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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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 NERFINISHED ⓘ
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 NERFINISHED ⓘ
relatedTo	Moser trick NERFINISHED ⓘ Weinstein neighborhood theorem NERFINISHED ⓘ 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 NERFINISHED ⓘ
usedIn	Hamiltonian mechanics NERFINISHED ⓘ canonical transformations theory ⓘ classical mechanics ⓘ geometric quantization ⓘ study of Poisson manifolds (via symplectic leaves) ⓘ

Referenced by (4)

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 ⓘ