Phase space

Phase space is the space of complete classical states. For one particle in three dimensions, a point in phase space contains position and momentum:

$$(x,y,z,p_x,p_y,p_z).$$

For $N$ particles it has $6N$ coordinates. Susskind’s compact slogan is: configuration space plus momentum space equals phase space.

A motion is not drawn as a path through ordinary space, but as a trajectory of the system point through phase space. Once the initial point is known, Hamilton’s equations in Hamiltonian mechanics determine the phase-space flow.

Why it matters

• Conservation laws restrict which regions of phase space can be reached.
• Poisson brackets describe how quantities change along the flow.
• The phase-space view prepares statistical mechanics and the quantum idea that position and momentum cannot both be specified arbitrarily sharply.

Mini example

A one-dimensional harmonic oscillator has phase coordinates $(x,p)$. Its trajectory in phase space is a closed loop: $x$ and $p$ trade off as energy moves between kinetic and potential form.

Common pitfalls

• Phase space is not physical space with extra decoration; each point is an entire state.
• A single particle in 3D gives a six-dimensional phase space, not a three-dimensional one.
• Momentum coordinates depend on the choice of generalized coordinates.

Source trail: Susskind The Theoretical Minimum index; reference book: Classical Mechanics: The Theoretical Minimum by Leonard Susskind and George Hrabovsky.