Liouville–Arnold theorem

E506853

mathematical theorem theorem in Hamiltonian mechanics

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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Statements (45)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in Hamiltonian mechanics ⓘ
alsoKnownAs	Liouville integrability theorem NERFINISHED ⓘ Liouville theorem on integrable systems NERFINISHED ⓘ
appliesTo	finite-dimensional Hamiltonian systems ⓘ
assumes	Poisson commutativity of integrals ⓘ compact connected regular level set of integrals ⓘ existence of n independent first integrals in involution ⓘ functional independence of integrals on a level set ⓘ smoothness of integrals ⓘ symplectic manifold phase space ⓘ
category	theorems in classical mechanics ⓘ theorems in symplectic topology ⓘ
concerns	action–angle variables ⓘ completely integrable systems ⓘ integrable Hamiltonian systems ⓘ invariant tori ⓘ quasi-periodic motion ⓘ
describes	local canonical transformation to action–angle variables ⓘ topological structure of invariant sets of integrable systems ⓘ
dimensionCondition	2n-dimensional symplectic manifold with n integrals in involution ⓘ
field	Hamiltonian mechanics NERFINISHED ⓘ classical mechanics ⓘ dynamical systems ⓘ symplectic geometry ⓘ
formalSetting	symplectic manifolds and Hamiltonian vector fields ⓘ
guarantees	complete integrability under its hypotheses ⓘ
historicalNote	Arnold formulated the modern geometric version in the 20th century NERFINISHED ⓘ Liouville proved an early version in the 19th century NERFINISHED ⓘ
implies	Hamiltonian flow is quasi-periodic on invariant tori ⓘ constants of motion become functions of action variables only ⓘ equations of motion become linear in angle variables ⓘ existence of action–angle coordinates near regular invariant tori ⓘ regular common level sets are n-dimensional tori ⓘ
namedAfter	Joseph Liouville NERFINISHED ⓘ Vladimir Arnold NERFINISHED ⓘ
relatedTo	Kolmogorov–Arnold–Moser theorem NERFINISHED ⓘ Liouville integrability NERFINISHED ⓘ Noether's theorem NERFINISHED ⓘ action–angle coordinates ⓘ
requires	Poisson bracket structure ⓘ
usedFor	analysis of planetary motion ⓘ construction of integrable models in classical mechanics ⓘ perturbation theory in celestial mechanics ⓘ study of near-integrable Hamiltonian systems ⓘ

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Carathéodory–Jacobi–Lie theorem → relatedTo → Liouville–Arnold theorem ⓘ