# Can Liouville's theorem apply to dissipative systems

#### enricfemi

form the proof in Hamiltonian, i didn't find any clue.

the problem is i cant understand it even i know how to prove it.

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#### dx

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Gold Member
Liouville's theorem applies to all mechanical systems if you don't ignore degrees of freedom.

#### atyy

Liouville's theorem applies to all mechanical systems if you don't ignore degrees of freedom.
Is this a correct expansion of what you are saying: Liouville's theorem says the phase space of a Hamiltonian system doesn't contract, so systems with attractors or dissipative systems can't be represented by Hamiltonians - but in principle the non-Hamiltonain behaviour comes from ignoring degrees of freedom?

#### dx

Homework Helper
Gold Member
Is this a correct expansion of what you are saying: Liouville's theorem says the phase space of a Hamiltonian system doesn't contract, so systems with attractors or dissipative systems can't be represented by Hamiltonians - but in principle the non-Hamiltonain behaviour comes from ignoring degrees of freedom?
Yes, as far as we know, all mechanical systems are Hamiltonian. Non-Hamiltonian behavior (like friction, for example) is assumed to be due to ignoring degrees of freedom.

#### enricfemi

indeed, i raise the problem because of the attractors.
but can you say it more clearly?
how do we ignore degrees of freedom while dealing with non-Hamiltonian systems?

#### atyy

Ok, let's suppose we have a damped pendulum. After a long time it will stop oscillating, no matter how hard you kicked it initially. So all trajectories in phase space end up at the same place, ie. the phase space volume has contracted.

But if we include all the air molecules which take energy away from the pendulum, then although the pendulum degrees of freedom eventually become identical for all trajectories, the air molecule degrees of freedom remain different, and those degrees of freedom preserve the phase space volume.

that's amazing!

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