On contact equivalence of systems of ordinary differential equations

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: $x^{(k+1)}=F(t,x,x^{\prime },...,x^{(k)})$, where $k>2$ or $k=2$ and $m>1$.