# Is it correct to say that the term "phase space" is valid only when applied to canonically conjugate coordinates and momenta?

Is it correct to say that the term phase space is valid only in relation to canonically conjugate coordinates and momenta?

Is there a simple/short way to terminologically distinguish the phase space of canonically conjugate coordinates and momenta from the phase space of coordinates and momentum that are not canonically conjugate?

Is it correct to say that the term "phase space" is valid only in relation to canonically conjugate coordinates and momenta ?

No. A phase space is a space in which all possible states of a system are represented, and each state corresponds to a unique point in the phase space. But there is no reason why the dimensions in phase space have to correspond to pairs of conjugate (physical) co-ordinates and momenta.

For a simple pendulum you could, for example, use a three-dimensional phase space where the dimensions are angle, horizontal speed and vertical speed. This is probably less useful than the usual two-dimensional angle/angular momentum phase space, but it is still a valid phase space since each state of the pendulum is represented by a unique point.

• What about my 2nd question?
– Igor
Commented Apr 3, 2023 at 9:46
• @Igor As far as I know, there is no special term for a phase space whose dimensions are conjugate co-ordinates and momenta. For the avoidance of doubt you should explicitly state what the dimensions of your phase space are. Commented Apr 3, 2023 at 9:55
• My main interest is simple plasma with point electrons and ions.
– Igor
Commented Apr 3, 2023 at 9:59
• According to my understanding, the statement "each state corresponds to a unique point in the phase space" is not valid in quantum mechanics. Commented Apr 3, 2023 at 10:19
• I would disagree, the standard terminology for a phase space with conjugate coordinates and momenta general is a Hamiltonian phase space (i.e. the phase space of a Hamiltonian system) Commented Apr 3, 2023 at 13:29

There are a few terminologies used specifically for phase spaces of canonically conjugate coordinates (and their generalisations). Such a phase space can either be called a Hamiltonian phase space or the phase space of a Hamiltonian system. The general theory of such spaces, including those with curvature, is called symplectic geometry, and in this field such spaces are referred to as symplectic manifolds.