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2013-12-29T10:47:08Z

91.157.210.180: Clarified the caption of the small cone section: A = r2 in the picture is correct only if the solid angle is one steradian.

In [[mathematics]], a '''symplectic manifold''' is a [[smooth manifold]], ''M'', equipped with a [[Closed and exact differential forms|closed]] [[nondegenerate form|nondegenerate]] differential [[two-form|2-form]], ω, called the [[symplectic form]]. The study of symplectic manifolds is called [[symplectic geometry]] or [[symplectic topology]]. Symplectic manifolds arise naturally in abstract formulations of [[classical mechanics]] and [[analytical mechanics]] as the [[cotangent bundle]]s of manifolds, e.g., in the [[Hamiltonian mechanics|Hamiltonian formulation]] of classical mechanics, which provides one of the major motivations for the field: The set of all possible configurations of a system is modelled as a manifold, and this manifold's [[cotangent bundle]] describes the [[phase space]] of the system.

Any real-valued [[differentiable function]], ''H'', on a symplectic manifold can serve as an '''[[energy function]]''' or '''Hamiltonian'''. Associated to any Hamiltonian is a [[Hamiltonian vector field]]; the [[integral curve]]s of the Hamiltonian vector field are solutions to [[Hamilton's equations]]. The Hamiltonian vector field defines a flow on the symplectic manifold, called a '''Hamiltonian flow''' or [[symplectomorphism]]. By [[Liouville's theorem (Hamiltonian)|Liouville's theorem]], Hamiltonian flows preserve the [[volume form]] on the phase space.

== Motivation ==

Symplectic manifolds arise from [[classical mechanics]], in particular, they are a generalization of the [[phase space]] of a closed system.<ref name="Webster">Ben Webster: ''What is a symplectic manifold, really?'' http://sbseminar.wordpress.com/2012/01/09/what-is-a-symplectic-manifold-really/</ref> In the same way the [[Hamilton equations]] allow one to derive the time evolution of a system from a set of [[differential equation]]s, the symplectic form should allow one to obtain a [[vector field]] describing the flow of the system from the differential ''dH'' of a Hamiltonian function ''H''. As [[Newton's laws of motion]] are linear differential equations, such a map should be linear as well.<ref name="Cohn">Henry Cohn: ''Why symplectic geometry is the natural setting for classical mechanics'' http://research.microsoft.com/en-us/um/people/cohn/thoughts/symplectic.html</ref> So we require a linear map ''T* M → TM'', or equivalently, an element of ''T* M'' ⊗ ''T* M''. Letting ω denote a [[Section (fiber bundle)|section]] of ''T* M'' ⊗ ''T* M'', the requirement that ω be [[Degenerate form|non-degenerate]] ensures that for every differential ''dH'' there is a unique corresponding vector field ''VH'' such that ''dH = ω(VH,· )''. Since one desires the Hamiltonian to be constant along flow lines, one should have ''dH(VH) = ω(VH, VH) = 0'', which implies that ''ω'' is [[Alternating form|alternating]] and hence a 2-form. Finally, one makes the requirement that ''ω'' should not change under flow lines, i.e. that the [[Lie derivative]] of ''ω'' along ''VH'' vanishes. Applying [[Cartan's formula]], this amounts to

:<math>\mathcal{L}_{V_H}(\omega) = d\omega(V_H) = 0</math>

which is equivalent to the requirement that ''ω'' should be [[Closed and exact differential forms|closed]].

== Definition ==

A symplectic form on a manifold ''M'' is a closed non-degenerate differential [[two-form|2-form]] ''ω''.<ref name="Gosson">Maurice de Gosson: ''Symplectic Geometry and Quantum Mechanics'' (2006) Birkhäuser Verlag, Basel ISBN 3-7643-7574-4. (page 10)
</ref><ref name="Arnold">{{Cite book|first=V. I.|last=Arnold|first2=A. N.|last2=Varchenko|first3=S. M.|last3=Gusein-Zade|title=The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1|publisher=Birkhäuser|year=1985|isbn=0-8176-3187-9|postscript=}}</ref>
Here, non-degenerate means that for all {{nowrap|1=''p'' ∈ ''M''}}, if there exists an {{nowrap|1=''X'' ∈ ''TpM''}} such that {{nowrap|1=''ω''(''X'',''Y'') = 0}} for all {{nowrap|1=''Y'' ∈ ''TpM''}}, then {{nowrap|1=''X'' = 0}}. The [[skew-symmetric]] condition (inherent in the definition of differential 2-form) means that for all {{nowrap|1=''p'' ∈ ''M''}} we have {{nowrap|1=''ω''(''X'',''Y'') = −''ω''(''Y'',''X'')}} for all {{nowrap|1=''X'',''Y'' ∈ ''TpM''.}} In odd dimensions, [[antisymmetric]] matrices are not invertible. Since ''ω'' is a differential two-form, the skew-symmetric condition implies that ''M'' has even dimension.<ref name="Gosson"/><ref name="Arnold"/> The closed condition means that the [[exterior derivative]] of ''ω ''vanishes, d''ω ''= 0. A symplectic manifold consists of a pair (''M'',''ω''), of a manifold ''M'' and a symplectic form ''ω''. Assigning a symplectic form ''ω'' to a manifold ''M'' is referred to as giving ''M'' a '''symplectic structure'''.

== Linear symplectic manifold ==

There is a standard linear model, namely a [[symplectic vector space]] '''R'''2''n''. Let '''R'''2''n'' have the basis {''v''1, ..., ''v''2''n''}. Then we define our symplectic form ''ω'' so that for all {{nowrap|1=1 ≤ ''i'' ≤ ''n''}} we have {{nowrap|1=''ω''(''v''''i'',''v''''n''+''i'') = 1,}} {{nowrap|1=''ω''(''v''''n''+''i'',''v''''i'') = −1,}} and ''ω'' is zero for all other pairs of basis vectors. In this case the symplectic form reduces to a simple [[quadratic form]]. If ''In'' denotes the {{nowrap|1=''n'' × ''n''}} [[identity matrix]] then the matrix, ''Ω'', of this quadratic form is given by the ({{nowrap|1=2''n'' × 2''n''}}) [[block matrix]]:

:<math> \Omega = \left(\begin{array}{c|c} 0 & I_n \\ \hline -I_n & 0 \end{array}\right). </math>

== Lagrangian and other submanifolds ==

There are several natural geometric notions of [[submanifold]] of a symplectic manifold.
*'''symplectic submanifolds''' (potentially of any even dimension) are submanifolds where the symplectic form is required to induce a symplectic form on them.
*'''isotropic submanifolds''' are submanifolds where the symplectic form restricts to zero, i.e. each tangent space is an isotropic subspace of the ambient manifold's tangent space. Similarly, if each tangent subspace to a submanifold is co-isotropic (the dual of an isotropic subspace), the submanifold is called '''co-isotropic'''.

The most important case of the isotropic submanifolds is that of '''Lagrangian submanifolds'''. A Lagrangian submanifold is, by definition, an isotropic submanifold of maximal dimension, namely half the dimension of the ambient symplectic manifold. One major example is that the graph of a [[symplectomorphism]] in the product symplectic manifold {{nowrap|1=(''M'' × ''M'', ω × −ω)}} is Lagrangian. Their intersections display rigidity properties not possessed by smooth manifolds; the [[Arnold conjecture]] gives the sum of the submanifold's [[Betti number]]s as a lower bound for the number of self intersections of a smooth Lagrangian submanifold, rather than the [[Euler characteristic]] in the smooth case.

== Lagrangian fibration ==

A '''Lagrangian fibration''' of a symplectic manifold ''M'' is a [[fibration]] where all of the [[Fiber_bundle#Formal_definition|fibres]] are Lagrangian submanifolds. Since ''M'' is even dimensional we can take local coordinates {{nowrap|1=(''p''1,…,''p''''n'',''q''1,…,''q''''n''),}} and by [[Darboux's theorem]] the symplectic form ''ω'' can be, at least locally, written as {{nowrap|1=''ω'' = ∑ d''p''''k'' &and; d''q''''k''}}, where d denotes the [[exterior derivative]] and ∧ denotes the [[exterior product]]. Using this set-up we can locally think of ''M'' as being the [[cotangent bundle]] T*'''R'''''n'', and the Lagrangian fibration as the trivial fibration {{nowrap|1=''π'' : T*'''R'''''n'' ↠ '''R'''''n''.}} This is the canonical picture.

== Lagrangian mapping ==
[[Image:TIKZ PICT FBN.png|thumb|<center>[[:File:TIKZ PICT FBN.png|Click to Enlarge]]</center>]]
Let ''L'' be a Lagrangian submanifold of a symplectic manifold (''K'',ω) given by an [[Immersion (mathematics)|immersion]] {{nowrap|1=''i'' : ''L'' ↪ ''K''}} (''i'' is called a '''Lagrangian immersion'''). Let {{nowrap|1=''π'' : ''K'' ↠ ''B''}} give a Lagrangian fibration of ''K''. The composite {{nowrap|1=(''π'' ○ ''i'') : ''L'' ↪ ''K'' ↠ ''B''}} is a '''Lagrangian mapping'''. The [[critical value|critical value set]] of ''π'' ○ ''i'' is called a [[Caustic (mathematics)|caustic]].

Two Lagrangian maps {{nowrap|1=(''π''1 ○ ''i''1) : ''L''1 ↪ ''K''1 ↠ ''B''1}} and {{nowrap|1=(''π''2 ○ ''i''2) : ''L''2 ↪ ''K''2 ↠ ''B''2}} are called '''Lagrangian equivalent''' if there exist [[diffeomorphism]]s σ, τ and ν such that both sides of the diagram given on the right [[commutative diagram|commute]], and τ preserves the symplectic form.<ref name="Arnold"/> Symbolically:
:<math> \tau \circ i_1 = i_2 \circ \sigma, \ \nu \circ \pi_1 = \pi_2 \circ \tau, \ \tau^*\omega_2 = \omega_1 \, , </math>
where τ*ω2 denotes the [[Pullback_(differential_geometry)#Pullback_of_differential_forms|pull back]] of ω2 by τ.

== Special cases and generalizations ==
* A symplectic manifold endowed with a [[metric tensor|metric]] that is [[Almost complex manifold#Compatible triples|compatible]] with the symplectic form is an [[almost Kähler manifold]] in the sense that the tangent bundle has an [[almost complex structure]], but this need not be [[integrability condition|integrable]].

* Symplectic manifolds are special cases of a [[Poisson manifold]]. The definition of a symplectic manifold requires that the symplectic form be non-degenerate everywhere, but if this condition is violated, the manifold may still be a Poisson manifold.

* A '''multisymplectic manifold''' of degree ''k'' is a manifold equipped with a closed nondegenerate ''k''-form.<ref>F. Cantrijn, L. A. Ibort and M. de León, J. Austral. Math. Soc. Ser. A 66 (1999), no. 3, 303-330.</ref>

* A '''polysymplectic manifold''' is a Legendre bundle provided with a polysymplectic tangent-valued <math>(n+2)</math>-form; it is utilized in Hamiltonian field theory.<ref>G. Giachetta, L. Mangiarotti and [[Sardanashvily|G. Sardanashvily]], Covariant Hamiltonian equations for field theory, Journal of Physics '''A32''' (1999) 6629-6642; [http://xxx.lanl.gov/abs/hep-th/9904062 arXiv: hep-th/9904062].</ref>

== See also ==
{{Portal|Mathematics}}
<div style="-moz-column-count:3; column-count:3;">
* [[Almost complex manifold]]
* [[Almost symplectic manifold]]
* [[Contact manifold]] − an odd-dimensional counterpart of the symplectic manifold.
* [[Fedosov manifold]]
* [[Poisson bracket]]
* [[Symplectic group]]
* [[Symplectic matrix]]
* [[Symplectic topology]]
* [[Symplectic vector space]]
* [[Symplectomorphism]]
* [[Tautological one-form]]
* [[Wirtinger inequality (2-forms)]]
* [[Covariant Hamiltonian field theory]]
</div>

==Notes==
{{Reflist}}

==References==
* [[Dusa McDuff]] and D. Salamon: ''Introduction to Symplectic Topology'' (1998) Oxford Mathematical Monographs, ISBN 0-19-850451-9.
* [[Ralph Abraham]] and [[Jerrold E. Marsden]], ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X ''See section 3.2''.
* [[Maurice A. de Gosson]]: ''Symplectic Geometry and Quantum Mechanics'' (2006) Birkhäuser Verlag, Basel ISBN 3-7643-7574-4.
* {{cite journal |author=Alan Weinstein |title=Symplectic manifolds and their lagrangian submanifolds |authorlink=Alan Weinstein |journal=Adv Math |volume=6 |issue=3 |year=1971 |pages=329–46 |doi=10.1016/0001-8708(71)90020-X }}

== External links ==
* {{Springer|author=Ü. Lumiste|title=Symplectic Structure|id=s/s091860}}
* [[Gennadi Sardanashvily|Sardanashvily, G.]], Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians,[http://xxx.lanl.gov/abs/0908.1886 arXiv: 0908.1886]
* {{planetmath reference|id=3672|title=Examples of symplectic manifolds}}

{{DEFAULTSORT:Symplectic Manifold}}
[[Category:Differential topology]]
[[Category:Symplectic geometry]]
[[Category:Hamiltonian mechanics]]
[[Category:Smooth manifolds]]

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