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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Superintegrable Hamiltonian system ： ウィキペディア英語版 Superintegrable Hamiltonian system In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a 2''n''-dimensional symplectic manifold for which the following conditions hold: (i) There exist ''n'' ≤ ''k'' independent integrals ''F'' ''i'' of motion. Their level surfaces (invariant submanifolds) form a fibered manifold $F:Z\backslash to\; N=F(Z)$ over a connected open subset $N\backslash subset\backslash mathbb\; R^k$. (ii) There exist smooth real functions $s\_$ on $N$ such that the Poisson bracket of integrals of motion reads $\backslash =\; s\_\backslash circ\; F$. (iii) The matrix function $s\_$ is of constant corank $m=2n-k$ on $N$. If $k=n$, 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 $F$ is a fiber bundle in tori $T^m$. Given its fiber $M$, there exists an open neighbourhood $U$ of $M$ which is a trivial fiber bundle provided with the bundle (generalized action-angle) coordinates $(I\_A,p\_i,q^i,\; \backslash phi^A)$, $A=1,\backslash ldots,\; m$, $i=1,\backslash ldots,n-m$ such that $(\backslash phi^A)$ are coordinates on $T^m$. These coordinates are the Darboux coordinates on a symplectic manifold $U$. A Hamiltonian of a superintegrable system depends only on the action variables $I\_A$ which are the Casimir functions of the coinduced Poisson structure on $F(U)$. 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 $T^\backslash times\backslash mathbb\; R^r$. == See also == *Integrable system *Action-angle coordinates 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Superintegrable Hamiltonian system」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.