Carathéodory–Jacobi–Lie theorem

E118708

mathematical theorem theorem in Hamiltonian mechanics theorem in symplectic geometry

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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All labels observed (1)

Label	Occurrences
Carathéodory–Jacobi–Lie theorem canonical	1

Statements (39)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in Hamiltonian mechanics ⓘ theorem in symplectic geometry ⓘ
appliesTo	Poisson manifolds ⓘ symplectic manifolds ⓘ
assumes	a set of pairwise Poisson-commuting functions ⓘ functional independence of the given functions on an open set ⓘ
category	theorem in geometry ⓘ theorem in mathematical analysis ⓘ
concerns	Poisson-commuting integrals of motion ⓘ normal forms of Hamiltonian systems near regular points ⓘ
concludes	existence of local canonical coordinates ⓘ the given commuting functions depend only on a subset of the canonical coordinates ⓘ
context	canonical coordinate systems in mechanics ⓘ local structure of symplectic manifolds ⓘ
field	Hamiltonian mechanics ⓘ differential geometry ⓘ mathematical physics ⓘ symplectic geometry ⓘ
generalizes	Darboux theorem ⓘ
guarantees	adaptation of canonical coordinates to a given integrable family of functions ⓘ existence of coordinates in which the symplectic form has standard canonical form ⓘ
namedAfter	Carl Gustav Jacob Jacobi ⓘ Constantin Carathéodory ⓘ Sophus Lie ⓘ
provides	canonical local coordinates adapted to a given set of commuting functions ⓘ
relatedTo	Darboux theorem ⓘ Liouville–Arnold theorem ⓘ Poisson brackets ⓘ symplectic form ⓘ
subjectOf	Darboux-type coordinate systems ⓘ action–angle variables ⓘ canonical local coordinates ⓘ commuting functions ⓘ integrable Hamiltonian systems ⓘ
usedIn	Hamiltonian mechanics ⓘ canonical transformations ⓘ classical mechanics ⓘ theory of integrable systems ⓘ

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Input

Subject: Carathéodory–Jacobi–Lie theorem
Description of subject: 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.

Referenced by (1)

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

Constantin Carathéodory → notableWork → Carathéodory–Jacobi–Lie theorem ⓘ