# New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries

## TL;DR

This paper advances the contact Hamiltonian and Lagrangian formalisms for dissipative systems, introducing new equations, analyzing symmetries, and establishing conserved quantities, with applications to classical dissipative models.

## Contribution

It presents a new form of contact dynamical equations, reviews and compares two Lagrangian formalisms, and develops symmetry and dissipation concepts for these systems.

## Key findings

- New contact dynamical equations introduced
- Equivalence of two Lagrangian formalisms established
- Method for constructing conserved quantities in dissipative systems

## Abstract

We provide new insights into the contact Hamiltonian and Lagrangian formulations of dissipative mechanical systems. In particular, we state a new form of the contact dynamical equations, and we review two recently presented Lagrangian formalisms, studying their equivalence. We define several kinds of symmetries for contact dynamical systems, as well as the notion of dissipation laws, prove a dissipation theorem and give a way to construct conserved quantities. Some well-known examples of dissipative systems are discussed.