Phase space has one position coordinate and one conjugate momentum for every degree of freedom; a point specifies an instantaneous mechanical state.
Velocity space is the space of velocity vectors at a fixed spatial position. Integrating a galactic distribution function over velocity space produces mass density; its normalized moments determine mean velocity and velocity dispersion.
The symplectic area enclosed by a closed orbit in a two-dimensional phase plane. With the orientation of Hamiltonian motion it is , where is the action variable.
An integral curve of a vector field in phase space, representing the positions and momenta of a moving system. A periodic one-dimensional Hamiltonian orbit encloses the area used to define the action variable.
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Phase space is a concept used in physics and mathematics to represent the state of a dynamic system. It is particularly useful in the fields of classical mechanics, statistical mechanics, and quantum mechanics. In phase space, each possible state of a system is represented by a point, with dimensions corresponding to the degrees of freedom of the system.
This idea comes up particularly in the phase space coordinate of Hamiltonian mechanics.