View source for Symplectic manifold

{{Use American English|date = March 2019}}
{{Short description|Type of manifold in differential geometry}}
In [[differential geometry]], a subject of [[mathematics]], a '''symplectic manifold''' is a [[Differentiable manifold#Definition|smooth manifold]], <math> M </math>, equipped with a [[Closed and exact differential forms|closed]] [[nondegenerate form|nondegenerate]] differential [[Differential form|2-form]] <math> \omega </math>, 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. For example, 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 modeled as a manifold, and this manifold's [[cotangent bundle]] describes the [[phase space]] of the system.

== Motivation ==

Symplectic manifolds arise from [[classical mechanics]]; in particular, they are a generalization of the [[phase space]] of a closed system.<ref name="Webster">{{cite web |first=Ben |last=Webster |title=What is a symplectic manifold, really? |url=https://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''.<ref name="Cohn">{{cite web |first=Henry |last=Cohn |title=Why symplectic geometry is the natural setting for classical mechanics |url=https://math.mit.edu/~cohn/Thoughts/symplectic.html }}</ref> So we require a linear map {{nowrap|''TM'' → ''T''<sup>∗</sup>''M''}}, or equivalently, an element of {{nowrap|''T''<sup>∗</sup>''M'' ⊗ ''T''<sup>∗</sup>''M''}}. Letting ''ω'' denote a [[Section (fiber bundle)|section]] of {{nowrap|''T''<sup>∗</sup>''M'' ⊗ ''T''<sup>∗</sup>''M''}}, the requirement that ''ω'' be [[Degenerate form|non-degenerate]] ensures that for every differential ''dH'' there is a unique corresponding vector field ''V<sub>H</sub>'' such that {{nowrap|1=''dH'' = ''ω''(''V<sub>H</sub>'', · )}}. Since one desires the Hamiltonian to be constant along flow lines, one should have {{nowrap|1=''dH''(''V<sub>H</sub>'') = ''ω''(''V<sub>H</sub>'', ''V<sub>H</sub>'') = 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 ''V<sub>H</sub>'' vanishes. Applying [[Cartan homotopy formula|Cartan's formula]], this amounts to (here <math> \iota_X</math> is the [[Interior product|interior product]]): 

:<math>\mathcal{L}_{V_H}(\omega) = 0\;\Leftrightarrow\;\mathrm d (\iota_{V_H} \omega) + \iota_{V_H} \mathrm d\omega= \mathrm d (\mathrm d\,H) + \mathrm d\omega(V_H) = \mathrm d\omega(V_H)=0</math>

so that, on repeating this argument for different smooth functions <math>H</math> such that the corresponding <math>V_H</math> span the tangent space at each point the argument is applied at, we see that the requirement for the vanishing Lie derivative along flows of <math>V_H</math> corresponding to arbitrary smooth <math>H</math> is equivalent to the requirement that ''ω'' should be [[Closed and exact differential forms|closed]].

== Definition ==

A '''symplectic form''' on a smooth [[manifold]] <math> M </math> is a closed non-degenerate differential [[two-form|2-form]] <math> \omega </math>.<ref name="Gosson">{{cite book |first=Maurice |last=de Gosson |title=Symplectic Geometry and Quantum Mechanics |year=2006 |publisher=Birkhäuser Verlag |location=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|authorlink1=Vladimir Arnold|authorlink3=Sabir Gusein-Zade|authorlink2=Alexander Varchenko|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=<!--None-->}}</ref> Here, non-degenerate means that for every point <math> p \in M </math>, the skew-symmetric pairing on the [[tangent space]] <math> T_p M </math> defined by <math> \omega </math> is non-degenerate. That is to say, if there exists an <math> X \in T_p M </math> such that <math> \omega( X, Y ) = 0 </math> for all <math> Y \in T_p M </math>, then <math> X = 0 </math>. Since in odd dimensions, [[skew-symmetric matrices]] are always singular, the requirement that <math> \omega </math> be nondegenerate implies that <math> M </math> has an even dimension.<ref name="Gosson"/><ref name="Arnold"/> The closed condition means that the [[exterior derivative]] of <math> \omega </math> vanishes. A '''symplectic manifold''' is a pair <math> (M, \omega) </math> where <math> M </math> is a smooth manifold and <math> \omega </math> is a symplectic form. Assigning a symplectic form to <math> M </math> is referred to as giving <math> M </math> a '''symplectic structure'''.

== Linear symplectic manifold ==

There is a standard linear model, namely a [[symplectic vector space]] <math>\R^{2n}.</math> Let <math>\{v_1, \ldots, v_{2n}\}</math> be a basis for <math>\R^{2n}.</math> We define our symplectic form ''ω'' on this basis as follows:

:<math>\omega(v_i, v_j) = \begin{cases} 1 & j-i =n \text{ with } 1 \leqslant i \leqslant n \\ -1 & i-j =n \text{ with } 1 \leqslant j \leqslant n \\ 0 & \text{otherwise} \end{cases}</math>

In this case the symplectic form reduces to a simple [[quadratic form]]. If ''I<sub>n</sub>'' denotes the ''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 = \begin{pmatrix} 0 & I_n  \\ -I_n & 0 \end{pmatrix}. </math>

== Lagrangian and other submanifolds ==

There are several natural geometric notions of [[submanifold]] of a symplectic manifold <math> (M, \omega) </math>.

* '''symplectic submanifolds''' of <math> M </math> (potentially of any even dimension) are submanifolds <math> S \subset M </math> such that <math> \omega|_S </math> is a symplectic form on <math> S </math>.

* '''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'''.

* '''Lagrangian submanifolds''' of a sympletic manifold <math>(M,\omega)</math> are submanifolds where the restriction of the symplectic form <math>\omega</math> to <math>L\subset M</math> is vanishing, i.e. <math>\omega|_L=0</math> and <math>\text{dim }L=\tfrac{1}{2}\dim M</math>. Langrangian submanifolds are the maximal isotropic submanifolds.

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.

=== Examples ===
Let <math>\R^{2n}_{\textbf{x},\textbf{y}}</math> have global coordinates labelled <math>(x_1, \ldots, x_n, y_1, \ldots, y_n).</math> Then, we can equip <math>\R_{\textbf{x},\textbf{y}}^{2n}</math> with the canonical symplectic form 

:<math>\omega =\mathrm{d}x_1\wedge \mathrm{d}y_1 + \cdots + \mathrm{d}x_n\wedge \mathrm{d}y_n.</math> 

There is a standard Lagrangian submanifold given by <math>\R^n_{\mathbf{x}} \to \R^{2n}_{\mathbf{x},\mathbf{y}}</math>. The form <math>\omega</math> vanishes on <math>\R^n_{\mathbf{x}}</math> because given any pair of tangent vectors <math>X= f_i(\textbf{x}) \partial_{x_i}, Y=g_i(\textbf{x})\partial_{x_i},</math> we have that <math>\omega(X,Y) = 0.</math> To elucidate, consider the case <math>n=1</math>. Then, <math>X = f(x)\partial_x, Y=g(x)\partial_x,</math> and <math>\omega = \mathrm{d}x\wedge \mathrm{d}y.</math> Notice that when we expand this out

:<math>\omega(X,Y) = \omega(f(x)\partial_x,g(x)\partial_x) = \frac{1}{2}f(x)g(x)(\mathrm{d}x(\partial_x)\mathrm{d}y(\partial_x) - \mathrm{d}y(\partial_x)\mathrm{d}x(\partial_x))</math>

both terms we have a <math>\mathrm{d}y(\partial_x)</math> factor, which is 0, by definition.

The cotangent bundle of a manifold is locally modeled on a space similar to the first example. It can be shown that we can glue these affine symplectic forms hence this bundle forms a symplectic manifold. A more non-trivial example of a Lagrangian submanifold is the zero section of the cotangent bundle of a manifold. For example, let 

:<math>X = \{(x,y) \in \R^2 : y^2 - x = 0\}.</math>

Then, we can present <math>T^*X</math> as

:<math>T^*X = \{(x,y,\mathrm{d}x,\mathrm{d}y) \in \R^4 : y^2 - x = 0, 2y\mathrm{d}y - \mathrm{d}x = 0\}</math>

where we are treating the symbols <math>\mathrm{d}x,\mathrm{d}y</math> as coordinates of <math>\R^4 = T^*\R^2.</math> We can consider the subset where the coordinates <math>\mathrm{d}x=0</math> and <math>\mathrm{d}y=0</math>, giving us the zero section. This example can be repeated for any manifold defined by the vanishing locus of smooth functions <math>f_1,\ldots,f_k</math> and their differentials <math>\mathrm{d}f_1,\ldots,df_k</math>.

Another useful class of Lagrangian submanifolds can be found using Morse theory. Given a Morse function <math>f:M\to\R</math> and for a small enough <math>\varepsilon</math> one can construct a Lagrangian submanifold given by the vanishing locus <math>\mathbb{V}(\varepsilon\cdot \mathrm{d}f) \subset T^*M</math>. For a generic morse function we have a Lagrangian intersection given by <math>M \cap \mathbb{V}(\varepsilon\cdot \mathrm{d}f) = \text{Crit}(f)</math>.

{{See also|symplectic category}}

=== Special Lagrangian submanifolds ===
In the case of [[Kahler manifold]]s (or [[Calabi-Yau manifolds]]) we can make a choice <math>\Omega=\Omega_1+\mathrm{i}\Omega_2</math> on <math>M</math> as a holomorphic n-form, where <math>\Omega_1</math> is the real part and <math>\Omega_2</math> imaginary. A Lagrangian submanifold <math>L</math> is called '''special''' if in addition to the above Lagrangian condition the restriction <math>\Omega_2</math> to <math>L</math> is vanishing. In other words, the real part <math>\Omega_1</math> restricted on <math>L</math> leads the volume form on <math>L</math>. The following examples are known as special Lagrangian submanifolds,
:# complex Lagrangian submanifolds of [[hyperKahler manifold]]s,
:# fixed points of a real structure of Calabi-Yau manifolds.
The [[SYZ conjecture]] has been proved for special Lagrangian submanifolds but in general, it is open, and brings a lot of impacts to the study of [[mirror symmetry (string theory)|mirror symmetry]]. see {{harv|Hitchin|1999}}

== 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''<sub>1</sub>,&hellip;,''p''<sub>''n''</sub>, ''q''<sub>1</sub>,&hellip;,''q''<sub>''n''</sub>),}} and by [[Darboux's theorem]] the symplectic form ''ω'' can be, at least locally, written as {{nowrap|1=''ω'' = &sum; d''p''<sub>''k''</sub> &and; d''q''<sub>''k''</sub>}}, 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]] <math>T^*\R^n,</math> and the Lagrangian fibration as the trivial fibration <math>\pi: T^*\R^n \to \R^n.</math> This is the canonical picture.

== Lagrangian mapping ==
[[File:TIKZ PICT FBN.png|thumb|]]
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=''&pi;'' : ''K'' ↠ ''B''}} give a Lagrangian fibration of ''K''. The composite {{nowrap|1=(''&pi;'' ∘ ''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=(''&pi;''<sub>1</sub> ∘ ''i''<sub>1</sub>) : ''L''<sub>1</sub> ↪ ''K''<sub>1</sub> ↠ ''B''<sub>1</sub>}} and {{nowrap|1=(''&pi;''<sub>2</sub> ∘ ''i''<sub>2</sub>) : ''L''<sub>2</sub> ↪ ''K''<sub>2</sub> ↠ ''B''<sub>2</sub>}} 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''<sup>∗</sup>''ω''<sub>2</sub> denotes the [[Pullback (differential geometry)#Pullback of differential forms|pull back]] of ''ω''<sub>2</sub> 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>{{cite journal |first=F. |last=Cantrijn |first2=L. A. |last2=Ibort |first3=M. |last3=de León |title=On the Geometry of Multisymplectic Manifolds |journal=J. Austral. Math. Soc. |series=Ser. A |volume=66 |year=1999 |issue=3 |pages=303–330 |doi=10.1017/S1446788700036636 }}</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>{{cite journal |first=G. |last=Giachetta |first2=L. |last2=Mangiarotti |first3=G. |last3=Sardanashvily |authorlink3=Gennadi Sardanashvily |title=Covariant Hamiltonian equations for field theory |journal=Journal of Physics |volume=A32 |year=1999 |issue= |pages=6629–6642 |doi=10.1088/0305-4470/32/38/302 |arxiv=hep-th/9904062 }}</ref>

== See also ==
{{Portal|Mathematics}}
{{colbegin|colwidth=22em}}
* [[Almost complex manifold]]
* [[Almost symplectic manifold]]
* [[Contact manifold]] &minus; 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]]
{{colend}}

==Notes==
{{Reflist}}

==References==
*{{cite book |first=Dusa |last=McDuff |authorlink=Dusa McDuff |first2=D. |last2=Salamon |title=Introduction to Symplectic Topology |year=1998 |series=Oxford Mathematical Monographs |isbn=0-19-850451-9 }}
*{{cite web |first=Denis |last=Auroux |authorlink=Denis Auroux |title=Seminar on Mirror Symmetry |url=https://math.berkeley.edu/~auroux/290s16.html }}
*{{cite web |first=Eckhard |last=Meinrenken |authorlink=Eckhard Meinrenken |title=Symplectic Geometry |url=https://www.math.toronto.edu/mein/teaching/LectureNotes/sympl.pdf }}
*{{cite book |first=Ralph |last=Abraham |authorlink=Ralph Abraham (mathematician) |first2=Jerrold E. |last2=Marsden |authorlink2=Jerrold E. Marsden |title=Foundations of Mechanics |year=1978 |publisher=Benjamin-Cummings |location=London |isbn=0-8053-0102-X |at=See Section 3.2 }}
* {{cite book |first=Maurice A. |last=de Gosson |authorlink=Maurice A. de Gosson |title=Symplectic Geometry and Quantum Mechanics |year=2006 |publisher=Birkhäuser Verlag |location=Basel |isbn=3-7643-7574-4 }}
* {{cite journal |author=Alan Weinstein |title=Symplectic manifolds and their lagrangian submanifolds |authorlink=Alan Weinstein |journal=Advances in Mathematics |volume=6 |issue=3 |year=1971 |pages=329–46 |doi=10.1016/0001-8708(71)90020-X }}

== External links ==
* {{cite web |title=How to find Lagrangian Submanifolds |work=[[Stack Exchange]] |date=December 17, 2014 |url=https://math.stackexchange.com/q/1072200 }}
* {{Springer|author=Ü. Lumiste|title=Symplectic Structure|id=s/s091860}}
* {{cite paper |authorlink=Gennadi Sardanashvily |last=Sardanashvily |first=G. |date=2009 |title=Fibre bundles, jet manifolds and Lagrangian theory |work=Lectures for theoreticians |arxiv=0908.1886 }}
* {{cite web |authorlink=Dusa McDuff |last=McDuff |first=D. |url=https://www.ams.org/notices/199808/mcduff.pdf |title=Symplectic Structures—A New Approach to Geometry |work=Notices of the AMS |date=November 1998 }}
* {{planetmath reference|id=3672|title=Examples of symplectic manifolds}}
* {{cite arXiv |first1=Nigel |last1=Hitchin |year=1999 |title=Lectures on Special Lagrangian Submanifolds |arxiv=math/9907034 |ref=harv}}

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