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[parent] Poincaré $1$-form (Definition)
Definition 1   Suppose $M$ is a manifold, and $T^\ast M$ is its cotangent bundle. Then the Poincaré $1$ -form, $\alpha \in \Omega^1(T^\ast M)$ , is locally defined as $$ \alpha = \sum_{i=1}^n y_i dx^i $$ where $x^i, y_i$ are canonical local coordinates for $T^\ast M$ .

Let us show that the Poincaré $1$ -form is globally defined. That is, $\alpha$ has the same expression in all local coordinates. Suppose $x^i, \tilde{x}^i$ are overlapping coordinates for $M$ . Then we have overlapping local coordinates $(x^i, y_i)$ , $(\tilde{x}^i, \tilde{y}_i)$ for $T^\ast M$ with the transformation rule $$ \tilde{y}_i = \frac{\partial \tilde{x}^j}{\partial x^i} y_j. $$ Hence \begin{eqnarray*} \sum_{i=1}^n \tilde{y}_i d\tilde{x}^i &=& \sum_{i=1}^n \tilde{y}_i \frac{\partial \tilde{x}^i}{\partial x^k} dx^k \\ &=& \sum_{i=1}^n \frac{\partial \tilde{x}^j}{\partial x^i} y_j \frac{\partial \tilde{x}^i}{\partial x^k} dx^k \\ &=& \sum_{k=1}^n y_k dx^k. \end{eqnarray*}

Properties

  1. The Poincaré $1$ -form play a crucial role in symplectic geometry. The form $d\alpha$ is the canonical symplectic form for $T^\ast M$ .
  2. Suppose $\pi\colon T^\ast M\to M$ is the canonical projection. Then $$ \alpha(w) = \xi( (D\pi)(w) ),\quad w\in T_\xi(T^\ast M), $$ which is an alternative definition of $\alpha$ without local coordinates.
  3. The restriction of this form to the unit cotangent bundle, is a contact form.




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"Poincaré $1$-form" is owned by matte. [ full author list (2) ]
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Other names:  Liouville one-form

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Cross-references: contact form, unit, restriction, canonical projection, symplectic form, geometry, transformation, coordinates, expression, Poincaré, local coordinates, canonical, cotangent bundle, manifold
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This is version 4 of Poincaré $1$-form, born on 2004-10-23, modified 2007-02-18.
Object id is 6405, canonical name is Poincare1Form.
Accessed 2783 times total.

Classification:
AMS MSC58A32 (Global analysis, analysis on manifolds :: General theory of differentiable manifolds :: Natural bundles)

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