Lagrangians Galore

TL;DR

This paper demonstrates that using Lie symmetries and the Jacobi last multiplier can generate numerous Lagrangians for simple mechanical models, simplifying the search for Lagrangians in classical mechanics.

Contribution

It introduces a method leveraging Lie symmetries and the Jacobi last multiplier to systematically produce multiple Lagrangians for classical mechanical systems.

Findings

The method yields many Lagrangians for simple models.

It applies to the harmonic oscillator and damped harmonic oscillator.

The approach simplifies the process of finding Lagrangians.

Abstract

Searching for a Lagrangian may seem either a trivial endeavour or an impossible task. In this paper we show that the Jacobi last multiplier associated with the Lie symmetries admitted by simple models of classical mechanics produces (too?) many Lagrangians in a simple way. We exemplify the method by such a classic as the simple harmonic oscillator, the harmonic oscillator in disguise [H Goldstein, {\it Classical Mechanics}, 2nd edition (Addison-Wesley, Reading, 1980)] and the damped harmonic oscillator. This is the first paper in a series dedicated to this subject.