Hamiltonian mechanics | Integrable systems | Dynamical systems

Superintegrable Hamiltonian system

In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a -dimensional symplectic manifold for which the following conditions hold: (i) There exist independent integrals of motion. Their level surfaces (invariant submanifolds) form a fibered manifold over a connected open subset . (ii) There exist smooth real functions on such that the Poisson bracket of integrals of motion reads. (iii) The matrix function is of constant corank on . If , this is the case of a completely integrable Hamiltonian system. The Mishchenko-Fomenko theorem for superintegrable Hamiltonian systems generalizes the Liouville-Arnold theorem on action-angle coordinates of completely integrable Hamiltonian system as follows. Let invariant submanifolds of a superintegrable Hamiltonian system be connected compact and mutually diffeomorphic. Then the fibered manifold is a fiber bundlein tori . There exists an open neighbourhood of which is a trivial fiber bundle provided with the bundle (generalized action-angle) coordinates ,, such that are coordinates on . These coordinates are the Darboux coordinates on a symplectic manifold . A Hamiltonian of a superintegrable system depends only on the action variables which are the Casimir functions of the coinduced Poisson structure on . The Liouville-Arnold theorem for completely integrable systems and the Mishchenko-Fomenko theorem for the superintegrable ones are generalized to the case of non-compact invariant submanifolds. They are diffeomorphic to a toroidal cylinder . (Wikipedia).

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Sarah Post: Rational extensions of superintegrable systems, exceptional polynomials & Painleve eq.s

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From playlist Integrable Systems 9th Workshop

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Differential Equations | First Order Linear System of DEs.

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From playlist Systems of Differential Equations

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From playlist Differential Equations

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From playlist Differential Equations

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From playlist Systems of Differential Equations

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From playlist Physics - Waves

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From playlist Data-Driven Dynamical Systems

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From playlist Differential Equations

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From playlist Hydrodynamics and fluctuations - microscopic approaches in condensed matter systems (ONLINE) 2021

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From playlist PHYSICS 69 ADVANCED MECHANICS: HAMILTONIAN MECHANICS

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Hamiltonian Simulation and Universal Quantum (...) - T. Cubitt - Main Conference - CEB T3 2017

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From playlist 2017 - T3 - Analysis in Quantum Information Theory - CEB Trimester

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From playlist Adiabatic Quantum Computing Conference 2016

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From playlist Graduate Summer School 2021: Mathematics of Topological Phases of Matter

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From playlist Differential Equations

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Related pages

Hamiltonian system | Darboux's theorem | Fiber bundle | Action-angle coordinates | Symplectic manifold | Integrable system | Nambu mechanics | Laplace–Runge–Lenz vector | Poisson manifold