Symplectic manifolds

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1 Introduction

A symplectic manifold is a smooth manifold $M$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; A '''symplectic manifold''' is a smooth manifold $M$ together with a differential two-form $\omega$ that is nondegenerate and closed. The form $\omega$ is called a '''symplectic form'''. The nondegeneracy means that the highest nonzero power of $\omega$ is a volume form on $M.$ It follows that a symplectic manifold is even dimensional. Symplectic manifolds originated from classical mechanics. The phase space of a dynamical system is the cotangent bundle of the configuration space and it is equipped with a symplectic form. This symplectic form is preserved by the flow of the system. </wikitex> == Examples == <wikitex>; $\bullet$ The most basic example of a symplectic manifold is $\mathbb R^{2n}$ equipped with the form $\omega_0:=dx^1\wedge dy^1 + \ldots + dx^n\wedge dy^n.$ A theorem of Darboux {{cite|McDuff-Salamon}} states that locally every symplectic manifold if of this form. More precisely, if $(M,\omega)$ is a symplectic n$-manifold then for every point $x\in M$ there exists an open neighbourhood $U\subset M$ of $p$ and a diffeomorphism $f\colon U\to f(U)\subset \mathbb R^{2n}$ such that the restriction of $\omega$ to $U$ is equal to the pull-back $f^*\omega_0.$ This implies that symplectic manifolds have no local invariants. $\bullet$ An area form on an oriented surface is symplectic. $\bullet$ Let $X$ be a smooth manifold and let $\lambda$ be a one-form on the cotangent bundle $T^*X$ defined as follows. If $V$ is a vector tangent to $T^*X$ at a point $\alpha$ then $\lambda_{\alpha}(X) = \alpha (\pi_*(X)),$ where $\pi\colon T^*X\to X$ is the projection. In local coordinates the form $\lambda$ can be expressed as $\sum y^idx^i.$ The differential $d\lambda$ is a symplectic form on the cotangent bundle $T^*X.$ $\bullet$ </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]M$ together with a differential two-form $\omega$$\omega$ that is nondegenerate and closed. The form $\omega$$\omega$ is called a symplectic form. The nondegeneracy means that the highest nonzero power of $\omega$$\omega$ is a volume form on $M.$$M.$ It follows that a symplectic manifold is even dimensional.

Symplectic manifolds originated from classical mechanics. The phase space of a dynamical system is the cotangent bundle of the configuration space and it is equipped with a symplectic form. This symplectic form is preserved by the flow of the system.

2 Examples

$\bullet$$\bullet$ The most basic example of a symplectic manifold is $\mathbb R^{2n}$$\mathbb R^{2n}$ equipped with the form $\omega_0:=dx^1\wedge dy^1 + \ldots + dx^n\wedge dy^n.$$\omega_0:=dx^1\wedge dy^1 + \ldots + dx^n\wedge dy^n.$

A theorem of Darboux [McDuff-Salamon] states that locally every symplectic manifold if of this form. More precisely, if $(M,\omega)$$(M,\omega)$ is a symplectic $2n$$2n$-manifold then for every point $x\in M$$x\in M$ there exists an open neighbourhood $U\subset M$$U\subset M$ of $p$$p$ and a diffeomorphism $f\colon U\to f(U)\subset \mathbb R^{2n}$$f\colon U\to f(U)\subset \mathbb R^{2n}$ such that the restriction of $\omega$$\omega$ to $U$$U$ is equal to the pull-back $f^*\omega_0.$$f^*\omega_0.$ This implies that symplectic manifolds have no local invariants.

$\bullet$$\bullet$ An area form on an oriented surface is symplectic.

$\bullet$$\bullet$ Let $X$$X$ be a smooth manifold and let $\lambda$$\lambda$ be a one-form on the cotangent bundle $T^*X$$T^*X$ defined as follows. If $V$$V$ is a vector tangent to $T^*X$$T^*X$ at a point $\alpha$$\alpha$ then $\lambda_{\alpha}(X) = \alpha (\pi_*(X)),$$\lambda_{\alpha}(X) = \alpha (\pi_*(X)),$ where $\pi\colon T^*X\to X$$\pi\colon T^*X\to X$ is the projection. In local coordinates the form $\lambda$$\lambda$ can be expressed as $\sum y^idx^i.$$\sum y^idx^i.$ The differential $d\lambda$$d\lambda$ is a symplectic form on the cotangent bundle $T^*X.$$T^*X.$

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3 Invariants

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4 Classification/Characterization

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5 Further discussion

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6 References

• [McDuff-Salamon] Template:McDuff-Salamon