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In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 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 bundles of manifolds. For example, in the 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.

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 curves 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 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.^[1] In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, 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.^[2] So we require a linear map TM → T^∗M, or equivalently, an element of T^∗M ⊗ T^∗M. Letting ω denote a section of T^∗M ⊗ T^∗M, the requirement that ω be non-degenerate ensures that for every differential dH there is a unique corresponding vector field V_H such that dH = ω(V_H, · ). Since one desires the Hamiltonian to be constant along flow lines, one should have dH(V_H) = ω(V_H, V_H) = 0, which implies that ω is 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_H vanishes. Applying Cartan's formula, this amounts to (here ιX{displaystyle iota _{X}} is the interior product):

LVH(ω)=0⇔d(ιVHω)+ιVHdω=d(dH)+dω(VH)=dω(VH)=0{displaystyle {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}

so that, on repeating this argument for different smooth functions H{displaystyle H} such that the corresponding VH{displaystyle V_{H}} 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 VH{displaystyle V_{H}} corresponding to arbitrary smooth H{displaystyle H} is equivalent to the requirement that ω should be closed.

Definition

A symplectic form on a manifold M is a closed non-degenerate differential 2-form ω.^[3][4] Here, non-degenerate means that for all p ∈ M, if there exists an X ∈ T_pM such that ω(X,Y) = 0 for all Y ∈ T_pM, then X = 0. The skew-symmetric condition (inherent in the definition of differential 2-form) means that for all p ∈ M we have ω(X,Y) = −ω(Y,X) for all X,Y ∈ T_pM. 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.^[3][4] The closed condition means that the exterior derivative of ω vanishes, dω = 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 R2n.{displaystyle mathbb {R} ^{2n}.} Let {v1,…,v2n}{displaystyle {v_{1},ldots ,v_{2n}}} be a basis for R2n.{displaystyle mathbb {R} ^{2n}.} We define our symplectic form ω on this basis as follows:

ω(vi,vj)={1j−i=n with 1⩽i⩽n−1i−j=n with 1⩽j⩽n0otherwise{displaystyle omega (v_{i},v_{j})={begin{cases}1&j-i=n{text{ with }}1leqslant ileqslant n\-1&i-j=n{text{ with }}1leqslant jleqslant n\0&{text{otherwise}}end{cases}}}

In this case the symplectic form reduces to a simple quadratic form. If I_n denotes the n × n identity matrix then the matrix, Ω, of this quadratic form is given by the 2n × 2n block matrix:

Ω=(0In−In0).{displaystyle Omega ={begin{pmatrix}0&I_{n}\-I_{n}&0end{pmatrix}}.}

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.

• 
  Lagrangian submanifolds of a sympletic manifold (M,ω){displaystyle (M,omega )} are submanifolds where the restriction of the symplectic form ω{displaystyle omega } to L⊂M{displaystyle Lsubset M} is vanishing, i.e. ω|L=0{displaystyle omega |_{L}=0} and dim L=12dim⁡M{displaystyle {text{dim }}L={tfrac {1}{2}}dim M}. 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 (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 numbers 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 Rx,y2n{displaystyle mathbb {R} _{{textbf {x}},{textbf {y}}}^{2n}} have global coordinates labelled (x1,…,xn,y1,…,yn).{displaystyle (x_{1},ldots ,x_{n},y_{1},ldots ,y_{n}).} Then, we can equip Rx,y2n{displaystyle mathbb {R} _{{textbf {x}},{textbf {y}}}^{2n}} with the canonical symplectic form

ω=dx1∧dy1+⋯+dxn∧dyn.{displaystyle omega =mathrm {d} x_{1}wedge mathrm {d} y_{1}+cdots +mathrm {d} x_{n}wedge mathrm {d} y_{n}.}

There is a standard Lagrangian submanifold given by Rxn→Rx,y2n{displaystyle mathbb {R} _{mathbf {x} }^{n}to mathbb {R} _{mathbf {x} ,mathbf {y} }^{2n}}. The form ω{displaystyle omega } vanishes on Rxn{displaystyle mathbb {R} _{mathbf {x} }^{n}} because given any pair of tangent vectors X=fi(x)∂xi,Y=gi(x)∂xi,{displaystyle X=f_{i}({textbf {x}})partial _{x_{i}},Y=g_{i}({textbf {x}})partial _{x_{i}},} we have that ω(X,Y)=0.{displaystyle omega (X,Y)=0.} To elucidate, consider the case n=1{displaystyle n=1}. Then, X=f(x)∂x,Y=g(x)∂x,{displaystyle X=f(x)partial _{x},Y=g(x)partial _{x},} and ω=dx∧dy.{displaystyle omega =mathrm {d} xwedge mathrm {d} y.} Notice that when we expand this out

ω(X,Y)=ω(f(x)∂x,g(x)∂x)=12f(x)g(x)(dx(∂x)dy(∂x)−dy(∂x)dx(∂x)){displaystyle 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}))}

both terms we have a dy(∂x){displaystyle mathrm {d} y(partial _{x})} 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

X={(x,y)∈R2:y2−x=0}.{displaystyle X={(x,y)in mathbb {R} ^{2}:y^{2}-x=0}.}

Then, we can present T∗X{displaystyle T^{*}X} as

T∗X={(x,y,dx,dy)∈R4:y2−x=0,2ydy−dx=0}{displaystyle T^{*}X={(x,y,mathrm {d} x,mathrm {d} y)in mathbb {R} ^{4}:y^{2}-x=0,2ymathrm {d} y-mathrm {d} x=0}}

where we are treating the symbols dx,dy{displaystyle mathrm {d} x,mathrm {d} y} as coordinates of R4=T∗R2.{displaystyle mathbb {R} ^{4}=T^{*}mathbb {R} ^{2}.} We can consider the subset where the coordinates dx=0{displaystyle mathrm {d} x=0} and dy=0{displaystyle mathrm {d} y=0}, giving us the zero section. This example can be repeated for any manifold defined by the vanishing locus of smooth functions f1,…,fk{displaystyle f_{1},ldots ,f_{k}} and their differentials df1,…,dfk{displaystyle mathrm {d} f_{1},ldots ,df_{k}}.

Another useful class of Lagrangian submanifolds can be found using Morse theory. Given a Morse function f:M→R{displaystyle f:Mto mathbb {R} } and for a small enough ε{displaystyle varepsilon } one can construct a Lagrangian submanifold given by the vanishing locus V(ε⋅df)⊂T∗M{displaystyle mathbb {V} (varepsilon cdot mathrm {d} f)subset T^{*}M}. For a generic morse function we have a Lagrangian intersection given by M∩V(ε⋅df)=Crit(f){displaystyle Mcap mathbb {V} (varepsilon cdot mathrm {d} f)={text{Crit}}(f)}.

Special Lagrangian submanifolds

In the case of Kahler manifolds (or Calabi-Yau manifolds) we can make a choice Ω=Ω1+iΩ2{displaystyle Omega =Omega _{1}+mathrm {i} Omega _{2}} on M{displaystyle M} as a holomorphic n-form, where Ω1{displaystyle Omega _{1}} is the real part and Ω2{displaystyle Omega _{2}} imaginary. A Lagrangian submanifold L{displaystyle L} is called special if in addition to the above Lagrangian condition the restriction Ω2{displaystyle Omega _{2}} to L{displaystyle L} is vanishing. In other words, the real part Ω1{displaystyle Omega _{1}} restricted on L{displaystyle L} leads the volume form on L{displaystyle L}. The following examples are known as special Lagrangian submanifolds,

1. complex Lagrangian submanifolds of hyperKahler manifolds,


2. 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. see (Hitchin 1999)

Lagrangian fibration

A Lagrangian fibration of a symplectic manifold M is a fibration where all of the fibres are Lagrangian submanifolds. Since M is even-dimensional we can take local coordinates (p₁,…,p_n, q₁,…,q_n), and by Darboux's theorem the symplectic form ω can be, at least locally, written as ω = ∑ dp_k ∧ dq_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∗Rn,{displaystyle T^{*}mathbb {R} ^{n},} and the Lagrangian fibration as the trivial fibration π:T∗Rn→Rn.{displaystyle pi :T^{*}mathbb {R} ^{n}to mathbb {R} ^{n}.} This is the canonical picture.

Lagrangian mapping

Let L be a Lagrangian submanifold of a symplectic manifold (K,ω) given by an immersion i : L ↪ K (i is called a Lagrangian immersion). Let π : K ↠ B give a Lagrangian fibration of K. The composite (π ∘ i) : L ↪ K ↠ B is a Lagrangian mapping. The critical value set of π ∘ i is called a caustic.

Two Lagrangian maps (π₁ ∘ i₁) : L₁ ↪ K₁ ↠ B₁ and (π₂ ∘ i₂) : L₂ ↪ K₂ ↠ B₂ are called Lagrangian equivalent if there exist diffeomorphisms σ, τ and ν such that both sides of the diagram given on the right commute, and τ preserves the symplectic form.^[4] Symbolically:

τ∘i1=i2∘σ, ν∘π1=π2∘τ, τ∗ω2=ω1,{displaystyle tau circ i_{1}=i_{2}circ sigma , nu circ pi _{1}=pi _{2}circ tau , tau ^{*}omega _{2}=omega _{1},,}

where τ^∗ω₂ denotes the pull back of ω₂ by τ.

Special cases and generalizations

• A symplectic manifold endowed with a metric that is 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 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.^[5]
  

• A polysymplectic manifold is a Legendre bundle provided with a polysymplectic tangent-valued (n+2){displaystyle (n+2)}-form; it is utilized in Hamiltonian field theory.^[6]
  

See also

Notes

References

• 
  Dusa McDuff and D. Salamon: Introduction to Symplectic Topology (1998) Oxford Mathematical Monographs,
  ISBN 0-19-850451-9.


• 
  Denis Auroux Seminar on Mirror Symmetry https://math.berkeley.edu/~auroux/290s16.html
  


• 
  Eckhard Meinrenken Symplectic Geometry http://www.math.toronto.edu/mein/teaching/LectureNotes/sympl.pdf
  


• 
  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.


• 
  Alan Weinstein (1971). "Symplectic manifolds and their lagrangian submanifolds". Advances in Mathematics. 6 (3): 329–46. doi:10.1016/0001-8708(71)90020-X.
  

External links

• How to find Lagrangian Submanifolds - Math.Stackexchange


• 
  Ü. Lumiste (2001) [1994], "Symplectic Structure", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  


• 
  Sardanashvily, G., Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians,arXiv: 0908.1886
  


• 
  McDuff, D., Symplectic structures - a new approach to geometry, Notices of the AMS, November 1998


• 
  "Examples of symplectic manifolds". PlanetMath.
  


• 
  Hitchin, Nigel (1999). "Lectures on Special Lagrangian Submanifolds". arXiv:math/9907034.