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.

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  • 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. (en)
  • En géométrie différentielle, la forme de Liouville est une 1-forme différentielle naturelle sur les variétés cotangentes. Sa différentielle est une forme symplectique Elle joue un rôle central en mécanique classique. L'étude de la géométrie des variétés cotangentes a son importance en géométrie symplectique, importance qui réside dans l'utilisation du théorème de Weinstein. (fr)
  • Niech (X, ω) będzie rozmaitością symplektyczną. 1-formę β spełniającą: nazywamy formą Liouville’a na X. Dla każdej pary ω i β istnieje jedno pole wektorowe η na X, takie że: gdzie oznacza zwężenie ω przez η. (pl)
  • 在数学中,重言 1-形式(Tautological one-form)是流形 Q 的余切丛 上一个特殊的 1-形式。这个形式的外导数定义了一个辛形式给出了 的辛流形结构。重言 1-形式在哈密顿力学与拉格朗日力学的形式化中起着重要的作用。重言 1-形式有时也称为刘维尔 1-形式,典范 1-形式,或者辛势能。一个类似的对象是切丛上的典范向量场。 在典范坐标中,重言 1-形式由下式给出: 在差一个全微分(恰当形式)的意义下,相空间中的任何“保持”典范 1-形式结构的坐标系,可以称之为典范坐标;不同典范坐标之间的变换称为典范变换。 典范辛形式由 给出。 (zh)
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  • En géométrie différentielle, la forme de Liouville est une 1-forme différentielle naturelle sur les variétés cotangentes. Sa différentielle est une forme symplectique Elle joue un rôle central en mécanique classique. L'étude de la géométrie des variétés cotangentes a son importance en géométrie symplectique, importance qui réside dans l'utilisation du théorème de Weinstein. (fr)
  • Niech (X, ω) będzie rozmaitością symplektyczną. 1-formę β spełniającą: nazywamy formą Liouville’a na X. Dla każdej pary ω i β istnieje jedno pole wektorowe η na X, takie że: gdzie oznacza zwężenie ω przez η. (pl)
  • 在数学中,重言 1-形式(Tautological one-form)是流形 Q 的余切丛 上一个特殊的 1-形式。这个形式的外导数定义了一个辛形式给出了 的辛流形结构。重言 1-形式在哈密顿力学与拉格朗日力学的形式化中起着重要的作用。重言 1-形式有时也称为刘维尔 1-形式,典范 1-形式,或者辛势能。一个类似的对象是切丛上的典范向量场。 在典范坐标中,重言 1-形式由下式给出: 在差一个全微分(恰当形式)的意义下,相空间中的任何“保持”典范 1-形式结构的坐标系,可以称之为典范坐标;不同典范坐标之间的变换称为典范变换。 典范辛形式由 给出。 (zh)
  • 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. (en)
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  • Tautological one-form (en)
  • Forme de Liouville (fr)
  • Forma Liouville’a (pl)
  • 重言1形式 (zh)
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