Liouville's theorem in Hamiltonian mechanics

E620665

conservation law result in classical mechanics theorem

Liouville's theorem in Hamiltonian mechanics states that the phase-space volume occupied by an ensemble of systems evolving under Hamiltonian dynamics is conserved over time, implying incompressible flow in phase space.

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All labels observed (4)

Label	Occurrences
Liouville's theorem in Hamiltonian mechanics canonical	2
Liouville's equation in Hamilton–Jacobi theory	1
Liouville's equation in statistical mechanics	1
Liouville’s theorem	1

Statements (48)

Predicate	Object
instanceOf	conservation law ⓘ result in classical mechanics ⓘ theorem ⓘ
appliesTo	Hamiltonian systems ⓘ autonomous Hamiltonian systems ⓘ canonical Hamiltonian equations of motion ⓘ
assumes	Hamilton's equations of motion ⓘ canonical coordinates and momenta ⓘ time-independent phase-space measure dq dp ⓘ
category	theorem in dynamical systems ⓘ theorem in physics ⓘ theorem in symplectic geometry ⓘ
concerns	invariance of the Liouville measure ⓘ phase space ⓘ phase-space distribution functions ⓘ phase-space volume ⓘ
expressedAs	the Poisson bracket of the distribution function with the Hamiltonian equals the negative time derivative of the distribution function ⓘ the divergence of the Hamiltonian flow in phase space is zero ⓘ
field	Hamiltonian mechanics NERFINISHED ⓘ analytical mechanics ⓘ classical mechanics ⓘ
formalStatement	dρ/dt = 0 along trajectories in phase space for Hamiltonian dynamics ⓘ ∂ρ/∂t + {ρ,H} = 0 ⓘ ∇·v = 0 in phase space for Hamiltonian flow ⓘ
historicalPeriod	19th century ⓘ
holdsFor	closed Hamiltonian systems ⓘ
implies	conservation of Gibbs entropy for isolated Hamiltonian systems ⓘ conservation of phase-space density along trajectories ⓘ incompressible flow in phase space ⓘ phase-space volume is invariant under canonical transformations ⓘ probability density in phase space is constant along trajectories ⓘ
mathematicalForm	volume-preserving flow on a symplectic manifold ⓘ
namedAfter	Joseph Liouville NERFINISHED ⓘ
relatedTo	Liouville equation NERFINISHED ⓘ Liouville's theorem in complex analysis NERFINISHED ⓘ canonical transformations ⓘ symplectic geometry ⓘ
requires	Hamiltonian flow to be differentiable ⓘ symplectic structure of phase space ⓘ
states	the phase-space volume occupied by an ensemble of Hamiltonian systems is conserved in time ⓘ
typicallyDoesNotHoldFor	non-Hamiltonian dissipative systems ⓘ systems with friction modeled as non-Hamiltonian forces ⓘ
usedIn	classical statistical mechanics ⓘ derivation of the microcanonical ensemble ⓘ ergodic theory ⓘ foundations of equilibrium statistical mechanics ⓘ statistical mechanics ⓘ
usedToJustify	uniform distribution on energy surfaces in microcanonical ensemble ⓘ

Referenced by (5)

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

Poincaré recurrence theorem → relatedConcept → Liouville's theorem in Hamiltonian mechanics ⓘ

Joseph Liouville → notableWork → Liouville's theorem in Hamiltonian mechanics ⓘ

this entity surface form: Liouville's equation in statistical mechanics

Joseph Liouville → notableWork → Liouville's theorem in Hamiltonian mechanics ⓘ

this entity surface form: Liouville's equation in Hamilton–Jacobi theory

Hamiltonian mechanics → relatedConcept → Liouville's theorem in Hamiltonian mechanics ⓘ

this entity surface form: Liouville’s theorem