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
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Proved By
Source Liouville's Theorem (Hamiltonian Mechanics) (Wikipedia)
Sources
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
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