Symmetries, constants of the motion and reduction of mechanical systems with external forces

TL;DR

This paper explores how symmetries and conserved quantities in mechanical systems are affected by external forces, providing a Noether's theorem extension, reduction methods, and practical examples within symplectic geometry.

Contribution

It extends Noether's theorem and reduction techniques to Lagrangian systems with external forces, including dissipation, within a symplectic geometric framework.

Findings

Derived a Noether's theorem for systems with external forces

Developed reduction methods for invariant external forces

Illustrated results with examples involving dissipation

Abstract

This paper is devoted to the study of mechanical systems subjected to external forces in the framework of symplectic geometry. We obtain a Noether's theorem for Lagrangian systems with external forces, among other results regarding symmetries and conserved quantities. We particularize our results for the so-called Rayleigh dissipation, i.e., external forces that are derived from a dissipation function, and illustrate them with some examples. Moreover, we present a theory for the reduction of Lagrangian systems subjected to external forces which are invariant under the action of a Lie group.