In Hamiltonian mechanics, I know that the time evolution is bijective, and it also conserves phase space volume (Liouville’s theorem). What I dont fully understand is why the volume conservation is necessary for reversibility.
Intuitively, it feels like bijectivity should already be enough: if every point in phase space maps uniquely forward and backward in time, then I should be able to reverse the dynamics right?
But apparently thats not the whole story. So my question is: Why is the conservation of phase space volume (and not just bijectivity) essential for the reversibility of a system in classical mechanics? Im not looking for answers from statistical mechanics or thermodynamics; Ive seen some answers that try to explain this in terms of information preservation for example, saying that volume conservation is connected to the idea that no information is lost during the evolution. But I find that a bit vague. Im not sure what kind of information is meant here, or how exactly this ties into mathematical reversibility. Also, how is that different from bijectivity, which already seems to prevent any forgetting of states?
