Difference between revisions of "Symplectic 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 | + | 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) |
Symplectic structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symplectic_structure&oldid=14696