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

Hamiltonians representing equations of motion with damping due to friction