Jacobi-Maupertuis metric of Lienard type equations and Jacobi last multiplier

Abstract

We present a construction of the Jacobi-Maupertuis (JM) principle for an equation of the Lienard type, $$ \ddot{x} + f(x) \dot{x}^2 + g(x) = 0, $$ 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.

Keywords

• Jacobi-Maupertuis metric
• position-dependent mass
• Jacobi's last multiplier