Liouville's Theorem

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Liouville's theorem

[′lyü‚vēlz ‚thir·əm]
(mathematics)
Every function of a complex variable which is bounded and analytic in the entire complex plane must be constant.

Liouville’s Theorem

(1) In mechanics, a theorem asserting that the volume in phase space of a system obeying the equations of mechanics in Hamiltonian form remains constant as the system moves. Liouville’s theorem was established in 1838 by the French scientist J. Liouville.

The state of a mechanical system defined by the generalized coordinates q1, q2, . . ., qN and the canonically conjugate generalized momenta p1, p2, . . ., pN (where N is the number of degrees of freedom of the system) can be considered as a point with rectangular Cartesian coordinates q1, q2, ..., qN; p1, p2, . . ., pN is a 2N-dimensional space called the phase space. The evolution of the system in time is represented as the motion of this phase point in the 27V-dimensional space. If phase points entirely fill some region of the phase space at the initial moment of time and pass over into another region of the space in the course of time, then the corresponding phase volume, according to Liouville’s theorem, will be the same. Thus the motion of the points that represent the state of the system in the phase space resembles that of an incompressible fluid.

Liouville’s theorem permits introduction of a distribution function of the particles of the system in phase space and is the basis of statistical physics.

REFERENCES

Synge, J. L. Klassicheskaia dinamika. Moscow, 1963. (Translated from English.)
Gibbs, J. Osnovnye printsipy statisticheskoi mekhaniki. Moscow, 1946. (Translated from English.)
Leontovich, M. A. Statisticheskaia fizika. Moscow-Leningrad, 1944.
D. N. ZUBAREV
(2) In the theory of analytic functions, a theorem asserting that every entire function that is finite on the entire complex plane is identically a constant. Liouville’s theorem is named after J. Liouville, who made it the basis of his lectures (1847) on the theory of elliptical functions. However, it was first formulated and proved by A. Cauchy in 1844.
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While the Liouville theorem has been applied in this context earlier, see [4], the theoretical results of turbulent mixing have not been invoked to address the problem of speckle intensity distribution.
Indeed, if we define the order and the type of the entire function f by equations (1), (2) and (3), then the quaternionic Liouville theorem proven in [10] generalizes as follows.
He covers defining stability, the Gelfand problems extremal solutions, the regularity theory of stable solutions, singular stable solutions, Liouville theorems for stable solutions, a conjecture of Di Giorgi, and further readings.
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