Question about Phase Space

[Muthumanimaran]

Can Phase Space be break down into different regions, different regions that are not mixed up with one another, if we do so, the different regions can raise to any conservation of physical quantity?

 

[Simon Bridge]

The phase space is the set of all possible states of the system... along with the rules for moving from one state to the next.
It may be that not all states will be available to every initial condition.

It is possible that the trajectories are manifestations of a conservation principle that acts locally to a region in the phase space.
But this is a pretty general topic ... part of the field of topology iirc. Did you have any particular kind of phase space in mind?

 

[wrobel]

just consider a particle in the field with potential $V(x)=e^x+1/x$. The phase space consists of two pieces $\{(x,\dot x)\mid x>0\}$ and $\{(x,\dot x)\mid x<0\}$