Jacobi-Maupertuis metric of Lienard type equations and Jacobi Last Multiplier

TL;DR

This paper develops a method to formulate Lienard type equations as geodesic problems using the Jacobi-Maupertuis metric, leveraging Jacobi's last multiplier, and demonstrates it with classical examples.

Contribution

It introduces a novel approach to derive the Jacobi-Maupertuis metric for Lienard equations using Jacobi's last multiplier, enabling geometric reformulation of these equations.

Findings

Reformulation of Lienard equations as geodesic problems.

Application to Painleve-Gambier, Jacobi, and Henon-Heiles systems.

Provides a geometric perspective on variable mass dynamics.

Abstract

We present a construction of the Jacobi-Maupertuis (JM) principle for an equation of the Lienard type, viz \ddot{x} + f(x)x^2 + g(x) = 0 using Jacobi's last multiplier. The JM metric allows us to reformulate the Newtonian equation of motion for a variable mass as a geodesic equation for a Riemannian metric. We illustrate the procedure with examples of Painleve-Gambier XXI, the Jacobi equation and the Henon-Heiles system.