Liouville measure

E898514

canonical measure measure symplectic invariant volume form

Liouville measure is a canonical volume measure on phase space in Hamiltonian mechanics and symplectic geometry that remains invariant under the system’s time evolution.

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Statements (45)

Predicate	Object
instanceOf	canonical measure ⓘ measure ⓘ symplectic invariant ⓘ volume form ⓘ
associatedWith	Hamiltonian flow ⓘ symplectic form ⓘ
characterizes	conservation of phase-space volume ⓘ
constructedFrom	top exterior power of the symplectic form ⓘ
definedOn	phase space ⓘ symplectic manifold ⓘ
domainDimension	2n for an n-degree-of-freedom Hamiltonian system ⓘ
ensures	phase-space density evolves via Liouville equation ⓘ probability is conserved along Hamiltonian trajectories ⓘ
expressedAs	dq^1 ∧ … ∧ dq^n ∧ dp_1 ∧ … ∧ dp_n in canonical coordinates ⓘ ω^n / n! for a 2n-dimensional symplectic manifold with symplectic form ω ⓘ
field	Hamiltonian mechanics NERFINISHED ⓘ ergodic theory ⓘ statistical mechanics ⓘ symplectic geometry ⓘ
hasProperty	absolutely continuous with respect to Lebesgue measure in canonical coordinates ⓘ canonical up to normalization ⓘ invariant under Hamiltonian flow ⓘ locally equivalent to Lebesgue measure in Darboux coordinates ⓘ preserved by time evolution ⓘ volume-preserving ⓘ
invariantUnder	Hamiltonian diffeomorphisms ⓘ canonical transformations ⓘ symplectomorphisms ⓘ
mathematicalNature	Borel measure on phase space ⓘ smooth measure induced by a volume form ⓘ
namedAfter	Joseph Liouville NERFINISHED ⓘ
normalization	can be scaled by a constant factor ⓘ
playsRoleIn	conservation laws in classical mechanics ⓘ measure-theoretic foundations of Hamiltonian dynamics ⓘ phase-space formulation of classical mechanics ⓘ
relatedConcept	invariant measure ⓘ microcanonical measure ⓘ symplectic volume ⓘ
relatedTo	Liouville’s theorem NERFINISHED ⓘ
usedIn	Hamiltonian statistical mechanics ⓘ classical ergodic theory ⓘ definition of invariant measures for dynamical systems ⓘ definition of microcanonical ensemble ⓘ formulation of Liouville’s theorem in Hamiltonian mechanics ⓘ phase-space integration of observables ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Joseph Liouville → hasEponym → Liouville measure ⓘ