# Phase Space

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## phase space

[′fāz ‚spās]
(mathematics)
In a dynamical system or transformation group, the topological space whose points are being moved about by the given transformations.
(statistical mechanics)
For a system with n degrees of freedom, a euclidean space with 2 n dimensions, one dimension for each of the generalized coordinates and one for each of the corresponding momenta.

## Phase Space

in classical and statistical mechanics, the multidimensional space of all generalized coordinates qi and generalized momenta pi (i = 1, 2,. .., N) of a mechanical system with N degrees of freedom. Thus, phase space has 2N dimensions and may be described by means of an orthogonal coordinate system with 2N axes; the number of axes corresponds to the number of generalized coordinates and momenta.

The state of a system is represented in phase space by a point with the coordinates q1, p1, . . ., qN, pN. A change in the state of the system with time is represented by the motion of the point along a line called a phase trajectory. For phase space, the concept of phase volume and other concepts in multidimensional geometry can be introduced.

Phase space is a fundamental concept in classical statistical mechanics, which studies the distribution functions of systems of many particles. Phase-space methods are also used in the theory of nonlinear oscillations.

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