Lie symmetries, Jacobi last multipliers and new non-standard Lagrangians for dissipative dynamical systems

TL;DR

This paper introduces a novel method leveraging Lie symmetries and Jacobi last multipliers to systematically derive multiple non-standard Lagrangians for dissipative dynamical systems, demonstrated on free particles and harmonic oscillators.

Contribution

It presents a new approach to find non-standard Lagrangians for dissipative systems using Lie symmetries and Jacobi last multipliers, enabling generation of multiple Lagrangians from a single system.

Findings

New non-standard Lagrangians for dissipative systems derived

Method applied successfully to free particle and harmonic oscillator

Simplifies obtaining Lagrangians for complex dissipative dynamics

Abstract

We present a new method based on Lie symmetries and Jacobi last multipliers which allows one to find many non-standard Lagrangians for dissipative dynamical systems. In particular, it is demonstrated that for every non-standard Lagrangian one can generate a new non-standard Lagrangian associated to a new equation of motion. We point out that the knowledge of Lie symmetries for a given dynamical system generates Jacobi last multipliers which can be used to obtain new non-standard Lagrangians for dissipative dynamical systems in a simple and straightforward way. We exemplify the new method by applying it to the case of the free particle and the simple harmonic oscillator in order to obtain new non-standard Lagrangians for dissipative systems.