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.