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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 Hamiltonian vector field ： ウィキペディア英語版 Hamiltonian vector field In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field, defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphisms of a symplectic manifold arising from the flow of a Hamiltonian vector field are known as canonical transformations in physics and (Hamiltonian) symplectomorphisms in mathematics. Hamiltonian vector fields can be defined more generally on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding to functions ''f'' and ''g'' on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the Poisson bracket of ''f'' and ''g''. == Definition == Suppose that (''M'',''ω'') is a symplectic manifold. Since the symplectic form ''ω'' is nondegenerate, it sets up a ''fiberwise-linear'' isomorphism : $\backslash omega:TM\backslash to\; T^$ *M, between the tangent bundle ''TM'' and the cotangent bundle ''T *M'', with the inverse : $\backslash Omega:T^$ *M\to TM, \quad \Omega=\omega^. Therefore, one-forms on a symplectic manifold ''M'' may be identified with vector fields and every differentiable function ''H'': ''M'' → R determines a unique vector field ''XH'', called the Hamiltonian vector field with the Hamiltonian ''H'', by requiring that for every vector field ''Y'' on ''M'', the identity :$\backslash mathrmH(Y)\; =\; \backslash omega(X\_H,Y)\backslash ,$ must hold. Note: Some authors define the Hamiltonian vector field with the opposite sign. One has to be mindful of varying conventions in physical and mathematical literature. 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「Hamiltonian vector field」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.