Jacobi vector fields for Lagrangian systems on algebroids

TL;DR

This paper explores the geometric properties of Jacobi vector fields in Lagrangian systems on algebroids, revealing their solutions' relation to Euler-Lagrange equations on non-standard algebroids and linking second variation nullity to conjugate points.

Contribution

It demonstrates that JVFs are solutions of EL equations on skew-symmetric algebroids, providing a new geometric perspective and a simplified proof of their relation to second variation nullity.

Findings

JVFs satisfy EL equations on non-standard algebroids

Connection established between null space of second variation and conjugate points

Provides a geometric, non-computational proof of key variational properties

Abstract

We study the geometric nature of the Jacobi equation. In particular we prove that Jacobi vector fields (JVFs) along a solution of the Euler-Lagrange (EL) equations are themselves solutions of the EL equations but considered on a non-standard algebroid (different from the tangent bundle Lie algebroid). As a consequence we obtain a simple non-computational proof of the relation between the null subspace of the second variation of the action and the presence of JVFs (and conjugate points) along an extremal. We work in the framework of skew-symmetric algebroids.