# Symplectic structure

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
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.