An infinitesimal structure of order one on an even-dimensional smooth orientable manifold which is defined by a non-degenerate -form on . Every tangent space has the structure of a symplectic space with skew-symmetric scalar product . All frames tangent to adapted to the symplectic structure (that is, frames with respect to which has the canonical form ) form a principal fibre bundle over whose structure group is the symplectic group . Specifying a symplectic structure on is equivalent to specifying an -structure on (cf. -structure).
Given a symplectic structure on , there is an isomorphism between the modules of vector fields and -forms on , under which a vector field is associated with a -form, . In this context, the image of the Lie bracket is called the Poisson bracket . In particular, when and are exact differentials, one obtains the concept of the Poisson bracket of two functions on , which generalizes the corresponding classical concept.
A symplectic structure is also called an almost-Hamiltonian structure, and if is closed, i.e. , a Hamiltonian structure, though the condition is sometimes included in the definition of a symplectic structure. These structures find application in global analytical mechanics, since the cotangent bundle of any smooth manifold admits a canonical Hamiltonian structure. It is defined by the form , where the -form on , called the Liouville form, is given by: for any tangent vector at the point , where is the projection . If one chooses local coordinates on , and , then , so that . In classical mechanics is interpreted as the configuration space and as the phase space.
A vector field on a manifold with a Hamiltonian structure is called a Hamiltonian vector field (or a Hamiltonian system) if the -form is closed. If, in addition, it is exact, that is, , then is called a Hamiltonian on and is a generalization of the corresponding classical concept.
|||S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)|
|||C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969)|
Mostly, for a symplectic structure on a manifold the defining -form is required to be closed (cf. [a1], p. 176, [a4], p. 36ff). If is not necessarily closed, one speaks of an almost-symplectic structure.
Let denote the vector field on a symplectic manifold that corresponds to the -form . Then the Poisson bracket on is defined by
This turns into a Lie algebra which satisfies the Leibniz property
More generally, an algebra which has an extra Lie bracket so that (*) is satisfied is called a Poisson algebra. A smooth manifold with a Poisson algebra structure on is called a Poisson manifold, [a4], p. 107ff.
|[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)|
Symplectic structure. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Symplectic_structure&oldid=14696