Infinitesimal symmetries in Contact Hamiltonian systems

TL;DR

This paper extends Noether's theorem to contact Hamiltonian systems, classifying infinitesimal symmetries and identifying dissipated quantities, revealing that symmetries lead to energy-dissipating functions rather than conserved quantities.

Contribution

It introduces a classification of infinitesimal symmetries in contact systems and shows how these relate to dissipated quantities, generalizing Noether's theorem.

Findings

Infinitesimal symmetries in contact systems lead to dissipated functions.

Quotients of dissipated functions and energy are conserved.

Extension of Noether's theorem to contact dynamics.

Abstract

In this paper, we extend the well-known Noether theorem for Lagrangian systems to contact Lagrangian systems. We introduce a classification of infinitesimal symmetries and obtain the corresponding dissipated quantities. We notice that in contact dynamics, the existence of infinitesimal symmetries does not produce conserved quantities, but functions that dissipate at the same rate than the energy; so, the corresponding quotients are true conserved quantities.