phase space

[penguin007]

Could anyone explain me what a phase space is and how we can build it??

Thanks in advance.

[0xDEADBEEF]

Phase space is a mathematical construct combining the position and the speed of an object in the same diagram.

[Claude Bile]

In mathematics a "space" is the name given to the range of values that a set of variables can take. A 2D scatter plot for instance is an example of a 2D space, however there is no limit on the number of variables that a space can encompass.

A phase space is a space used to describe a physical system. The state of the physical system can be represented by a coordinate in the phase space.

Claude.

[Bob S]

Look up Liouville's Theorem in Google. for example

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

In simple terms. phase space represents the area (product) of a transverse coordinate x times the slope dx/dz of a large ensemble of particles or rays moving in the z direction, such as a charged particle beam or optical rays (photons). Liouville's Theorem states that the area (product of x and dx/dz) is conserved as the ensemble of particles evolves.

Bob S

[penguin007]

Thanks for all your answers!
I also read a lot of stuff on the internet (amid them Liouville's theorem: even if I didn't understand everything it helped me a lot).
I retained that the coordinates of a phase space are constituted by INDEPENDENT parameters of the system (is that right?).

Thanks again.

Penguin007

[Claude Bile]

Generally speaking, yes, you would construct a phase space with independent variables as it yields the phase space with fewest dimensions.

Technically though the variables do not have to be independent. Chaos theorists for example sometimes work with systems where the number of independent variables (the dimension of the phase space) is unknown.

Claude.