- #1
homology
- 306
- 1
Suppose you have a dissipative system where
[tex] \dot{q}=p/m [/tex]
[tex] \dot{p}=-\gamma p -k\sin(q) [/tex]
So there isn't a Hamiltonian for this system and Louiville's theorem doesn't hold. But the equations of motion still give us a vector field on phase space and we can still take the Lie derivative of [tex] \omega = dp\wedge dq [/tex] along it.
If I do that I get [tex] L_X \omega = -\gamma dp\wedge dq [/tex] (let me know if you want to see the details, I'm omitting them here to keep things brief) where X is the dynamical vector field. I'm still new to the geometric understanding of mechanics and so the above 'looks' like a differential equation, but I'm not sure what to make of this? I suppose I'd like to see how to use this to interpret [tex] \omega [/tex] as a function of time, but I'm stuck.
Any ideas are welcome.
[tex] \dot{q}=p/m [/tex]
[tex] \dot{p}=-\gamma p -k\sin(q) [/tex]
So there isn't a Hamiltonian for this system and Louiville's theorem doesn't hold. But the equations of motion still give us a vector field on phase space and we can still take the Lie derivative of [tex] \omega = dp\wedge dq [/tex] along it.
If I do that I get [tex] L_X \omega = -\gamma dp\wedge dq [/tex] (let me know if you want to see the details, I'm omitting them here to keep things brief) where X is the dynamical vector field. I'm still new to the geometric understanding of mechanics and so the above 'looks' like a differential equation, but I'm not sure what to make of this? I suppose I'd like to see how to use this to interpret [tex] \omega [/tex] as a function of time, but I'm stuck.
Any ideas are welcome.
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