On equivalence of systems of differential equations
Abstract
We consider the problem of equivalence for systems of ordinary differential equations as a special case of a more general problem of equivalence of distributions $\mathcal V $ on a differentiable manifold $M$ in the presence of a line field $\mathcal X $. We prove that any pair $(\mathcal X ,\mathcal V )$ defines a canonical frame on a certain bundle over $M$. The construction is direct and we provide explicit normalization conditions for the frame. In special cases, when $\mathcal V $ is integrable and an additional set of conditions is satisfied, the pairs encode systems of ordinary differential equations. We show that in this case, the canonical frame defines a Cartan connection. We also provide a construction of higher-order invariants by introducing Schwarzian-like derivatives that act on endomorphisms of vector bundles.
