Can Liouville's theorem apply to dissipative systems

AI Thread Summary
Liouville's theorem states that the phase space of a Hamiltonian system does not contract, which implies that systems with attractors or dissipative behaviors cannot be accurately represented by Hamiltonians. Non-Hamiltonian behaviors, such as friction, are attributed to the neglect of certain degrees of freedom in the system. For example, in a damped pendulum, while the pendulum's motion eventually ceases, including the air molecules as degrees of freedom maintains the phase space volume. This highlights the importance of considering all relevant degrees of freedom to fully understand mechanical systems. The discussion emphasizes that all mechanical systems can be viewed as Hamiltonian when all degrees of freedom are accounted for.
enricfemi
Messages
195
Reaction score
0
form the proof in Hamiltonian, i didn't find any clue.

the problem is i can't understand it even i know how to prove it.
 
Physics news on Phys.org
Liouville's theorem applies to all mechanical systems if you don't ignore degrees of freedom.
 
dx said:
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?
 
atyy said:
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.
 
Thanks for reply!
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?
 
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!:eek:
 
Back
Top