Superintegrable Hamiltonian system

In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a 2 n {displaystyle 2n} -dimensional symplectic manifold for which the following conditions hold: In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a 2 n {displaystyle 2n} -dimensional symplectic manifold for which the following conditions hold: (i) There exist k > n {displaystyle k>n} independent integrals F i {displaystyle F_{i}} of motion. Their level surfaces (invariant submanifolds) form a fibered manifold F : Z → N = F ( Z ) {displaystyle F:Z o N=F(Z)} over a connected open subset N ⊂ R k {displaystyle Nsubset mathbb {R} ^{k}} . (ii) There exist smooth real functions s i j {displaystyle s_{ij}} on N {displaystyle N} such that the Poisson bracket of integrals of motion reads { F i , F j } = s i j ∘ F {displaystyle {F_{i},F_{j}}=s_{ij}circ F} . (iii) The matrix function s i j {displaystyle s_{ij}} is of constant corank m = 2 n − k {displaystyle m=2n-k} on N {displaystyle N} . If k = n {displaystyle k=n} , 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 F {displaystyle F} is a fiber bundlein tori T m {displaystyle T^{m}} . There exists an open neighbourhood U {displaystyle U} of F {displaystyle F} which is a trivial fiber bundle provided with the bundle (generalized action-angle) coordinates ( I A , p i , q i , ϕ A ) {displaystyle (I_{A},p_{i},q^{i},phi ^{A})} , A = 1 , … , m {displaystyle A=1,ldots ,m} , i = 1 , … , n − m {displaystyle i=1,ldots ,n-m} such that ( ϕ A ) {displaystyle (phi ^{A})} are coordinates on T m {displaystyle T^{m}} . These coordinates are the Darboux coordinates on a symplectic manifold U {displaystyle U} . A Hamiltonian of a superintegrable system depends only on the action variables I A {displaystyle I_{A}} which are the Casimir functions of the coinduced Poisson structure on F ( U ) {displaystyle F(U)} . 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 T m − r × R r {displaystyle T^{m-r} imes mathbb {R} ^{r}} .