The volume of a region of phase space is preserved as it evolves under Hamiltonian dynamics. Named for Joseph Liouville, it is a foundational conservation law of classical mechanics and statistical mechanics, distinct from the same mathematician's theorem in complex analysis.
Facts
StatementThe theorem states that the phase space distribution function of a Hamiltonian system is constant along the system's trajectories: the density of system points near any given point stays constant with time as the system evolves. 1 Classification
Statement Form Liouville's Theorem (Hamiltonian Mechanics)
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Source Joseph Liouville (Wikipedia)
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Source Liouville's Theorem (Hamiltonian Mechanics) (Wikipedia)
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1. Liouville's Theorem (Hamiltonian Mechanics) (Wikipedia)
Wikimedia FoundationWikipedia, Liouville's theorem (Hamiltonian), lead section
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.
Proved By: Joseph Liouville, Lead paragraph
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian
In Group: Classical Mechanics and Dynamics, Lead section
Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that the phase-space distribution
View the Source Joseph Liouville (Wikipedia)
Principal Result Of: Joseph Liouville, Article textQuote, Principal Result Of: Joseph Liouville, Article text
Many mathematical concepts are named after him, including Liouville's theorem in Hamiltonian mechanics and Liouville's theorem for complex numbers.
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