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 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."