Using an old method of Jacobi to derive Lagrangians: a nonlinear dynamical system with variable coefficients

TL;DR

This paper revisits Jacobi's classical method of using the Jacobi Last Multiplier to derive Lagrangians for second-order differential equations, demonstrating its efficiency and ability to produce multiple Lagrangians, including new ones.

Contribution

It applies Jacobi's method to a nonlinear dynamical system with variable coefficients, revealing multiple Lagrangians and showcasing its advantages over previous methods.

Findings

Jacobi's method efficiently derives multiple Lagrangians.

The method reproduces known Lagrangians and finds new ones.

It simplifies the process compared to previous lengthy calculations.

Abstract

We present a method devised by Jacobi to derive Lagrangians of any second-order differential equation: it consists in finding a Jacobi Last Multiplier. We illustrate the easiness and the power of Jacobi's method by applying it to the same equation studied by Musielak et al. with their own method [Musielak ZE, Roy D and Swift LD. Method to derive Lagrangian and Hamiltonian for a nonlinear dynamical system with variable coefficients. Chaos, Solitons & Fractals, 2008;58:894-902]. While they were able to find one particular Lagrangian after lengthy calculations, Jacobi Last Multiplier method yields two different Lagrangians (and many others), of which one is that found by Musielak et al, and the other(s) is(are) quite new.