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I'm trying to understand the relationship between the two conservation laws. As I understand, Liouville's result is a weaker condition: it relies merely on the particular form assumed by Hamilton's equations and their correspondent hamiltonian vector field, while conservation of energy in the hamiltonian formalism requires the Hamiltonian to be explicitly independent of time. I'd go as far as saying it's a deeper principle in this respect.

On the other hand, its direct implications for the trajectories and the motion of particles are quite elusive to me. I've seen mostly physical systems where both energy and phase space volume get conserved and I've seen a bunch of dissipative systems, say a damped oscillator, where both of them aren't conserved. I'm looking for examples somewhere in the middle in order to capture their respective uniqueness: phase space volume preservation but no energy conservation, a hypothetical situation where energy is conserved but Liouville's theorem doesn't hold and so on. I'd rather know an example out of classical mechanics than statistical mechanics, which I know it's where Liouville's theorem comes particularly in handy but I know almost nothing about it, but please go ahead if you think it can shed some light on the matter.

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2 Answers 2

Actually, the Liouville theorem is more general - it is valid even if the distribution function depends on time, and even if the Hamiltonian depends on time.

http://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian)

-> phase space volume preservation but no energy conservation: any Hamiltonian which depends on time, but you already know that. For example, system of free particles under action of prescribed time-dependent forces.

-> energy is conserved but Liouville's theorem doesn't hold : this is harder to find. Liouville theorem is valid for every normal Hamiltonian, so we have to look for non-Hamiltonian system which nevertheless has energy and this is conserved. The only thing that may have such behaviour that comes to my mind is non-holonomic system with some nasty moving constraints, like ball on a plane with no slipping. Based on what Goldstein says in the 2nd chapter of his book, I think that for such systems there may not be Hamiltonian, ergo no Liouville theorem. One has rather the basic Newtonian equations of motion and constraint equations and inequalities - energy then can be defined as sum of kinetic energies and may be conserved.

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Liouville's theorem not only depends on the form of Hamilton's equations but also on the fact that $\partial\rho/\partial t = 0$, where $\rho$ is the statistical distribution function of the system. This is strictly true only for closed systems and is approximately true for quasi-closed systems when not observed for too long a time.

Energy of a system is conserved when its Lagrangian, and therefore also the Hamiltonian, is not explicitly dependent on time. For macroscopic bodies, this will be true only for a closed system and approximately true for a quasi-closed system over a short enough duration of time.

A duration of time is considered long or short on how it compares with the relaxation time of the system. Relaxation time is, roughly, the time the system takes to adjust to its environment.

Thus, both Liouville's theorem and energy conservation are true over the same time scale and under the same conditions. In that sense, they are equivalent.

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