Electron. J. Diff. Equ.,
Vol. 2014 (2014), No. 89, pp. 110.
Hamiltonians representing equations of motion with damping due to friction
Stephen MontgomerySmith
Abstract:
Suppose that
is a Hamiltonian on a manifold M, and
,
the Rayleigh dissipation function, satisfies
the same hypotheses as a Lagrangian on the manifold M.
We provide a Hamiltonian framework that gives the equation
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 2, 2014.
Math Subject Classifications: 70H25.
Key Words: Hamiltonian; Lagrangian; Rayleigh dissipation function;
friction; Nother's Theorem.
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Stephen MontgomerySmith
Department of Mathematics, University of Missouri
Columbia, MO 65211, USA
email: stephen@missouri.edu

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