System and phase space trajectory

[Logic Cloud]

To what extent do phase space trajectories describe a system? I often see classical systems being identified with (trajectories in) phase space, from which I get the impression these trajectories are supposed to completely specify a system. However, if you take for example the trajectory x^2+p^2=1 for a one-dimensional harmonic oscillator, it is still left open if x(t=0)=0 or x(t=0)=1 which corresponds to two different parameterizations of the circle. This leads me to ask: what is the role of phase space trajectories in the description of physical systems?

 

[VantagePoint72]

A phase trajectory, by definition, includes a particular parametrization in its specification. It's a path through configuration space, with a path being defined as a continuous map from an interval in the real numbers to the path's range. So, $x^2 + y^2 = 1$ isn't a trajectory, it's just a curve. A corresponding trajectory would be $t\in[0,1) \rightarrow (\cos t, \sin t)$, etc. You can always reparametrize, but then you have a different trajectory.

 

[Logic Cloud]

I see. So there really isn't any need for the concept of phase space for describing physical systems, since the trajectory can be found by just solving the equation of motion directly. My question was motivated by the classical variant of the Dirac-Von Neumann axioms where a classical system is associated with phase space, but maybe I'm reading too much into it.

 

[VantagePoint72]

Trajectories in phase space are just geometric representations of the solutions to the equations of motion. It's not one or the other, they're interchangeable.