Liouville's theorem in Hamiltonian mechanics
E620665
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.
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 |
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.
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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.
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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.
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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
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 ⓘ |
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Input
Subject: Liouville's theorem in Hamiltonian mechanics Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Liouville's equation in statistical mechanics
this entity surface form:
Liouville's equation in Hamilton–Jacobi theory
this entity surface form:
Liouville’s theorem