# Do forces always work in the hamiltonian mechanics?

1. Sep 1, 2010

### nonequilibrium

Do forces always "work" in the hamiltonian mechanics?...

Hello,

I'll phrase my question two times: once for the people in a hurry, and a second time in a broader way:

1) short If I write down the equation of a force, can it be that Hamiltonian mechanics doesn't 'apply' to it? I was reading a paper where they wrote down $$\overline{F} = \overline{v} \times \left( \overline{x} \times \overline{v} \right)$$ and there they concluded that if such a force works on some particles in a torus, the phase space volume of their macrostate would change, and as Liouville's Theorem says such a volume is constant in Hamiltonian Mechanics, this would mean this force somehow doesn't fit in the Hamiltonian Mechanics system? Is this weird? I thought it applied to all particles and force fields (in classical mechanics)...

2) longer I was reading up one some papers on Maxwell's demon (the one about the 2nd law that controls the microstate with all his knowledge) and in the paper "Mechanical models of Maxwell’s demon with noninvariant phase volume" (it can be found on google, didn't know if I was allowed to link) they wanted to try and build such a demon by constructing a force field that would act on all the particles in such a way as to lower the entropy. Anyway, in part B they define $$\overline{F} = \overline{v} \times \left( \overline{x} \times \overline{v} \right)$$. They then notice the phase volume is not constant. They apparently do not find this odd, because as they say they don't work with the Hamiltonian Mechanics. I was under the impression the Hamiltonian Mechanics is (one equivalent formulation of) the whole of classical mechanics; does this imply that I'm wrong and that forces have to fulfill a certain requirement before you can work with them in Hamiltonian Mechanics? What else is "out there"?

Thank you.