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	3
Liouville's theorem in Hamiltonian mechanics canonical	2
Liouville's equation in Hamilton–Jacobi theory	1
Liouville's equation in statistical mechanics	1

How this entity was disambiguated

This entity first appeared as the object of triple T6801183 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Liouville's theorem in Hamiltonian mechanics
Context triple: [Poincaré recurrence theorem, relatedConcept, Liouville's theorem in Hamiltonian mechanics]

A. Liouville–Arnold theorem
The Liouville–Arnold theorem is a fundamental result in Hamiltonian mechanics that guarantees the integrability of a system with sufficiently many conserved quantities and describes its motion as quasi-periodic on invariant tori in phase space.
B. Kolmogorov–Arnold–Moser theory
Kolmogorov–Arnold–Moser theory is a fundamental result in dynamical systems that explains the persistence of quasi-periodic motions in nearly integrable Hamiltonian systems under small perturbations.
C. Carathéodory–Jacobi–Lie theorem
The Carathéodory–Jacobi–Lie theorem is a fundamental result in symplectic geometry and Hamiltonian mechanics that provides canonical local coordinates adapted to a given set of commuting functions.
D. On a General Method in Dynamics
"On a General Method in Dynamics" is a foundational 19th-century paper by William Rowan Hamilton that introduced Hamiltonian mechanics, reformulating classical dynamics in terms of generalized coordinates and conjugate momenta.
E. Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that describes physical systems in terms of generalized coordinates and conjugate momenta using a Hamiltonian function, providing a powerful framework for both classical and quantum physics.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Liouville's theorem in Hamiltonian mechanics
Target entity description: 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.

A. Liouville–Arnold theorem
The Liouville–Arnold theorem is a fundamental result in Hamiltonian mechanics that guarantees the integrability of a system with sufficiently many conserved quantities and describes its motion as quasi-periodic on invariant tori in phase space.
B. Kolmogorov–Arnold–Moser theory
Kolmogorov–Arnold–Moser theory is a fundamental result in dynamical systems that explains the persistence of quasi-periodic motions in nearly integrable Hamiltonian systems under small perturbations.
C. Carathéodory–Jacobi–Lie theorem
The Carathéodory–Jacobi–Lie theorem is a fundamental result in symplectic geometry and Hamiltonian mechanics that provides canonical local coordinates adapted to a given set of commuting functions.
D. On a General Method in Dynamics
"On a General Method in Dynamics" is a foundational 19th-century paper by William Rowan Hamilton that introduced Hamiltonian mechanics, reformulating classical dynamics in terms of generalized coordinates and conjugate momenta.
E. Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that describes physical systems in terms of generalized coordinates and conjugate momenta using a Hamiltonian function, providing a powerful framework for both classical and quantum physics.
F. None of above. chosen

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 ⓘ 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 ⓘ
relatedTo	Liouville equation ⓘ Liouville's theorem in complex analysis ⓘ linked to: Liouville's theorem 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 ⓘ

How these facts were elicited

Referenced by (7)

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 ⓘ

Joseph Liouville → notableWork → Liouville's equation in statistical mechanics ⓘ