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.

 

[PJC01]

Phase space trajectories play a crucial role in the description of physical systems, but they do not completely specify a system. Phase space, which is a mathematical space that represents all possible states of a system, is a useful tool for analyzing and understanding the behavior of a system. The trajectories in phase space show the evolution of a system over time, and they provide valuable information about the system's dynamics.

However, as the example of the one-dimensional harmonic oscillator mentioned, the phase space trajectory alone does not fully describe the system. The initial conditions, such as the position and momentum of the oscillator at t=0, also play a significant role in determining the behavior of the system. These initial conditions are not explicitly shown in the phase space trajectory, but they are essential for understanding the system's behavior.

Furthermore, in more complex systems, the phase space trajectory can become chaotic or unpredictable, making it impossible to fully determine the system's future behavior from the trajectory alone. In these cases, other factors such as external influences or quantum effects must also be considered.

In summary, phase space trajectories are a valuable tool for understanding the behavior of physical systems, but they do not provide a complete description of the system. Other factors, such as initial conditions and external influences, must also be taken into account for a comprehensive understanding of a system.