Why Can't Flows in Phase Space Cross?

[Eidos]

Hello ladies and gentlemen

Why can't flows in phase space cross?
Would it imply that the system may be at the same state at some future time and then follow a different trajectory? That is to say that the identical initial condition gives a different final condition.

To my mind, flows in phase space would only not cross if the system is time invariant.

Slightly related, non-dissipative systems have their volumes preserved in phase space (Liouville's Theorem), is that the total volume of the phase space or any selectable portion of it?

Thanks for any replies

 

[Parlyne]

If the Hamiltonian has explicit time dependence, you can always extend the phase space to include t and E as extra dimensions (in fact, in relativity, one should always include these - conservation of energy will, then, define a surface in the phase space which the system is constrained to remain on). With this extension, it should be clear that even without time invariance, the phase space flows will not cross unless the system is not deterministic.

 

[Eidos]

Cool thanks that clears up a number of things that have been troubling me.