Electronic Journal of Differential Equations,
Vol. 2014 (2014), No. 89, pp. 1-10.

Title: Hamiltonians representing equations of motion with damping due to friction

Author: Stephen Montgomery-Smith (Univ. of Missouri, Columbia, MO, USA)

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.''

Submitted  January 23, 2014. Published April 02, 2014.
Math Subject Classifications: 70H25.
Key Words: Hamiltonian; Lagrangian; Rayleigh dissipation function;
           friction; Nother's Theorem.