I have some doubts about Liouville theorem, probably its just something conceptual.

So: I know that for a system in which Liouville’s theorem holds, the volume in the phase space is conserved.

But the conservation of the volume does immediately imply the absence of asymptotically stable points.

However, if the Hamiltonian is time dependent and in particular of its time derivative is negative along the phase curves, a system does possess asymptotically stable points.

For that system, Hamilton equations still hold, hence it’s an Hamiltonian system.

But Liouville’s theorem doesn’t hold anymore.

My question: for what kind of systems does Liouville’s theorem hold? For example: the dumped oscillator has one asymptotically stable point.

  • $\begingroup$ I see, we are using different notions of asymptotically stable points! I am deleting my answer because it is wrong! $\endgroup$ – Valter Moretti Aug 6 '18 at 11:58
  • $\begingroup$ What is the dumped harmonic oscillator? Or did you mean the damped harmonic oscillator? $\endgroup$ – Walter Aug 6 '18 at 17:22
  • $\begingroup$ I don't think the damped harmonic oscillator (with equation of motion $\ddot{x}+a\dot{x}+bx=0$) is a Hamiltonian system. Dissipation is inherently non-Hamiltonian and violates Liouville's theorem. $\endgroup$ – Walter Aug 6 '18 at 17:23

Liouville's theorem holds for all Hamiltonian systems.

If your definition of an asymptotically stable point $\boldsymbol{x}^*$ means that trajectories from points $\boldsymbol{x}$ in some neighbourhood of $\boldsymbol{x}^*$ tend to $\boldsymbol{x}^*$ as $t\to\infty$, then

  1. phase-space volume and/or density near $\boldsymbol{x}^*$ are not conserved
  2. and the system cannot be Hamiltonian (because of Liouville's theorem).

An example of such a non-Hamiltonian system is indeed the unforced damped harmonic oscillator

$$ \ddot{x} + c\dot{x} + k x = 0 $$

with $c\neq0$.


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