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In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics.It asserts that the phase-space distribution function is constant along the trajectories of the system—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time.
Liouville's formula is a generalization of Abel's identity and can be used to prove it. Since Liouville's formula relates the different linearly independent solutions of the system of differential equations, it can help to find one solution from the other(s), see the example application below.
This is a special case of the general problem of Sturm–Liouville theory. If a and b are positive, the eigenvalues are all positive, and the solutions are trigonometric functions. A solution that satisfies square-integrable initial conditions for u and u t can be obtained from expansion of these functions in the appropriate trigonometric series.
In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation. Liouville theory is defined for all complex values of the central charge c {\displaystyle c} of its Virasoro symmetry algebra , but it is unitary only if
In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844 [1]), states that every bounded entire function must be constant. That is, every holomorphic function f {\displaystyle f} for which there exists a positive number M {\displaystyle M} such that | f ( z ) | ≤ M ...
Liouville's theorem has various meanings, all mathematical results named after Joseph Liouville: In complex analysis, see Liouville's theorem (complex analysis) There is also a related theorem on harmonic functions
For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelfand equation. In differential geometry, Liouville's equation, named after Joseph Liouville, [1] [2] is the nonlinear partial differential equation satisfied by the conformal factor f of a metric f 2 (dx 2 + dy 2) on a surface of constant Gaussian curvature K:
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]