Liouville’s theorem follows from the continuity of transport of some conserved quantity. If this quantity is not energy, then you don’t need time-translation invariance. For example, forced oscillations (with explicitly time-dependent force, like first pushing a child on a swing harder and harder and then letting the swing relax to a stop) still obey the theorem.
Liouville’s theorem follows from the continuity of transport of some conserved quantity. If this quantity is not energy, then you don’t need time-translation invariance. For example, forced oscillations (with explicitly time-dependent force, like first pushing a child on a swing harder and harder and then letting the swing relax to a stop) still obey the theorem.