Phase Space

From Vectorcalcumb

What is a Phase Space?

Given any mechanical system, we have a notion of the state of the system at a particular point in time. The state of a system is the set of parameters which specify enough information about the system that given the rules of how the system evolves, we can determine all future states of the system. These rules come from the laws of physics, and are called the kinematics of the system.

Consider a particle constrained to a move along a straight line. This is a wondefully simple mechanical system. It's state is completely determined by only two parameters, position(<m>r</m>), and linear momentum (<m>p</m>). This system's evolution over time is governed completely by Newton's Second Law, <m> F = dp/dt </m>, that is, the rate of change of the linear momentum of the system is equal to the external force acting upon it.

If there is no force acting on the particle<m>F=0</m>, by Newton's law <m> F = dp/dt right dp/dt = 0</m> , it's linear momentum(<m>p</m>) will not change, and therefore, if we know the position(<m>r</m>) of the particle and it's momentum (<m>p</m>) at any point in time, we can determine the momentum and position of the particle at any point in the future. If we take the momentum as one coordinate, and the position as another, we have a 2-dimensional space. This space is what we call the Phase Space of the system. As we have just seen here, the Phase Space for mechanical system that can move in a single dimension is a two dimensional one. We have the position of the particle along the line, and it's momentum at that position, and this determines a unique point in the phase space<m> r * p </m>. This notion generalizes, and for any n-dimensional mechanical system, we have a 2n-dimensional phase space. This turns out to be quite important.

Over the course of the semester, we have developed the idea of a <m> Manifold </m>. Casually, a manifold can be viewed as an n-dimensional object which locally looks like <

2 examples:

1) particle on line:2D plane w/ area form (pg.20)

2) n particles on line: R^(2n) w/ symplectic form (pg. 39)