Darboux theorem

E506852

mathematical theorem

The Darboux theorem is a fundamental result in symplectic geometry stating that all symplectic manifolds are locally symplectomorphic to the standard symplectic space, implying that the symplectic form can always be put into a canonical local normal form.

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All labels observed (2)

Label	Occurrences
Darboux theorem canonical	5
Darboux coordinates	1

How this entity was disambiguated

This entity first appeared as the object of triple T5256390 — 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: Darboux theorem
Context triple: [Carathéodory–Jacobi–Lie theorem, generalizes, Darboux theorem]

A. Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
B. 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.
C. Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
D. Poincaré lemma
The Poincaré lemma is a fundamental result in differential geometry and topology stating that every closed differential form on a star-shaped (or more generally, contractible) domain is locally exact.
E. Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
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: Darboux theorem
Target entity description: The Darboux theorem is a fundamental result in symplectic geometry stating that all symplectic manifolds are locally symplectomorphic to the standard symplectic space, implying that the symplectic form can always be put into a canonical local normal form.

A. Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
B. 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.
C. Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
D. Poincaré lemma
The Poincaré lemma is a fundamental result in differential geometry and topology stating that every closed differential form on a star-shaped (or more generally, contractible) domain is locally exact.
E. Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
F. None of above. chosen

Statements (46)

Predicate	Object
instanceOf	mathematical theorem ⓘ
appliesTo	finite-dimensional symplectic manifolds ⓘ
assumes	nondegenerate closed 2-form ⓘ
category	results about canonical forms ⓘ
concerns	local structure of symplectic manifolds ⓘ symplectic forms ⓘ symplectic manifolds ⓘ
conclusion	existence of local coordinates in which the symplectic form is standard ⓘ
contrastWith	Riemannian geometry where curvature gives local invariants ⓘ
coordinateName	Darboux coordinates ⓘ
dimensionCondition	manifold dimension is even ⓘ
doesNotDependOn	choice of Riemannian metric ⓘ global topology of the manifold ⓘ
domain	smooth manifolds ⓘ
field	differential geometry ⓘ symplectic geometry ⓘ
formalization	for any point of a symplectic manifold there exist local coordinates making the symplectic form standard ⓘ
guaranteesExistenceOf	Darboux coordinates ⓘ linked to: Darboux theorem
historicalPeriod	19th century mathematics ⓘ
implies	every symplectic form admits a canonical local normal form ⓘ there are no local invariants of symplectic manifolds other than dimension ⓘ
influenceOn	mathematical formulation of classical mechanics ⓘ modern symplectic topology ⓘ
involves	diffeomorphisms preserving the symplectic form ⓘ symplectomorphisms ⓘ
locality	purely local result ⓘ
localModel	standard symplectic vector space (R^{2n}, sum dq_i wedge dp_i) ⓘ
namedAfter	Jean Gaston Darboux ⓘ linked to: Johannes G. G. Darboux
relatedTo	Moser trick ⓘ Weinstein neighborhood theorem ⓘ canonical coordinates in Hamiltonian mechanics ⓘ normal form theorems in differential geometry ⓘ
requires	closedness of the symplectic form ⓘ nondegeneracy of the symplectic form ⓘ
shows	symplectic geometry has no local curvature-type invariants ⓘ
standardForm	sum of dq_i wedge dp_i in suitable local coordinates ⓘ
statement	all symplectic manifolds are locally symplectomorphic to the standard symplectic space ⓘ
symplecticFormLocalExpression	omega = sum_{i=1}^n dq_i wedge dp_i in Darboux coordinates ⓘ
teaches	all symplectic manifolds are locally indistinguishable as symplectic spaces ⓘ
type	canonical form result ⓘ local normal form theorem ⓘ
usedIn	Hamiltonian mechanics ⓘ canonical transformations theory ⓘ classical mechanics ⓘ geometric quantization ⓘ study of Poisson manifolds (via symplectic leaves) ⓘ

How these facts were elicited

Referenced by (6)

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

Carathéodory–Jacobi–Lie theorem → generalizes → Darboux theorem ⓘ

Carathéodory–Jacobi–Lie theorem → relatedTo → Darboux theorem ⓘ

Johannes G. G. Darboux → notableWork → Darboux theorem ⓘ

Johannes G. G. Darboux → hasNotableConceptNamedAfter → Darboux theorem ⓘ

Darboux theorem → guaranteesExistenceOf → Darboux coordinates ⓘ

linked to: Darboux theorem

Introduction to Symplectic Topology → topic → Darboux theorem ⓘ

All labels observed (2)

How this entity was disambiguated Show

Statements (46)

How these facts were elicited Show

Referenced by (6)

How this entity was disambiguated

How these facts were elicited