  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."
