Jacobi equations using a variational principle

TL;DR

This paper introduces a variational principle that simultaneously derives the Jacobi equations and Lagrange equations for systems with a Lagrangian, aiding in stability analysis and constant of motion identification.

Contribution

It presents a novel variational approach to obtain Jacobi equations alongside Lagrange equations, enhancing analysis of system stability and conserved quantities.

Findings

Identified a constant of motion in Jacobi equations for autonomous systems

Recovered classical stability conditions for two-dimensional orbits

Provided a method to analyze system stability using the variational principle

Abstract

A variational principle is proposed for obtaining the Jacobi equations in systems admitting a Lagrangian description. The variational principle gives simultaneously the Lagrange equations of motion and the Jacobi variational equations for the system. The approach can be of help in finding constants of motion in the Jacobi equations as well as in analysing the stability of the systems and can be related to the vertical extension of the Lagrangian formalism. To exemplify two of such aspects, we uncover a constant of motion in the Jacobi equations of autonomous systems and we recover the well-known sufficient conditions of stability of two dimensional orbits in classical mechanics.