| Summary: | 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.''
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