# Phase Space

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## phase space

[′fāz ‚spās]*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.

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Phase Space

in classical and statistical mechanics, the multidimensional space of all generalized coordinates *q _{i}* and generalized momenta

*p*(

_{i}*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 *q _{1}, p_{1}, . . ., q_{N}, p_{N}*. 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.