phase space?

Question

Could anybody give me a precise mathematical definition of a phase space?

I don't need an explanation of it, just the definition. Thanks :)

@PMP: thanx, I saw that on Wikipedia but I'd prefer an exact formal definition. Wikipedia just explains it

Answer 1

Suppose you have a classical 3-dimensional system with N particles. At each point in time, each particle has a position that can be described by 3 coordinates and a momentum that can be described by 3 coordindates. So each particle can be described by 6 numbers. If there are N particles then every particle can be described by 6N numbers.

This means that you can completely define the system by 6N numbers at each point in time. This is referred to as phase space. Each point in phase space describes a 6N coordinate system for at a particular point in time.

Answer 2

If you have a function involving x, and it's 1'st and 2'nd time-derivatives such as
F(x'', x', x)=0
which do not depend explicitly on t, then by substituting something such as v=x' you can usually hammer it into the form of
f(v(dv/dx), v, x)
then the (x, v) plane is called the 'phase-plane' (or phase-space) of f and the curves of f graphed on it are called the 'phase-trajectories' of F.

This can also be generalized to higher orders.
You'll run into it when you take a course on differential equations.

HTH

Doug

Answer 3

For definition see
http://mathworld.wolfram.com/PhaseSpace.html

Answer 4

Wikipedia knows all.