are:
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.
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:
where , are functions of only. In this case and
The trajectories of this equation are Lyapunov stable if , and unstable if ;
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.
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|[a12]||D. Kosambi, "Systems of differential equations of second order" Quart. J. Math. Oxford , 6 (1935) pp. 1–12|
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