Hamiltonians representing equations of motion with damping due to friction
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 $$ \dot q = \frac{\partial H}{\partial p}(q,p) ,...
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| Format: | Article |
| Language: | English |
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Texas State University
2014-04-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2014/89/abstr.html |
| _version_ | 1818015520218152960 |
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| author | Stephen Montgomery-Smith |
| author_facet | Stephen Montgomery-Smith |
| author_sort | Stephen Montgomery-Smith |
| collection | DOAJ |
| description | Suppose that $H(q,p)$ is a Hamiltonian on a manifold M, and
$\tilde L(q,\dot q)$, the Rayleigh dissipation function, satisfies
the same hypotheses as a Lagrangian on the manifold M.
We provide a Hamiltonian framework that gives the equation
$$
\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)
$$
The method is to embed M into a larger framework where the motion drives
a wave equation on the negative half line, where the energy in the wave
represents heat being carried away from the motion.
We obtain a version of Nother's Theorem that is valid for dissipative systems.
We also show that this framework fits the widely held view of how Hamiltonian
dynamics can lead to the ``arrow of time.'' |
| first_indexed | 2024-04-14T06:58:13Z |
| format | Article |
| id | doaj.art-8b39324e3f7d42a297451dbe545537c2 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-04-14T06:58:13Z |
| publishDate | 2014-04-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-8b39324e3f7d42a297451dbe545537c22022-12-22T02:06:49ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912014-04-01201489,110Hamiltonians representing equations of motion with damping due to frictionStephen Montgomery-Smith0 Univ. of Missouri, Columbia, MO, USA Suppose that $H(q,p)$ is a Hamiltonian on a manifold M, and $\tilde L(q,\dot q)$, the Rayleigh dissipation function, satisfies the same hypotheses as a Lagrangian on the manifold M. We provide a Hamiltonian framework that gives the equation $$ \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.''http://ejde.math.txstate.edu/Volumes/2014/89/abstr.htmlHamiltonianLagrangianRayleigh dissipation functionfrictionNother's Theorem |
| spellingShingle | Stephen Montgomery-Smith Hamiltonians representing equations of motion with damping due to friction Electronic Journal of Differential Equations Hamiltonian Lagrangian Rayleigh dissipation function friction Nother's Theorem |
| title | Hamiltonians representing equations of motion with damping due to friction |
| title_full | Hamiltonians representing equations of motion with damping due to friction |
| title_fullStr | Hamiltonians representing equations of motion with damping due to friction |
| title_full_unstemmed | Hamiltonians representing equations of motion with damping due to friction |
| title_short | Hamiltonians representing equations of motion with damping due to friction |
| title_sort | hamiltonians representing equations of motion with damping due to friction |
| topic | Hamiltonian Lagrangian Rayleigh dissipation function friction Nother's Theorem |
| url | http://ejde.math.txstate.edu/Volumes/2014/89/abstr.html |
| work_keys_str_mv | AT stephenmontgomerysmith hamiltoniansrepresentingequationsofmotionwithdampingduetofriction |