Phase space

Phase space has one position coordinate

q_{i}

and one conjugate momentum

p_{i}

for every degree of freedom; a point

(q, p)

specifies an instantaneous mechanical state.

Velocity space ( $v \in R^{3}$ )

d^{3} v = 2 π v_{⊥} d v_{⊥} d v_{∥}

μ = v_{⊥}^{2} / (2 B)

, this gives the measure in the title. When the magnetic field changes, both the measure and any explicit factor

μ B

in a moment must be perturbed.

The symplectic area enclosed by a closed orbit in a two-dimensional phase plane. With the orientation of Hamiltonian motion it is

\oint p d q = 2 π I

, where

I

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.