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An [[Infinitesimal structure|infinitesimal structure]] of order one on an even-dimensional smooth orientable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918601.png" /> which is defined by a non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918602.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918603.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918604.png" />. Every tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918605.png" /> has the structure of a symplectic space with skew-symmetric scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918606.png" />. All frames tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918607.png" /> adapted to the symplectic structure (that is, frames with respect to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918608.png" /> has the canonical form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918609.png" />) form a principal fibre bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186010.png" /> whose structure group is the [[Symplectic group|symplectic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186011.png" />. Specifying a symplectic structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186012.png" /> is equivalent to specifying an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186013.png" />-structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186014.png" /> (cf. [[G-structure(2)|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186015.png" />-structure]]).
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An [[Infinitesimal structure|infinitesimal structure]] of order one on an even-dimensional smooth orientable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918601.png" /> which is defined by a non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918602.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918603.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918604.png" />. Every tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918605.png" /> has the structure of a symplectic space with skew-symmetric scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918606.png" />. All frames tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918607.png" /> adapted to the symplectic structure (that is, frames with respect to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918608.png" /> has the canonical form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s0918609.png" />) form a principal fibre bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186010.png" /> whose structure group is the [[Symplectic group|symplectic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186011.png" />. Specifying a symplectic structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186012.png" /> is equivalent to specifying an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186013.png" />-structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186014.png" /> (cf. [[G-structure|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186015.png" />-structure]]).
  
 
Given a symplectic structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186016.png" />, there is an isomorphism between the modules of vector fields and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186017.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186018.png" />, under which a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186019.png" /> is associated with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186020.png" />-form, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186021.png" />. In this context, the image of the Lie bracket <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186022.png" /> is called the Poisson bracket <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186023.png" />. In particular, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186025.png" /> are exact differentials, one obtains the concept of the Poisson bracket of two functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186026.png" />, which generalizes the corresponding classical concept.
 
Given a symplectic structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186016.png" />, there is an isomorphism between the modules of vector fields and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186017.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186018.png" />, under which a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186019.png" /> is associated with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186020.png" />-form, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186021.png" />. In this context, the image of the Lie bracket <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186022.png" /> is called the Poisson bracket <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186023.png" />. In particular, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186025.png" /> are exact differentials, one obtains the concept of the Poisson bracket of two functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091860/s09186026.png" />, which generalizes the corresponding classical concept.

Revision as of 08:31, 19 October 2014

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.

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 -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.

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=33889
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article