Third Order ODEs Systems and Its Characteristic Connections

TL;DR

This paper computes a unique characteristic Cartan connection for third order ODE systems, enabling the determination of fundamental invariants and trivializability, with applications to equations on circles in Euclidean space.

Contribution

It introduces a distinct characteristic connection for third order ODE systems, facilitating the computation of invariants and trivializability criteria.

Findings

Derived the characteristic Cartan connection for third order ODE systems.

Identified all fundamental invariants of such systems.

Applied invariants to equations on circles in Euclidean space.

Abstract

We compute the characteristic Cartan connection associated with a system of third order ODEs. Our connection is different from Tanaka normal one, but still is uniquely associated with the system of third order ODEs. This allows us to find all fundamental invariants of a system of third order ODEs and, in particular, determine when a system of third order ODEs is trivializable. As application differential invariants of equations on circles in ${\mathbb R}^n$ are computed.