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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.
In complex analysis, see Liouville's theorem (complex analysis) There is also a related theorem on harmonic functions; In conformal mappings, see Liouville's theorem (conformal mappings) In Hamiltonian mechanics, see Liouville's theorem (Hamiltonian) and Liouville–Arnold theorem; In linear differential equations, see Liouville's formula
The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696). [2] It immediately occupied the attention of Jacob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733.
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 ...
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
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:
In mathematical physics, Liouville made two fundamental contributions: the Sturm–Liouville theory, which was joint work with Charles François Sturm, and is now a standard procedure to solve certain types of integral equations by developing into eigenfunctions, and the fact (also known as Liouville's theorem) that time evolution is measure ...
newton meter squared per kilogram squared (N⋅m 2 /kg 2) shear modulus: pascal (Pa) or newton per square meter (N/m 2) gluon field strength tensor: inverse length squared (1/m 2) acceleration due to gravity: meters per second squared (m/s 2), or equivalently, newtons per kilogram (N/kg) magnetic field strength