An old Method of Jacobi to find Lagrangians

TL;DR

This paper revisits Jacobi's classical method for deriving Lagrangians of second-order differential equations, demonstrating its effectiveness and broader applicability compared to a recent modern approach.

Contribution

It introduces Jacobi's method based on Jacobi Last Multipliers for finding multiple Lagrangians, expanding on recent methods by Ibragimov.

Findings

Jacobi's method can derive many Lagrangians for second-order equations.

The Lagrangians from Ibragimov are special cases of Jacobi's broader set.

Jacobi's method is simple and elegant for obtaining Lagrangians.

Abstract

In a recent paper by Ibragimov [N. H. Ibragimov, Invariant Lagrangians and a new method of integration of nonlinear equations, J. Math. Anal. Appl. 304 (2005) 212--235] a method was presented in order to find Lagrangians of certain second-order ordinary differential equations admitting a two-dimensional Lie symmetry algebra. We present a method devised by Jacobi which enables to derive (many) Lagrangians of any second-order differential equation. The method is based on the search of the Jacobi Last Multipliers of the equations. We exemplify the simplicity and elegance of Jacobi's method by applying it to the same two equations as did Ibragimov. We show that the Lagrangians obtained by Ibragimov are particular cases of some of the many Lagrangians that can be obtained by Jacobi's method.