# Symplectic manifold

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Type of manifold in differential geometry

In [differential geometry](/source/Differential_geometry), a **symplectic manifold** is a [smooth manifold](/source/Differentiable_manifold#Definition), M {\displaystyle M} , equipped with a [closed](/source/Closed_and_exact_differential_forms) [nondegenerate](/source/Nondegenerate_form) [differential 2-form](/source/Differential_form) ω {\displaystyle \omega } , called the symplectic form. The study of symplectic manifolds is called [symplectic geometry](/source/Symplectic_geometry) or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of [classical mechanics](/source/Classical_mechanics) and [analytical mechanics](/source/Analytical_mechanics) as the [cotangent bundles](/source/Cotangent_bundle) of manifolds. For example, in the [Hamiltonian formulation](/source/Hamiltonian_mechanics) 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](/source/Phase_space) of the system.

## Motivation

Symplectic manifolds arise from [classical mechanics](/source/Classical_mechanics); in particular, they are a generalization of the [phase space](/source/Phase_space) of a closed system.[1][2] In the same way the [Hamilton equations](/source/Hamilton_equations) allow one to derive the time evolution of a system from a set of [differential equations](/source/Differential_equation), the symplectic form should allow one to obtain a [vector field](/source/Vector_field) describing the flow of the system from the differential d H {\displaystyle dH} of a Hamiltonian function H {\displaystyle H} .[3] So we require a linear map T M → T ∗ M {\displaystyle TM\rightarrow T^{*}M} from the [tangent manifold](/source/Tangent_manifold) T M {\displaystyle TM} to the [cotangent manifold](/source/Cotangent_manifold) T ∗ M {\displaystyle T^{*}M} , or equivalently, an element of T ∗ M ⊗ T ∗ M {\displaystyle T^{*}M\otimes T^{*}M} . Letting ω {\displaystyle \omega } denote a [section](/source/Section_(fiber_bundle)) of T ∗ M ⊗ T ∗ M {\displaystyle T^{*}M\otimes T^{*}M} , the requirement that ω {\displaystyle \omega } be [non-degenerate](/source/Degenerate_form) ensures that for every differential d H {\displaystyle dH} there is a unique corresponding vector field V H {\displaystyle V_{H}} such that d H = ω ( V H , ⋅ ) {\displaystyle dH=\omega (V_{H},\cdot )} . Since one desires the Hamiltonian to be constant along flow lines, one should have ω ( V H , V H ) = d H ( V H ) = 0 {\displaystyle \omega (V_{H},V_{H})=dH(V_{H})=0} , which implies that ω {\displaystyle \omega } is [alternating](/source/Alternating_form) and hence a 2-form. Finally, one makes the requirement that ω {\displaystyle \omega } should not change under flow lines, i.e. that the [Lie derivative](/source/Lie_derivative) of ω {\displaystyle \omega } along V H {\displaystyle V_{H}} vanishes. Applying [Cartan's formula](/source/Cartan_homotopy_formula), this amounts to (here ι X {\displaystyle \iota _{X}} is the [interior product](/source/Interior_product)):

- L V H ( ω ) = 0 ⇔ d ( ι V H ω ) + ι V H d ω = d ( d H ) + d ω ( V H ) = d ω ( V H ) = 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 V H {\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 V H {\displaystyle V_{H}} corresponding to arbitrary smooth H {\displaystyle H} is equivalent to the requirement that *ω* should be [closed](/source/Closed_and_exact_differential_forms).

## Definition

Let M {\displaystyle M} be a smooth [manifold](/source/Manifold). A **symplectic form** on M {\displaystyle M} is a closed non-degenerate differential [2-form](/source/2-form) ω {\displaystyle \omega } .[4][5] Here, non-degenerate means that for every point p ∈ M {\displaystyle p\in M} , the skew-symmetric pairing on the [tangent space](/source/Tangent_space) T p M {\displaystyle T_{p}M} defined by ω {\displaystyle \omega } is non-degenerate. That is to say, if there exists an X ∈ T p M {\displaystyle X\in T_{p}M} such that ω ( X , Y ) = 0 {\displaystyle \omega (X,Y)=0} for all Y ∈ T p M {\displaystyle Y\in T_{p}M} , then X = 0 {\displaystyle X=0} . The closed condition means that the [exterior derivative](/source/Exterior_derivative) of ω {\displaystyle \omega } vanishes.[4][5]

A **symplectic manifold** is a pair ( M , ω ) {\displaystyle (M,\omega )} where M {\displaystyle M} is a smooth manifold and ω {\displaystyle \omega } is a symplectic form. Assigning a symplectic form to M {\displaystyle M} is referred to as giving M {\displaystyle M} a **symplectic structure**. Since in odd dimensions, [skew-symmetric matrices](/source/Skew-symmetric_matrices) are always singular, nondegeneracy implies that dim ⁡ M {\displaystyle \dim M} is even.[6]

By nondegeneracy, ω {\displaystyle \omega } can be used to define a pair of [**musical isomorphisms**](/source/Musical_isomorphism) ω ♭ : T M → T ∗ M , ω ♯ : T ∗ M → T M {\displaystyle \omega ^{\flat }:TM\rightarrow T^{*}M,\omega ^{\sharp }:T^{*}M\rightarrow TM} , such that ω ( X , Y ) = ω ♭ ( X ) ( Y ) {\displaystyle \omega (X,Y)=\omega ^{\flat }(X)(Y)} for any two vector fields X , Y {\displaystyle X,Y} , and ω ♯ ∘ ω ♭ = Id {\displaystyle \omega ^{\sharp }\circ \omega ^{\flat }=\operatorname {Id} } .

A symplectic manifold ( M , ω ) {\displaystyle (M,\omega )} is **exact** iff the symplectic form ω {\displaystyle \omega } is [exact](/source/Closed_and_exact_differential_forms), i.e. equal to ω = − d θ {\displaystyle \omega =-d\theta } for some 1-form θ {\displaystyle \theta } . The symplectic form on any compact symplectic manifold without boundary is inexact, by [Stokes' theorem](/source/Stokes'_theorem).[7]

By [Darboux's theorem](/source/Darboux's_theorem), around any point p {\displaystyle p} there exists a local coordinate system, in which ω = Σ i d p i ∧ d q i {\displaystyle \omega =\Sigma _{i}dp_{i}\wedge dq^{i}} , where d denotes the [exterior derivative](/source/Exterior_derivative) and ∧ denotes the [exterior product](/source/Exterior_product).[8] This form is called the [Poincaré two-form](/source/Poincar%C3%A9_two-form) or the **canonical two-form**. Thus, we can locally think of *M* as being the [cotangent bundle](/source/Cotangent_bundle) T ∗ R n {\displaystyle T^{*}\mathbb {R} ^{n}} and generated by the corresponding [tautological 1-form](/source/Tautological_one-form) θ = Σ i p i d q i , ω = d θ {\displaystyle \theta =\Sigma _{i}p_{i}dq^{i},\;\omega =d\theta } .

A (local) **Liouville form** is any (locally defined) λ {\displaystyle \lambda } such that ω = d λ {\displaystyle \omega =d\lambda } . A vector field X {\displaystyle X} is (locally) **Liouville** iff L X ω = ω {\displaystyle {\mathcal {L}}_{X}\omega =\omega } . By [Cartan's magic formula](/source/Cartan's_magic_formula), this is equivalent to d ( ω ( X , ⋅ ) ) = ω {\displaystyle d(\omega (X,\cdot ))=\omega } . A Liouville vector field can thus be interpreted as a way to recover a (local) Liouville form. By Darboux's theorem, around any point there exists a local Liouville form, though it might not exist globally.

On a symplectic manifold, every smooth function H : M → R {\displaystyle H:M\to \mathbb {R} } determines a [Hamiltonian vector field](/source/Hamiltonian_vector_field) X H {\displaystyle X_{H}} by ι X H ω = d H {\displaystyle \iota _{X_{H}}\omega =dH} , up to sign convention.[9] The integral curves of X H {\displaystyle X_{H}} are the Hamiltonian flow of H {\displaystyle H} . In classical mechanics, H {\displaystyle H} is the energy function and the symplectic form encodes Hamilton's equations. The set of all Hamiltonian vector fields make up a [Lie algebra](/source/Lie_algebra), and is written as ( Ham ⁡ ( M ) , [ ⋅ , ⋅ ] ) {\displaystyle (\operatorname {Ham} (M),[\cdot ,\cdot ])} where [ ⋅ , ⋅ ] {\displaystyle [\cdot ,\cdot ]} is the [Lie bracket](/source/Lie_bracket_of_vector_fields).

Given any two smooth functions f , g : M → R {\displaystyle f,g:M\to \mathbb {R} } , their **[Poisson bracket](/source/Poisson_bracket)** is defined by { f , g } = ω ( X g , X f ) {\displaystyle \{f,g\}=\omega (X_{g},X_{f})} .[10] This makes any symplectic manifold into a [Poisson manifold](/source/Poisson_manifold).[11] The **Poisson bivector** is a [bivector](/source/Bivector) field π {\displaystyle \pi } defined by { f , g } = π ( d f ∧ d g ) {\displaystyle \{f,g\}=\pi (df\wedge dg)} , or equivalently, by π := ω − 1 {\displaystyle \pi :=\omega ^{-1}} . The Poisson bracket and Lie bracket are related by X { f , g } = [ X f , X g ] {\textstyle X_{\{f,g\}}=[X_{f},X_{g}]} .

## Basic properties

If ( M , ω ) {\displaystyle (M,\omega )} is a symplectic manifold of dimension 2 n {\displaystyle 2n} , then ω n {\displaystyle \omega ^{n}} is a nowhere-vanishing top-degree form. Thus every symplectic manifold is orientable and has a natural volume form, called the symplectic volume form.[6]

Unlike a Riemannian metric, a symplectic form does not define lengths or angles. By [Darboux's theorem](/source/Darboux's_theorem), all symplectic manifolds of the same dimension are locally symplectomorphic. Consequently, symplectic geometry has no local curvature invariant analogous to the Riemannian curvature tensor; many of its main questions are global.[2][8]

## Submanifolds

There are several natural geometric notions of [submanifold](/source/Submanifold) of a symplectic manifold ( M , ω ) {\displaystyle (M,\omega )} . Let N ⊂ M {\displaystyle N\subset M} be a submanifold. It is[12][7]

- **symplectic** iff ω | N {\displaystyle \omega |_{N}} is a symplectic form on N {\displaystyle N} ;

- **isotropic** iff ω | N = 0 {\displaystyle \omega |_{N}=0} , equivalently, iff T p N ⊂ T p N ω {\displaystyle T_{p}N\subset T_{p}N^{\omega }} for any p ∈ N {\displaystyle p\in N} ;

- **coisotropic** iff T p N ω ⊂ T p N {\displaystyle T_{p}N^{\omega }\subset T_{p}N} for any p ∈ N {\displaystyle p\in N} ;

- **Lagrangian** iff it is both **isotropic** and **coisotropic**, i.e. ω | N = 0 {\displaystyle \omega |_{N}=0} and dim N = 1 2 dim ⁡ M {\displaystyle {\text{dim }}N={\tfrac {1}{2}}\dim M} . By the nondegeneracy of ω {\displaystyle \omega } , Lagrangian submanifolds are the maximal isotropic submanifolds and minimal coisotropic submanifolds.

## Lagrangian submanifolds

Lagrangian submanifolds are the most important submanifolds. [Weinstein](/source/Alan_Weinstein) proposed the "symplectic creed": *Everything is a Lagrangian submanifold.* By that, he means that everything in symplectic geometry is most naturally expressed in terms of Lagrangian submanifolds.[13]

A **Lagrangian fibration** of a symplectic manifold *M* is a [fibration](/source/Fibration) where all of the [fibers](/source/Fiber_bundle#Formal_definition) are Lagrangian submanifolds.

Given a submanifold N ⊂ M {\displaystyle N\subset M} of codimension 1, the **characteristic line distribution** on it is the duals to its tangent spaces: T p N ω {\displaystyle T_{p}N^{\omega }} . If there also exists a Liouville vector field X {\displaystyle X} in a neighborhood of it that is [transverse](/source/Transversality) to it. In this case, let α := ω ( X , ⋅ ) | N {\displaystyle \alpha :=\omega (X,\cdot )|_{N}} , then ( N , α ) {\displaystyle (N,\alpha )} is a [contact manifold](/source/Contact_manifold), and we say it is a **contact type** submanifold. In this case, the [Reeb vector field](/source/Reeb_vector_field) is tangent to the characteristic line distribution.

An *n*-submanifold is locally specified by a smooth function u : R n → M {\displaystyle u:\mathbb {R} ^{n}\to M} . It is a Lagrangian submanifold if ω ( ∂ i , ∂ j ) = 0 {\displaystyle \omega (\partial _{i},\partial _{j})=0} for all i , j ∈ 1 : n {\displaystyle i,j\in 1:n} . If locally there is a canonical coordinate system ( q , p ) {\displaystyle (q,p)} , then the condition is equivalent to [ u , v ] p , q = ∑ i = 1 n ( ∂ q i ∂ u ∂ p i ∂ v − ∂ p i ∂ u ∂ q i ∂ v ) = 0 , ∀ i , j ∈ 1 : n {\displaystyle [u,v]_{p,q}=\sum _{i=1}^{n}\left({\frac {\partial q_{i}}{\partial u}}{\frac {\partial p_{i}}{\partial v}}-{\frac {\partial p_{i}}{\partial u}}{\frac {\partial q_{i}}{\partial v}}\right)=0,\quad \forall i,j\in 1:n} where [ ⋅ , ⋅ ] p , q {\displaystyle [\cdot ,\cdot ]_{p,q}} is the [Lagrange bracket](/source/Lagrange_bracket) in this coordinate system.

The graph of a closed 1-form on M {\displaystyle M} is a Lagrangian submanifold of T ∗ M {\displaystyle T^{*}M} . In particular, the graph of d f {\displaystyle df} is Lagrangian. Conversely, if a Lagrangian submanifold L ⊂ T ∗ M {\displaystyle L\subset T^{*}M} projects diffeomorphically to M {\displaystyle M} , then it is the graph of a closed 1-form.[12] It is globally the graph of d f {\displaystyle df} only when that closed 1-form is exact.

### Lagrangian mapping

See also: [symplectic category](/source/Symplectic_category)

Let *L* be a Lagrangian submanifold of a symplectic manifold (*K*,ω) given by an [immersion](/source/Immersion_(mathematics)) *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](/source/Caustic_(mathematics)).[14]

Two Lagrangian maps (*π*1 ∘ *i*1) : *L*1 ↪ *K*1 ↠ *B*1 and (*π*2 ∘ *i*2) : *L*2 ↪ *K*2 ↠ *B*2 are called **Lagrangian equivalent** if there exist [diffeomorphisms](/source/Diffeomorphism) *σ*, *τ* and *ν* such that both sides of the diagram given on the right [commute](/source/Commutative_diagram), and *τ* preserves the symplectic form.[5] Symbolically:

- τ ∘ i 1 = i 2 ∘ σ , ν ∘ π 1 = π 2 ∘ τ , τ ∗ ω 2 = ω 1 , {\displaystyle \tau \circ i_{1}=i_{2}\circ \sigma ,\ \nu \circ \pi _{1}=\pi _{2}\circ \tau ,\ \tau ^{*}\omega _{2}=\omega _{1}\,,}

where *τ*∗*ω*2 denotes the [pull back](/source/Pullback_(differential_geometry)#Pullback_of_differential_forms) of *ω*2 by *τ*.

## Symmetries

Main article: [Symplectomorphism](/source/Symplectomorphism)

A map f : ( M , ω ) → ( M ′ , ω ′ ) {\displaystyle f:(M,\omega )\to (M',\omega ')} between symplectic manifolds is a [symplectomorphism](/source/Symplectomorphism) when it preserves the symplectic structure, i.e. the [pullback](/source/Pullback_(differential_geometry)) is the same f ∗ ω ′ = ω {\displaystyle f^{*}\omega '=\omega } . The most important symplectomorphisms are symplectic flows, i.e. ones generated by integrating a vector field on ( M , ω ) {\displaystyle (M,\omega )} .

Given a vector field X {\displaystyle X} on ( M , ω ) {\displaystyle (M,\omega )} , it generates a symplectic flow iff L X ω = 0 {\displaystyle {\mathcal {L}}_{X}\omega =0} . Such vector fields are called **symplectic**. Any Hamiltonian vector field is symplectic, and conversely, any symplectic vector field is *locally* Hamiltonian.

A property that is preserved under all symplectomorphisms is a **symplectic invariant**. In the spirit of [Erlangen program](/source/Erlangen_program), symplectic geometry is the study of symplectic invariants.

## Examples

### The standard symplectic structure

Main article: [Symplectic vector space](/source/Symplectic_vector_space)

Let { v 1 , … , v 2 n } {\displaystyle \{v_{1},\ldots ,v_{2n}\}} be a basis for R 2 n . {\displaystyle \mathbb {R} ^{2n}.} We define our symplectic form ω {\displaystyle \omega } on this basis as follows:

- ω ( v i , v j ) = { 1 j − i = n with 1 ⩽ i ⩽ n − 1 i − j = n with 1 ⩽ j ⩽ n 0 otherwise {\displaystyle \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}}}

In this case the symplectic form reduces to a simple [bilinear form](/source/Bilinear_form). If I n {\displaystyle I_{n}} denotes the n × n {\displaystyle n\times n} [identity matrix](/source/Identity_matrix) then the matrix, Ω {\displaystyle \Omega } , of this bilinear form is given by the 2 n × 2 n {\displaystyle 2n\times 2n} [block matrix](/source/Block_matrix):

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

That is,

- ω = d x 1 ∧ d y 1 + ⋯ + d x n ∧ d y n . {\displaystyle \omega =\mathrm {d} x_{1}\wedge \mathrm {d} y_{1}+\dotsb +\mathrm {d} x_{n}\wedge \mathrm {d} y_{n}.}

It has a fibration by Lagrangian submanifolds with fixed value of y {\displaystyle y} , i.e. { R n × { y } : y ∈ R n } {\displaystyle \{\mathbb {R} ^{n}\times \{y\}:y\in \mathbb {R} ^{n}\}} .

A Liouville form for this is λ = 1 2 ∑ i ( x i d y i − y i d x i ) {\textstyle \lambda ={\frac {1}{2}}\sum _{i}\left(x_{i}dy_{i}-y_{i}dx_{i}\right)} and ω = d λ {\textstyle \omega =d\lambda } , the Liouville vector field is Y = 1 2 ∑ i ( x i ∂ x i + y i ∂ y i ) , {\displaystyle Y={\frac {1}{2}}\sum _{i}\left(x_{i}\partial _{x_{i}}+y_{i}\partial _{y_{i}}\right),} the radial field. Another Liouville form is Σ i x i d y i {\displaystyle \Sigma _{i}x_{i}dy_{i}} , with Liouville vector field Y = ∑ i x i ∂ x i {\textstyle Y=\sum _{i}x_{i}\partial _{x_{i}}} .

### Surfaces

Every oriented smooth surface with an area form is a symplectic manifold. In dimension two, the closedness condition is automatic for any 2-form.

### Cotangent bundles

If Q {\displaystyle Q} is a smooth manifold, its cotangent bundle T ∗ Q {\displaystyle T^{*}Q} carries a [canonical 1-form](/source/Canonical_1-form) λ {\displaystyle \lambda } , also called the tautological or Liouville 1-form. The exterior derivative ω = d λ {\displaystyle \omega =d\lambda } , up to sign convention, is the [canonical symplectic form](/source/Canonical_symplectic_form) on T ∗ Q {\displaystyle T^{*}Q} , also called the Poincaré two-form.

The canonical 1-form is defined by the property that, for any v ∈ T x , α T ∗ Q {\displaystyle v\in T_{x,\alpha }T^{*}Q} , λ ( v ) = α ( π ∗ v ) {\displaystyle \lambda (v)=\alpha (\pi _{*}v)} where π : T ∗ Q → Q {\displaystyle \pi :T^{*}Q\to Q} is the bundle projection. In local coordinates q i {\displaystyle q^{i}} on Q {\displaystyle Q} , the canonical 1-form is λ = ∑ i = 1 n p i d q i {\displaystyle \lambda =\sum _{i=1}^{n}p_{i}dq^{i}} where p i {\displaystyle p_{i}} are fiber coordinates on the cotangent bundle such that α = ∑ i = 1 n p i ( α ) d q i {\displaystyle \alpha =\sum _{i=1}^{n}p_{i}(\alpha )dq^{i}} . In these coordinates, the canonical symplectic form is

- ω = ∑ i = 1 n d p i ∧ d q i {\displaystyle \omega =\sum _{i=1}^{n}dp_{i}\wedge dq^{i}}

The tautological 1-form λ = ∑ i p i d q i {\displaystyle \lambda =\sum _{i}p_{i}dq^{i}} has Liouville vector field Y = ∑ i p i ∂ p i {\displaystyle Y=\sum _{i}p_{i}\partial _{p_{i}}} , the fiberwise radial field. Its flow dilates covectors: ( q , p ) ↦ ( q , e t p ) {\textstyle (q,p)\mapsto \left(q,e^{t}p\right)} .

The zero section of the cotangent bundle is Lagrangian.

### Kähler manifolds

A [Kähler manifold](/source/K%C3%A4hler_manifold) is a symplectic manifold equipped with a compatible integrable complex structure. They form a particular class of [complex manifolds](/source/Complex_manifold). A large class of examples come from complex [algebraic geometry](/source/Algebraic_geometry). Any smooth complex [projective variety](/source/Projective_variety) V ⊂ C P n {\displaystyle V\subset \mathbb {CP} ^{n}} has a symplectic form which is the restriction of the [Fubini—Study form](/source/Fubini-Study_metric) on the [projective space](/source/Projective_space) C P n {\displaystyle \mathbb {CP} ^{n}} .

A symplectic manifold endowed with a [metric](/source/Metric_tensor) that is [compatible](/source/Almost_complex_manifold#Compatible_triples) with the symplectic form is an [almost Kähler manifold](/source/Almost_K%C3%A4hler_manifold) in the sense that the tangent bundle has an [almost complex structure](/source/Almost_complex_structure), but this need not be [integrable](/source/Integrability_condition). A compatible almost-complex structure is an endomorphism J {\displaystyle J} of the tangent space such that J 2 = − I {\displaystyle J^{2}=-I} , ω ( X , J Y ) = − ω ( J X , Y ) {\displaystyle \omega (X,JY)=-\omega (JX,Y)} , and ω ( X , J X ) ≥ 0 {\displaystyle \omega (X,JX)\geq 0} for all X {\displaystyle X} . For such a compatible almost complex structure, g ( X , Y ) = ω ( X , J Y ) {\displaystyle g(X,Y)=\omega (X,JY)} defines a Riemannian metric. When J {\displaystyle J} is integrable, the resulting symplectic manifold is Kähler.[15]

### Coadjoint orbits

[Coadjoint orbits](/source/Coadjoint_orbit) of [Lie groups](/source/Lie_group) carry natural symplectic forms. If O ⊂ g ∗ {\displaystyle {\mathcal {O}}\subset {\mathfrak {g}}^{*}} is the coadjoint orbit through ξ {\displaystyle \xi } , then tangent vectors at ξ {\displaystyle \xi } have the form ad X ∗ ⁡ ξ {\displaystyle \operatorname {ad} _{X}^{*}\xi } , and the symplectic form is given, up to sign convention, by

- ω ξ ( ad X ∗ ⁡ ξ , ad Y ∗ ⁡ ξ ) = ⟨ ξ , [ X , Y ] ⟩ . {\displaystyle \omega _{\xi }(\operatorname {ad} _{X}^{*}\xi ,\operatorname {ad} _{Y}^{*}\xi )=\langle \xi ,[X,Y]\rangle .}

Coadjoint orbits also arise naturally in [moment map](/source/Moment_map) theory and [symplectic reduction](/source/Symplectic_reduction).[16]

### Lagrangian correspondences

A [symplectomorphism](/source/Symplectomorphism) can be described as a Lagrangian submanifold. If ϕ : ( M , ω M ) → ( N , ω N ) {\displaystyle \phi :(M,\omega _{M})\to (N,\omega _{N})} is a symplectomorphism, then its graph is a Lagrangian submanifold of M ¯ × N {\displaystyle {\overline {M}}\times N} , where M ¯ {\displaystyle {\overline {M}}} denotes M {\displaystyle M} equipped with the symplectic form − ω M {\displaystyle -\omega _{M}} .[17]

More generally, a **Lagrangian correspondence** from M {\displaystyle M} to N {\displaystyle N} is a Lagrangian submanifold of M ¯ × N {\displaystyle {\overline {M}}\times N} . Lagrangian correspondences are used in formulations of the [symplectic category](/source/Symplectic_category) and in [Floer homology](/source/Floer_homology).

## Generalizations

- [Presymplectic manifolds](/source/Presymplectic_form) generalize the symplectic manifolds by only requiring ω {\displaystyle \omega } to be closed, but possibly degenerate. Any submanifold of a symplectic manifold inherits a presymplectic structure.

- [Poisson manifolds](/source/Poisson_manifold) generalize the symplectic manifolds by preserving only the [differential-algebraic](/source/Differential_algebra) structures of a symplectic manifold.

- [Dirac manifolds](/source/Dirac_structure) generalize Poisson manifolds and presymplectic manifolds by preserving even less structure. The definition is designed so that any submanifold of a Poisson manifold induces a Dirac manifold. They can be called "pre-Poisson" manifolds.

- A **multisymplectic manifold** of degree *k* is a manifold equipped with a closed nondegenerate *k*-form.[18]

- 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](/source/Hamiltonian_field_theory).[19]

## See also

- [Mathematics portal](https://en.wikipedia.org/wiki/Portal:Mathematics)

- [Almost symplectic manifold](/source/Almost_symplectic_manifold)

- [Contact manifold](/source/Contact_manifold) – Branch of geometryPages displaying short descriptions of redirect targets—an odd-dimensional counterpart of the symplectic manifold.

- [Covariant Hamiltonian field theory](/source/Covariant_Hamiltonian_field_theory) – Formalism in classical field theory based on Hamiltonian mechanicsPages displaying short descriptions of redirect targets

- [Fedosov manifold](/source/Fedosov_manifold)

- [Poisson bracket](/source/Poisson_bracket) – Operation in Hamiltonian mechanics

- [Symplectic group](/source/Symplectic_group) – Mathematical group

- [Symplectic matrix](/source/Symplectic_matrix) – Mathematical concept

- [Symplectic topology](/source/Symplectic_topology) – Branch of differential geometry and differential topologyPages displaying short descriptions of redirect targets

- [Symplectic vector space](/source/Symplectic_vector_space) – Mathematical concept

- [Symplectomorphism](/source/Symplectomorphism) – Isomorphism of symplectic manifolds

- [Tautological one-form](/source/Tautological_one-form) – Canonical differential form

- [Wirtinger inequality (2-forms)](/source/Wirtinger_inequality_(2-forms))

## Citations

1. **[^](#cite_ref-Webster_1-0)** Webster, Ben (9 January 2012). ["What is a symplectic manifold, really?"](https://sbseminar.wordpress.com/2012/01/09/what-is-a-symplectic-manifold-really/).

1. ^ [***a***](#cite_ref-McDuffSalamon2017Intro_2-0) [***b***](#cite_ref-McDuffSalamon2017Intro_2-1) [McDuff, Dusa](/source/Dusa_McDuff); Salamon, Dietmar (2017). "Introduction". *Introduction to Symplectic Topology* (3rd ed.). Oxford University Press. pp. 1–7. [ISBN](/source/ISBN_(identifier)) [978-0-19-879489-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-879489-9).

1. **[^](#cite_ref-Cohn_3-0)** Cohn, Henry. ["Why symplectic geometry is the natural setting for classical mechanics"](https://math.mit.edu/~cohn/Thoughts/symplectic.html).

1. ^ [***a***](#cite_ref-Gosson_4-0) [***b***](#cite_ref-Gosson_4-1) de Gosson, Maurice (2006). *Symplectic Geometry and Quantum Mechanics*. Basel: Birkhäuser Verlag. p. 10. [ISBN](/source/ISBN_(identifier)) [3-7643-7574-4](https://en.wikipedia.org/wiki/Special:BookSources/3-7643-7574-4).

1. ^ [***a***](#cite_ref-Arnold_5-0) [***b***](#cite_ref-Arnold_5-1) [***c***](#cite_ref-Arnold_5-2) [Arnold, V. I.](/source/Vladimir_Arnold); [Varchenko, A. N.](/source/Alexander_Varchenko); [Gusein-Zade, S. M.](/source/Sabir_Gusein-Zade) (1985). *The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1*. Birkhäuser. [ISBN](/source/ISBN_(identifier)) [0-8176-3187-9](https://en.wikipedia.org/wiki/Special:BookSources/0-8176-3187-9).

1. ^ [***a***](#cite_ref-CannasSymplecticForms_6-0) [***b***](#cite_ref-CannasSymplecticForms_6-1) Cannas da Silva, Ana (2001). *Lectures on Symplectic Geometry*. Lecture Notes in Mathematics. Vol. 1764. Springer. secs. 1.1–1.4. [ISBN](/source/ISBN_(identifier)) [978-3-540-42195-5](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-42195-5).

1. ^ [***a***](#cite_ref-McDuffSalamon2017Ch3_7-0) [***b***](#cite_ref-McDuffSalamon2017Ch3_7-1) [McDuff, Dusa](/source/Dusa_McDuff); Salamon, Dietmar (2017). "3. Symplectic manifolds". *Introduction to Symplectic Topology* (3rd ed.). Oxford University Press. pp. 94–151. [ISBN](/source/ISBN_(identifier)) [978-0-19-879489-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-879489-9).

1. ^ [***a***](#cite_ref-CannasDarboux_8-0) [***b***](#cite_ref-CannasDarboux_8-1) Cannas da Silva, Ana (2001). *Lectures on Symplectic Geometry*. Lecture Notes in Mathematics. Vol. 1764. Springer. sec. 8.1. [ISBN](/source/ISBN_(identifier)) [978-3-540-42195-5](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-42195-5).

1. **[^](#cite_ref-AbrahamMarsden1978_9-0)** [Abraham, Ralph](/source/Ralph_Abraham_(mathematician)); [Marsden, Jerrold E.](/source/Jerrold_E._Marsden) (1978). *Foundations of Mechanics* (2nd ed.). Benjamin/Cummings. ch. 3, sec. 3.2. [ISBN](/source/ISBN_(identifier)) [0-8053-0102-X](https://en.wikipedia.org/wiki/Special:BookSources/0-8053-0102-X).

1. **[^](#cite_ref-CannasHamiltonian_10-0)** Cannas da Silva, Ana (2001). *Lectures on Symplectic Geometry*. Lecture Notes in Mathematics. Vol. 1764. Springer. sec. 18. [ISBN](/source/ISBN_(identifier)) [978-3-540-42195-5](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-42195-5).

1. **[^](#cite_ref-WeinsteinPoisson1983_11-0)** [Weinstein, Alan](/source/Alan_Weinstein) (1983). "The local structure of Poisson manifolds". *Journal of Differential Geometry*. **18** (3): 523–557. [doi](/source/Doi_(identifier)):[10.4310/jdg/1214437787](https://doi.org/10.4310%2Fjdg%2F1214437787).

1. ^ [***a***](#cite_ref-CannasLagrangian_12-0) [***b***](#cite_ref-CannasLagrangian_12-1) Cannas da Silva, Ana (2001). *Lectures on Symplectic Geometry*. Lecture Notes in Mathematics. Vol. 1764. Springer. sec. 3. [ISBN](/source/ISBN_(identifier)) [978-3-540-42195-5](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-42195-5).

1. **[^](#cite_ref-13)** Weinstein, Alan (1981). ["Symplectic geometry"](https://www.ams.org/bull/1981-05-01/S0273-0979-1981-14911-9/). *Bulletin of the American Mathematical Society*. **5** (1): 1–13. [doi](/source/Doi_(identifier)):[10.1090/S0273-0979-1981-14911-9](https://doi.org/10.1090%2FS0273-0979-1981-14911-9). [ISSN](/source/ISSN_(identifier)) [0273-0979](https://search.worldcat.org/issn/0273-0979).

1. **[^](#cite_ref-Arnold1990_14-0)** [Arnold, V. I.](/source/Vladimir_Arnold) (1990). "1. Symplectic geometry". *Singularities of Caustics and Wave Fronts*. Mathematics and Its Applications. Vol. 62. Springer. [doi](/source/Doi_(identifier)):[10.1007/978-94-011-3330-2](https://doi.org/10.1007%2F978-94-011-3330-2).

1. **[^](#cite_ref-CannasKahler_15-0)** Cannas da Silva, Ana (2001). *Lectures on Symplectic Geometry*. Lecture Notes in Mathematics. Vol. 1764. Springer. secs. 12–17. [ISBN](/source/ISBN_(identifier)) [978-3-540-42195-5](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-42195-5).

1. **[^](#cite_ref-CannasMomentMaps_16-0)** Cannas da Silva, Ana (2001). *Lectures on Symplectic Geometry*. Lecture Notes in Mathematics. Vol. 1764. Springer. secs. 21–22. [ISBN](/source/ISBN_(identifier)) [978-3-540-42195-5](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-42195-5).

1. **[^](#cite_ref-WeinsteinCategory2010_17-0)** [Weinstein, Alan](/source/Alan_Weinstein) (2010). "Symplectic categories". *Portugaliae Mathematica*. **67** (2): 261–278. [arXiv](/source/ArXiv_(identifier)):[0911.4133](https://arxiv.org/abs/0911.4133). [doi](/source/Doi_(identifier)):[10.4171/PM/1866](https://doi.org/10.4171%2FPM%2F1866).

1. **[^](#cite_ref-18)** Cantrijn, F.; Ibort, L. A.; de León, M. (1999). ["On the Geometry of Multisymplectic Manifolds"](https://doi.org/10.1017%2FS1446788700036636). *J. Austral. Math. Soc*. Ser. A. **66** (3): 303–330. [doi](/source/Doi_(identifier)):[10.1017/S1446788700036636](https://doi.org/10.1017%2FS1446788700036636).

1. **[^](#cite_ref-19)** Giachetta, G.; Mangiarotti, L.; [Sardanashvily, G.](/source/Gennadi_Sardanashvily) (1999). "Covariant Hamiltonian equations for field theory". *Journal of Physics*. **A32** (38): 6629–6642. [arXiv](/source/ArXiv_(identifier)):[hep-th/9904062](https://arxiv.org/abs/hep-th/9904062). [Bibcode](/source/Bibcode_(identifier)):[1999JPhA...32.6629G](https://ui.adsabs.harvard.edu/abs/1999JPhA...32.6629G). [doi](/source/Doi_(identifier)):[10.1088/0305-4470/32/38/302](https://doi.org/10.1088%2F0305-4470%2F32%2F38%2F302). [S2CID](/source/S2CID_(identifier)) [204899025](https://api.semanticscholar.org/CorpusID:204899025).

## General and cited references

- [McDuff, Dusa](/source/Dusa_McDuff); Salamon, D. (1998). *Introduction to Symplectic Topology*. Oxford Mathematical Monographs. [ISBN](/source/ISBN_(identifier)) [0-19-850451-9](https://en.wikipedia.org/wiki/Special:BookSources/0-19-850451-9).

- Hofer, Helmut; Zehnder, Eduard (2011). *Symplectic Invariants and Hamiltonian Dynamics*. Modern Birkhäuser Classics. Basel: Springer Basel AG Springer e-books. [ISBN](/source/ISBN_(identifier)) [978-3-0348-0104-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-0348-0104-1).

- [Auroux, Denis](/source/Denis_Auroux). ["Seminar on Mirror Symmetry"](https://math.berkeley.edu/~auroux/290s16.html).

- [Meinrenken, Eckhard](/source/Eckhard_Meinrenken). ["Symplectic Geometry"](https://www.math.toronto.edu/mein/teaching/LectureNotes/sympl.pdf) (PDF).

- [Abraham, Ralph](/source/Ralph_Abraham_(mathematician)); [Marsden, Jerrold E.](/source/Jerrold_E._Marsden) (1978). *Foundations of Mechanics*. London: Benjamin-Cummings. See Section 3.2. [ISBN](/source/ISBN_(identifier)) [0-8053-0102-X](https://en.wikipedia.org/wiki/Special:BookSources/0-8053-0102-X).

- [de Gosson, Maurice A.](/source/Maurice_A._de_Gosson) (2006). *Symplectic Geometry and Quantum Mechanics*. Basel: Birkhäuser Verlag. [ISBN](/source/ISBN_(identifier)) [3-7643-7574-4](https://en.wikipedia.org/wiki/Special:BookSources/3-7643-7574-4).

- [Alan Weinstein](/source/Alan_Weinstein) (1971). ["Symplectic manifolds and their lagrangian submanifolds"](https://doi.org/10.1016%2F0001-8708%2871%2990020-X). *[Advances in Mathematics](/source/Advances_in_Mathematics)*. **6** (3): 329–46. [doi](/source/Doi_(identifier)):[10.1016/0001-8708(71)90020-X](https://doi.org/10.1016%2F0001-8708%2871%2990020-X).

- Arnold, V. I. (1990). "Ch.1, Symplectic geometry". [*Singularities of Caustics and Wave Fronts*](http://link.springer.com/10.1007/978-94-011-3330-2). Mathematics and Its Applications. Vol. 62. Dordrecht: Springer Netherlands. [doi](/source/Doi_(identifier)):[10.1007/978-94-011-3330-2](https://doi.org/10.1007%2F978-94-011-3330-2). [ISBN](/source/ISBN_(identifier)) [978-1-4020-0333-2](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4020-0333-2). [OCLC](/source/OCLC_(identifier)) [22509804](https://search.worldcat.org/oclc/22509804).

## Further reading

- Dunin-Barkowski, Petr (2024). "Symplectic duality for topological recursion". *Transactions of the American Mathematical Society*. [arXiv](/source/ArXiv_(identifier)):[2206.14792](https://arxiv.org/abs/2206.14792). [doi](/source/Doi_(identifier)):[10.1090/tran/9352](https://doi.org/10.1090%2Ftran%2F9352).

- ["How to find Lagrangian Submanifolds"](https://math.stackexchange.com/q/1072200). *[Stack Exchange](/source/Stack_Exchange)*. December 17, 2014.

- [Lumist, Ü.](/source/%C3%9Clo_Lumiste) (2001) [1994], ["Symplectic Structure"](https://www.encyclopediaofmath.org/index.php?title=Symplectic_Structure), *[Encyclopedia of Mathematics](/source/Encyclopedia_of_Mathematics)*, [EMS Press](/source/European_Mathematical_Society)

- [Sardanashvily, G.](/source/Gennadi_Sardanashvily) (2009). "Fibre bundles, jet manifolds and Lagrangian theory". *Lectures for Theoreticians*. [arXiv](/source/ArXiv_(identifier)):[0908.1886](https://arxiv.org/abs/0908.1886).

- [McDuff, D.](/source/Dusa_McDuff) (November 1998). ["Symplectic Structures—A New Approach to Geometry"](https://www.ams.org/notices/199808/mcduff.pdf) (PDF). *Notices of the AMS*.

- [Hitchin, Nigel](/source/Nigel_Hitchin) (1999). "Lectures on Special Lagrangian Submanifolds". [arXiv](/source/ArXiv_(identifier)):[math/9907034](https://arxiv.org/abs/math/9907034).

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