On contact equivalence of systems of ordinary differential equations

TL;DR

This paper develops a canonical geometric framework to determine when pairs of distributions on manifolds are equivalent, providing new solutions to contact equivalence problems for systems of ordinary differential equations.

Contribution

It constructs a canonical bundle and frame for pairs of distributions, establishing a criterion for their equivalence, and applies this to solve contact equivalence of certain ODE systems.

Findings

Canonical bundle with a canonical frame constructed

Equivalence of pairs characterized by diffeomorphic frames

New solution to contact equivalence of systems of ODEs

Abstract

We consider a problem of equivalence of generic pairs $(X,V)$ on a manifold $M$, where $V$ is a distribution of rank $m$ and $X$ is a distribution of rank one. We construct a canonical bundle with a canonical frame. We prove that two pairs are equivalent if and only if the corresponding frames are diffeomorphic. As a particular case, with $V$ integrable, we provide a new solution to the problem of contact equivalence of systems of $m$ ordinary differential equations: $x^{(k+1)}=F(t,x,x',...,x^{(k)})$, where $k>2$ or $k=2$ and $m>1$.