# Phase Space

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Related to Phase Space: Liouville's theorem

## 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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from 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 It might be outdated or ideologically biased.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 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.

References in periodicals archive ?
For each respondent, there were 200,000 positions of the electric field in the phase space collected during 200 seconds, where the position of the individual point was defined by the coordinates expressed in millivolts (mV).
Following Table 1, Figure 5 represents the dynamical behaviors and phase space trajectory of the three fish populations against time, beginning with the initial values x(0) = 1.01, y(0) = 1.02, and z(0) = 0.01.
Williams, "Phase space analysis of EEG in temporal lobe epilepsy," in Proceedings of the Annual International Conference of the IEEE Engineering in Medicine and Biology Society, pp.
If d is the true/estimated embedding dimension of the system (i.e., number of variables governing the dynamics of the system), then each state of the system can be represented in the phase space by the d-dimensional vectors of the form [[bar.v].sub.i] given as follows:
Yang, "Probing the noncommutative effects of phase space in the time-dependent Aharonov-Bohm effect," Annals of Physics, vol.
Here N, [lambda] are the normalization and phase constants, respectively; the parameters [p.sub.x],[p.sub.y],[q.sub.x],[q.sub.y] specify a point in the classical phase space of dimension 2n.
So we cannot reconstruct a complete system phase space using only one dimension.
C(r) is the cumulative distribution function, which represents the probability that the distance between two phase points is less than r in phase space.
The plot is very similar (besides a phase shift) to the phase space plot shown in Figure 1(b).
When such system evolves in time, the sequence of all its states forms a trajectory in the phase space, that is, a multidimensional space whose dimension depends on the number of the variables of the system.
Therefore, an innovative technique, phase space-based using a single sensor, for damage detection of a bridge structure excited by a moving mass is here proposed, and the damage index extracted from the response phase space is presented.

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