Hamiltonians representing equations of motion with damping due to friction

TL;DR

This paper develops a Hamiltonian framework for systems with damping due to friction, incorporating dissipation into Hamiltonian mechanics and extending Noether's theorem to dissipative systems.

Contribution

It introduces a novel Hamiltonian approach that models damping via wave equations and provides a version of Noether's theorem applicable to dissipative systems.

Findings

Embedded damping in Hamiltonian systems via wave equations

Extended Noether's theorem for dissipative dynamics

Framework aligns Hamiltonian mechanics with the arrow of time

Abstract

Suppose that $H(q,p)$ is a Hamiltonian on a manifold $M$, and $\tilde L(q,\dot q)$, the Rayleigh dissipation function, satisfies the same hypotheses as a Lagrangian on the manifold $M$. We provide a Hamiltonian framework that gives the equation $\dot q = \frac{\partial H}{\partial p}(q,p), \quad \dot p = - \frac{\partial H}{\partial q}(q,p) - \frac{\partial \tilde L}{\partial \dot q}(q,\dot q)$. The method is to embed $M$ into a larger framework where the motion drives a wave equation on the negative half line, where the energy in the wave represents heat being carried away from the motion. We obtain a version of N\"other's Theorem that is valid for dissipative systems. We also show that this framework fits the widely held view of how Hamiltonian dynamics can lead to the "arrow of time."