PlanetMath: Poincaré $1$-form

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Poincaré

-form

(Definition)

Definition 1 Suppose is a manifold, and $T^\ast M$ is its cotangent bundle. Then the Poincaré -form, $\alpha \in \Omega^1(T^\ast M)$ , is locally defined as

$\displaystyle \alpha = \sum_{i=1}^n y_i dx^i$

where are canonical local coordinates for $T^\ast M$ .

Let us show that the Poincaré -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 . Then we have overlapping local coordinates , $(\tilde{x}^i, \tilde{y}_i)$ for $T^\ast M$ with the transformation rule

$\displaystyle \tilde{y}_i = \frac{\partial \tilde{x}^j}{\partial x^i} y_j.$

Hence

$\displaystyle \sum_{i=1}^n \tilde{y}_i d\tilde{x}^i$	$\displaystyle =$	$\displaystyle \sum_{i=1}^n \tilde{y}_i \frac{\partial \tilde{x}^i}{\partial x^k} dx^k$
	$\displaystyle =$	$\displaystyle \sum_{i=1}^n \frac{\partial \tilde{x}^j}{\partial x^i} y_j \frac{\partial \tilde{x}^i}{\partial x^k} dx^k$
	$\displaystyle =$	$\displaystyle \sum_{k=1}^n y_k dx^k.$

Properties

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