I Can a phase space be subject to change, and if so, what are the implications?

[bahamagreen]

I'm asking if space space is subject to change, if not, why not, and if so, then would it be subject to a subsequent phase space that describes that?

Let us make a phase space for a deterministic time evolved dynamic system of three spatial dimensions.
Since the actions of the system and the phase space are totally correlated, is there a sense in which one is the cause of the other?
It looks to me like the system, being deterministic, cannot independently deviate from its time evolution, so it cannot change its phase space; but if the phase space itself were subject to change, this would alter the system (changing the system's evolution throughout all time)...

First, are there any concepts of phase space that allow itself to be subject to change? I know this raises questions about what kind of "time" within which the phase space would change, being a different concept of time from the system's time.

If phase space were subject to change, it would seem that the next step would be to make a higher phase space "II" to describe the "time" evolution of the first phase space. And if this phase space II were also subject to change, this process might be indefinitely iterative up to phase space "c" where no further changes occur (as in taking the derivative of a function over and over that reaches a point where all subsequent higher derivatives return zero...)

Anyone ever look at this or is it fundamentally preempted by the problem of defining a concept of "time" for a phase space?
Is there anything that just prevents a phase space from being subject to change?

 

[mfb]

If you want to model external influences, you can make a time-dependent phase space. If the internal dynamics plays a role for that, it is probably easier to include the external influence in the phase space. You can always find a system large enough to have a time-independent phase space, unless the laws of physics itself change over time.

 

[bahamagreen]

If you want to model external influences, you can make a time-dependent phase space.
If I am understanding how you understand me, you are equating external influences (external to the space and time dimensions of the system) to a changing phase space (time-dependent in its own dimensions). That is the way I am thinking.

If the internal dynamics plays a role for that, it is probably easier to include the external influence in the phase space.
Some dynamics of the system may be migrated into the phase space, especially dynamics whose fundamental character are subject to change?

You can always find a system large enough to have a time-independent phase space, unless the laws of physics itself change over time.
This also sounds like the way I am thinking. Sounds like it is actually OK for a phase space to have its own kind of time dimension (time-dependent phase space) separate from the system time dimension. Does it then follow, overall, that for any time-dependent phase space we may make for it a phase space which may also be time-dependent, iteratively up to a phase space which is time-independent? I have wondered along these lines before, but was vaguely recalling a hard block with assigning a time evolution to a phase space (maybe because of a similar argument that spacetime is not subject to time evolution - thinking of spacetime as a kind of phase space)... maybe I'm confused about one of both of those?

 

[mfb]

Some dynamics of the system may be migrated into the phase space, especially dynamics whose fundamental character are subject to change?

Yes. Make a larger system with a larger phase space if it helps.
Most tools in physics work better with a time-independent phase space.

Does it then follow, overall, that for any time-dependent phase space we may make for it a phase space which may also be time-dependent, iteratively up to a phase space which is time-independent?

Sure.