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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 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
One of these examples is Liouville's constant L = 0.110001000000000000000001 … , {\displaystyle L=0.110001000000000000000001\ldots ,} in which the n th digit after the decimal point is 1 if n {\displaystyle n} is the factorial of a positive integer and 0 otherwise.
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 ...
The Liouville lambda function, denoted by λ(n) and named after Joseph Liouville, is an important arithmetic function. Its value is +1 if n is the product of an even number of prime numbers , and −1 if it is the product of an odd number of primes.
Symbol Name Meaning SI unit of measure alpha: alpha particle: angular acceleration: radian per second squared (rad/s 2) fine-structure constant: unitless beta: velocity in terms of the speed of light c: unitless beta particle: gamma: Lorentz factor: unitless photon: gamma ray: shear strain: radian
This is a Liouville dynamical system if ξ and η are taken as φ 1 and φ 2, respectively; thus, the function Y equals Y = cosh 2 ξ − cos 2 η {\displaystyle Y=\cosh ^{2}\xi -\cos ^{2}\eta }
In classical mechanics, for a body with constant mass, the (vector) acceleration of the body's center of mass is proportional to the net force vector (i.e. sum of all forces) acting on it (Newton's second law): = =, where F is the net force acting on the body, m is the mass of the body, and a is the center-of-mass acceleration.