Symplectic manifold

In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, ${\displaystyle M}$, equipped with a closed nondegenerate differential 2-form ${\displaystyle \omega }$, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.

Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the symplectic form should allow one to obtain a vector field describing the flow of the system from the differential ${\displaystyle dH}$ of a Hamiltonian function ${\displaystyle H}$. So we require a linear map ${\displaystyle TM\rightarrow T^{*}M}$ from the tangent manifold ${\displaystyle TM}$ to the cotangent manifold ${\displaystyle T^{*}M}$, or equivalently, an element of ${\displaystyle T^{*}M\otimes T^{*}M}$. Letting ${\displaystyle \omega }$ denote a section of ${\displaystyle T^{*}M\otimes T^{*}M}$, the requirement that ${\displaystyle \omega }$ be non-degenerate ensures that for every differential ${\displaystyle dH}$ there is a unique corresponding vector field ${\displaystyle V_{H}}$ such that ${\displaystyle dH=\omega (V_{H},\cdot )}$. Since one desires the Hamiltonian to be constant along flow lines, one should have ${\displaystyle \omega (V_{H},V_{H})=dH(V_{H})=0}$, which implies that ${\displaystyle \omega }$ is alternating and hence a 2-form. Finally, one makes the requirement that ${\displaystyle \omega }$ should not change under flow lines, i.e. that the Lie derivative of ${\displaystyle \omega }$ along ${\displaystyle V_{H}}$ vanishes. Applying Cartan's formula, this amounts to (here ${\displaystyle \iota _{X}}$ is the interior product):

so that, on repeating this argument for different smooth functions ${\displaystyle H}$ such that the corresponding ${\displaystyle V_{H}}$ span the tangent space at each point the argument is applied at, we see that the requirement for the vanishing Lie derivative along flows of ${\displaystyle V_{H}}$ corresponding to arbitrary smooth ${\displaystyle H}$ is equivalent to the requirement that ω should be closed.

Let ${\displaystyle M}$ be a smooth manifold. A symplectic form on a is a closed non-degenerate differential 2-form ${\displaystyle \omega }$. Here, non-degenerate means that for every point ${\displaystyle p\in M}$, the skew-symmetric pairing on the tangent space ${\displaystyle T_{p}M}$ defined by ${\displaystyle \omega }$ is non-degenerate. That is to say, if there exists an ${\displaystyle X\in T_{p}M}$ such that ${\displaystyle \omega (X,Y)=0}$ for all ${\displaystyle Y\in T_{p}M}$, then ${\displaystyle X=0}$. The closed condition means that the exterior derivative of ${\displaystyle \omega }$ vanishes.

A symplectic manifold is a pair ${\displaystyle (M,\omega )}$ where ${\displaystyle M}$ is a smooth manifold and ${\displaystyle \omega }$ is a symplectic form. Assigning a symplectic form to ${\displaystyle M}$ is referred to as giving ${\displaystyle M}$ a symplectic structure. Since in odd dimensions, skew-symmetric matrices are always singular, nondegeneracy implies that ${\displaystyle \dim M}$ is even.

By nondegeneracy, ${\displaystyle \omega }$ can be used to define a pair of musical isomorphisms ${\displaystyle \omega ^{\flat }:TM\rightarrow T^{*}M,\omega ^{\sharp }:T^{*}M\rightarrow TM}$, such that ${\displaystyle \omega (X,Y)=\omega ^{\flat }(X)(Y)}$ for any two vector fields ${\displaystyle X,Y}$, and ${\displaystyle \omega ^{\sharp }\circ \omega ^{\flat }=\operatorname {Id} }$.

A symplectic manifold ${\displaystyle (M,\omega )}$ is exact iff the symplectic form ${\displaystyle \omega }$ is exact, i.e. equal to ${\displaystyle \omega =-d\theta }$ for some 1-form ${\displaystyle \theta }$. The area 2-form on the 2-sphere is an inexact symplectic form, by the hairy ball theorem.

By Darboux's theorem, around any point ${\displaystyle p}$ there exists a local coordinate system, in which ${\displaystyle \omega =\Sigma _{i}dp_{i}\wedge dq^{i}}$, where d denotes the exterior derivative and ∧ denotes the exterior product. This form is called the Poincaré two-form or the canonical two-form. Thus, we can locally think of M as being the cotangent bundle ${\displaystyle T^{*}\mathbb {R} ^{n}}$ and generated by the corresponding tautological 1-form ${\displaystyle \theta =\Sigma _{i}p_{i}dq^{i},\;\omega =d\theta }$.