Liouville's theorem states that the phase space volume of a system is conserved over time. Intuitively, this seems to imply that if a system is at some time constrained to, say, a curve in phase space, the length of that curve should be conserved over time. Is this true? If it isn't, is there a comparable theorem for dealing with systems like this?

I am confused by this because:

  1. Phase space is not a metric space, so I'm not sure if it is sensible to talk about the "length" of a curve in phase space.
  2. It does not seem to be true for any of the simple systems that I can think of. For example, see the phase space plot of a Zeldovich plane wave collapse on page two of these lecture notes, a common test for N-body simulations. The "length" of the curve representing the system seems to grow over time.
  3. It certainly seems like relations which rely on the correctness of the Liouville theorem in their derivations are still true for systems like these.

1 Answer 1


Comments to the question (v2):

  1. Liouville's theorem is about a full $2n$-dimensional volume (of a region) of phase space, not 1D lengths. It may fail in projected versions of phase space.

  2. Note that phase space is used in several meanings in physics. Here we mean a symplectic manifold $(M,\omega)$, such as, e.g. a cotangent-bundle. A tangent-bundle may not work. Liouville's theorem may also fail if the system is not on Hamiltonian form.

  • $\begingroup$ Okay, that's what I was afraid of. Is there a reason, then, that things like the Boltzmann equation still appear to hold for phase sheets and the like? $\endgroup$
    – mansfield
    Jan 25, 2015 at 23:42
  • $\begingroup$ Which system are you thinking of? What is the Hamiltonian? $\endgroup$
    – Qmechanic
    Jan 26, 2015 at 0:07
  • $\begingroup$ Cold dark matter distributions are frequently approximated as three-dimensional phase sheets since the velocity dispersion as a given point is vastly smaller than any other characteristic scale. (That is what is being done in the notes I linked to earlier.) $\endgroup$
    – mansfield
    Jan 26, 2015 at 2:02

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