Lagrangians for dissipative nonlinear oscillators: the method of Jacobi Last Multiplier

TL;DR

This paper introduces Jacobi's method for deriving Lagrangians of second-order differential equations, demonstrating its effectiveness on various dissipative nonlinear oscillators and comparing it to Musielak's approach.

Contribution

It presents a systematic application of Jacobi's method to find Lagrangians for dissipative nonlinear oscillators, including Liènard and Riccati equations, showcasing its simplicity and power.

Findings

Successfully derived Lagrangians for several nonlinear oscillators

Demonstrated the method's efficiency compared to existing approaches

Applied the method to complex equations like Liènard and Riccati types

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 several equations and also a class of equations studied by Musielak with his own method [Musielak ZE, Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients. J. Phys. A: Math. Theor. 41 (2008) 055205 (17pp)], and in particular to a Li\`enard type nonlinear oscillator, and a second-order Riccati equation.