Can Liouville's theorem apply to dissipative systems

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SUMMARY

Liouville's theorem applies to all mechanical systems that do not ignore degrees of freedom, asserting that the phase space of a Hamiltonian system remains constant. In dissipative systems, such as a damped pendulum, the phase space volume appears to contract due to the loss of energy, but this contraction is resolved by considering additional degrees of freedom, such as those of air molecules. Thus, while non-Hamiltonian behavior like friction arises from neglecting certain degrees of freedom, the underlying Hamiltonian nature of mechanical systems remains intact. This discussion clarifies the relationship between Hamiltonian mechanics and dissipative systems.

PREREQUISITES
  • Understanding of Liouville's theorem in Hamiltonian mechanics
  • Familiarity with phase space concepts
  • Knowledge of dissipative systems and their characteristics
  • Basic principles of degrees of freedom in mechanical systems
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  • Research the implications of Liouville's theorem in non-Hamiltonian systems
  • Study the role of degrees of freedom in dissipative systems
  • Explore examples of Hamiltonian and non-Hamiltonian systems in classical mechanics
  • Learn about the impact of friction and damping on phase space trajectories
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Physicists, mechanical engineers, and students of classical mechanics seeking to deepen their understanding of Hamiltonian systems and the implications of Liouville's theorem in the context of dissipative phenomena.

enricfemi
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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.
 
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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:
 

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