Hamiltonians representing equations of motion with damping due to friction

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 Nother'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.''

Keywords

• Hamiltonian
• Lagrangian
• Rayleigh dissipation function
• friction
• Nother's Theorem