# Symplectic structure

An infinitesimal structure of order one on an even-dimensional smooth orientable manifold $M ^ {2n}$ which is defined by a non-degenerate $2$- form $\Phi$ on $M ^ {2n}$. Every tangent space $T _ {x} ( M ^ {2n} )$ has the structure of a symplectic space with skew-symmetric scalar product $\Phi ( X, Y)$. All frames tangent to $M ^ {2n}$ adapted to the symplectic structure (that is, frames with respect to which $\Phi$ has the canonical form $\Phi = 2 \sum _ {\alpha = 1 } ^ {n} \omega ^ \alpha \wedge \omega ^ {n + \alpha }$) form a principal fibre bundle over $M ^ {2n}$ whose structure group is the symplectic group $\mathop{\rm Sp} ( n)$. Specifying a symplectic structure on $M ^ {2n}$ is equivalent to specifying an $\mathop{\rm Sp} ( n)$- structure on $M ^ {2n}$( cf. $G$- structure).
Given a symplectic structure on $M ^ {2n}$, there is an isomorphism between the modules of vector fields and $1$- forms on $M ^ {2n}$, under which a vector field $X$ is associated with a $1$- form, $\omega _ {X} : Y \mapsto \Phi ( X, Y)$. In this context, the image of the Lie bracket $[ X, Y]$ is called the Poisson bracket $[ \omega _ {X} , \omega _ {Y} ]$. In particular, when $\omega _ {X}$ and $\omega _ {Y}$ are exact differentials, one obtains the concept of the Poisson bracket of two functions on $M ^ {2n}$, which generalizes the corresponding classical concept.
A symplectic structure is also called an almost-Hamiltonian structure, and if $\Phi$ is closed, i.e. $d \Phi = 0$, a Hamiltonian structure, though the condition $d \Phi = 0$ is sometimes included in the definition of a symplectic structure. These structures find application in global analytical mechanics, since the cotangent bundle $T ^ {*} ( M)$ of any smooth manifold $M$ admits a canonical Hamiltonian structure. It is defined by the form $\Phi = d \theta$, where the $1$- form $\theta$ on $T ^ {*} ( M)$, called the Liouville form, is given by: $\theta _ {u} ( X _ {u} ) = u ( \pi _ {*} X _ {u} )$ for any tangent vector $X _ {u}$ at the point $u \in T ^ {*} ( M)$, where $\pi$ is the projection $T ^ {*} ( M) \rightarrow M$. If one chooses local coordinates $x ^ {1} \dots x ^ {n}$ on $M$, and $u = y _ {i} ( u) dx _ {\pi ( u) } ^ {i}$, then $\theta = y _ {i} dx ^ {i}$, so that $\Phi = dy _ {i} \wedge dx ^ {i}$. In classical mechanics $M$ is interpreted as the configuration space and $T ^ {*} ( M)$ as the phase space.
A vector field $X$ on a manifold $M ^ {2n}$ with a Hamiltonian structure is called a Hamiltonian vector field (or a Hamiltonian system) if the $1$- form $\omega _ {X}$ is closed. If, in addition, it is exact, that is, $\omega _ {X} = - dH$, then $H$ is called a Hamiltonian on $M ^ {2n}$ and is a generalization of the corresponding classical concept.