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:
- 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.
- 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.
- It certainly seems like relations which rely on the correctness of the Liouville theorem in their derivations are still true for systems like these.