Conjugate points for systems of second-order ordinary differential equations

TL;DR

This paper extends the concept of conjugate points and Jacobi fields to systems of second-order ODEs, exploring their properties in generalized symmetric spaces and Lagrangian systems with symmetry, with examples and applications.

Contribution

It generalizes the notion of conjugate points and Jacobi fields to systems of second-order ODEs and applies these concepts to symmetric spaces and Lagrangian systems with symmetry groups.

Findings

Characterization of conjugate points via eigendistributions of the Jacobi endomorphism.

Analysis of conjugate points along relative equilibria in symmetric Lagrangian systems.

Examples illustrating the theoretical concepts and potential applications.

Abstract

We recall the notion of Jacobi fields, as it was extended to systems of second-order ordinary differential equations. Two points along a base integral curve are conjugate if there exists a non-trivial Jacobi field along that curve that vanishes on both points. Based on arguments that involve the eigendistributions of the Jacobi endomorphism, we discuss conjugate points for a certain generalization (to the current setting) of locally symmetric spaces. Next, we study conjugate points along relative equilibria of Lagrangian systems with a symmetry Lie group. We end the paper with some examples and applications.