# 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.

#### References

 [1] S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) [2] C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969)

Mostly, for a symplectic structure on a manifold the defining $2$- form $\Phi$ is required to be closed (cf. [a1], p. 176, [a4], p. 36ff). If $\Phi$ is not necessarily closed, one speaks of an almost-symplectic structure.

Let $\Phi ( \omega )$ denote the vector field on a symplectic manifold $M$ that corresponds to the $1$- form $\omega$. Then the Poisson bracket on $C ^ \infty ( M)$ is defined by

$$\{ f, g \} = \Phi ( \phi ( df), \phi ( dg)) .$$

This turns $C ^ \infty ( M)$ into a Lie algebra which satisfies the Leibniz property

$$\tag{* } \{ f, gh \} = \{ f, g \} h + g \{ f, h \} .$$

More generally, an algebra $A$ which has an extra Lie bracket $\{ , \}$ so that (*) is satisfied is called a Poisson algebra. A smooth manifold $M$ with a Poisson algebra structure on $C ^ \infty ( M)$ is called a Poisson manifold, [a4], p. 107ff.

#### References

 [a1] R. Abraham, J.E. Marsden, "Foundations of mechanics" , Benjamin/Cummings (1978) [a2] W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German) [a3] J.M. Souriau, "Structures des systèmes dynamiques" , Dunod (1969) [a4] P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) (Translated from French) [a5] V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian) [a6] V.I. [V.I. Arnol'd] Arnold, A.B. [A.B. Givent'al] Giventhal, "Symplectic geometry" , Dynamical Systems , IV , Springer (1990) (Translated from Russian) [a7] A. Crumeyrolle (ed.) J Grifone (ed.) , Symplectic geometry , Pitman (1983)
How to Cite This Entry:
Symplectic structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symplectic_structure&oldid=48935
This article was adapted from an original article by Ãœ. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article