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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 Tautological one-form ： ウィキペディア英語版 Tautological one-form In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle ''T'' *''Q'' of a manifold ''Q''. The exterior derivative of this form defines a symplectic form giving ''T'' *''Q'' the structure of a symplectic manifold. The tautological one-form plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics. The tautological one-form is sometimes also called the Liouville one-form, the Poincaré one-form, the canonical one-form, or the symplectic potential. A similar object is the canonical vector field on the tangent bundle. In algebraic geometry and complex geometry the term "canonical" is discouraged, due to confusion with the canonical class, and the term "tautological" is preferred, as in tautological bundle. In canonical coordinates, the tautological one-form is given by :$\backslash theta\; =\; \backslash sum\_i\; p\_i\; dq^i$ Equivalently, any coordinates on phase space which preserve this structure for the canonical one-form, up to a total differential (exact form), may be called canonical coordinates; transformations between different canonical coordinate systems are known as canonical transformations. The canonical symplectic form, also known as the Poincaré two-form, is given by :$\backslash omega\; =\; -d\backslash theta\; =\; \backslash sum\_i\; dq^i\; \backslash wedge\; dp\_i$ The extension of this concept to general fibre bundles is known as the solder form. ==Coordinate-free definition== The tautological 1-form can also be defined rather abstractly as a form on phase space. Let $Q$ be a manifold and $M=T^$ *Q be the cotangent bundle or phase space. Let :$\backslash pi:M\backslash to\; Q$ be the canonical fiber bundle projection, and let :$T\_\backslash pi:TM\; \backslash to\; TQ$ be the induced tangent map. Let ''m'' be a point on ''M''. Since ''M'' is the cotangent bundle, we can understand ''m'' to be a map of the tangent space at $q=\backslash pi(m)$: :$m:T\_qQ\; \backslash to\; \backslash mathbb$. That is, we have that ''m'' is in the fiber of ''q''. The tautological one-form $\backslash theta\_m$ at point ''m'' is then defined to be :$\backslash theta\_m\; =\; m\; \backslash circ\; T\_\backslash pi$. It is a linear map :$\backslash theta\_m:T\_mM\; \backslash to\; \backslash mathbb$ and so :$\backslash theta:M\; \backslash to\; T^$ *M. 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「Tautological one-form」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.