A geometric setting for systems of ordinary differential equations

TL;DR

This paper extends the geometric framework for second order ODEs to higher order systems, defining a canonical nonlinear connection and analyzing its curvature to understand the behavior of solutions.

Contribution

It develops a geometric setting that assigns a canonical nonlinear connection to higher order ODE systems, including explicit curvature computations and applications.

Findings

Derived a Jacobi equation for higher order ODEs

Computed all curvature components of the nonlinear connection

Expressed invariants related to third and fourth order ODEs

Abstract

To a system of second order ordinary differential equations (SODE) one can assign a canonical nonlinear connection that describes the geometry of the system. In this work we develop a geometric setting that allows us to assign a canonical nonlinear connection also to a system of higher order ordinary differential equations (HODE). For this nonlinear connection we develop its geometry, and explicitly compute all curvature components of the corresponding Jacobi endomorphism. Using these curvature components we derive a Jacobi equation that describes the behavior of nearby geodesics to a HODE. We motivate the applicability of this nonlinear connection using examples from the equivalence problem, the inverse problem of the calculus of variations, and biharmonicity. For example, using components of the Jacobi endomorphism we express two Wuenschmann-type invariants that appear in the study of scalar third or fourth order ordinary differential equations.