|
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 : 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 : 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 be a manifold and be the cotangent bundle or phase space. Let : be the canonical fiber bundle projection, and let : 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 : :. That is, we have that ''m'' is in the fiber of ''q''. The tautological one-form at point ''m'' is then defined to be :. It is a linear map : and so :. 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
|