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)

Comments

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)