Symplectic manifold

In differential geometry, a symplectic manifold is a smooth manifold, $M$, equipped with a closed nondegenerate differential 2-form $\omega$, 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.

Motivation

Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dH} of a Hamiltonian function $H$. So we require a linear map Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle TM\rightarrow T^{*}M} from the tangent manifold Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle TM} to the cotangent manifold $T^{*}M$, or equivalently, an element of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T^{*}M\otimes T^{*}M} . Letting $\omega$ denote a section of $T^{*}M\otimes T^{*}M$, the requirement that $\omega$ be non-degenerate ensures that for every differential Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dH} there is a unique corresponding vector field $V_{H}$ such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dH=\omega (V_{H},\cdot )} . Since one desires the Hamiltonian to be constant along flow lines, one should have Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \omega (V_{H},V_{H})=dH(V_{H})=0} , which implies that $\omega$ is alternating and hence a 2-form. Finally, one makes the requirement that $\omega$ should not change under flow lines, i.e. that the Lie derivative of $\omega$ along $V_{H}$ vanishes. Applying Cartan's formula, this amounts to (here Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \iota _{X}} is the interior product):

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 $H$ such that the corresponding $V_{H}$ 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 $V_{H}$ corresponding to arbitrary smooth $H$ is equivalent to the requirement that ω should be closed.

Definition

Let $M$ be a smooth manifold. A symplectic form on $M$ is a closed non-degenerate differential 2-form $\omega$. Here, non-degenerate means that for every point Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p\in M} , the skew-symmetric pairing on the tangent space Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{p}M} defined by $\omega$ is non-degenerate. That is to say, if there exists an Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle X\in T_{p}M} such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \omega (X,Y)=0} for all Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Y\in T_{p}M} , then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle X=0} . The closed condition means that the exterior derivative of $\omega$ vanishes.

A symplectic manifold is a pair $(M,\omega )$ where $M$ is a smooth manifold and $\omega$ is a symplectic form. Assigning a symplectic form to $M$ is referred to as giving $M$ a symplectic structure. Since in odd dimensions, skew-symmetric matrices are always singular, nondegeneracy implies that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \dim M} is even.

By nondegeneracy, $\omega$ can be used to define a pair of musical isomorphisms Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \omega ^{\flat }:TM\rightarrow T^{*}M,\omega ^{\sharp }:T^{*}M\rightarrow TM} , such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \omega (X,Y)=\omega ^{\flat }(X)(Y)} for any two vector fields Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle X,Y} , and $\omega ^{\sharp }\circ \omega ^{\flat }=\operatorname {Id}$.

A symplectic manifold $(M,\omega )$ is exact iff the symplectic form $\omega$ is exact, i.e. equal to Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \omega =-d\theta } for some 1-form $\theta$. The symplectic form on any compact symplectic manifold without boundary is inexact, by Stokes' theorem.

By Darboux's theorem, around any point $p$ there exists a local coordinate system, in which Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \omega =\Sigma _{i}dp_{i}\wedge dq^{i}} , where d denotes the exterior derivative and ∧ denotes the exterior product. This form is called the Poincaré two-form or the canonical two-form. Thus, we can locally think of M as being the cotangent bundle Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T^{*}\mathbb {R} ^{n}} and generated by the corresponding tautological 1-form Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \theta =\Sigma _{i}p_{i}dq^{i},\;\omega =d\theta } .

A (local) Liouville form is any (locally defined) $\lambda$ such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \omega =d\lambda } . A vector field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is (locally) Liouville iff Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal L_X \omega = \omega} . By Cartan's magic formula, this is equivalent to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H:M\to\mathbb R} determines a Hamiltonian vector field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_H} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iota_{X_H}\omega=dH} , up to sign convention. The integral curves of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_H} are the Hamiltonian flow of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} . In classical mechanics, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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, and is written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\operatorname{Ham}(M), [\cdot, \cdot])} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\cdot, \cdot]} is the Lie bracket.

Given any two smooth functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f, g : M \to \R } , their Poisson bracket is defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{f,g\} = \omega (X_g,X_f) } . This makes any symplectic manifold into a Poisson manifold. The Poisson bivector is a bivector field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi } defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ f,g \} = \pi(df \wedge dg) } , or equivalently, by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi := \omega^{-1} } . The Poisson bracket and Lie bracket are related by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle X_{\{f,g\}} = [X_f,X_g]} .

Basic properties

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M,\omega)} is a symplectic manifold of dimension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2n} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.

Unlike a Riemannian metric, a symplectic form does not define lengths or angles. By 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.

Submanifolds

There are several natural geometric notions of submanifold of a symplectic manifold Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, \omega) } . Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \subset M} be a submanifold. It is

• symplectic iff Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega|_N } is a symplectic form on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N } ;
• isotropic iff Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega|_N = 0 } , equivalently, iff Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_p N \subset T_p N^\omega } for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \in N } ;
• coisotropic iff Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_p N^\omega \subset T_p N } for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \in N } ;
• Lagrangian iff it is both isotropic and coisotropic, i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega|_N=0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{dim }N=\tfrac{1}{2}\dim M} . By the nondegeneracy of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega } , Lagrangian submanifolds are the maximal isotropic submanifolds and minimal coisotropic submanifolds.

Lagrangian submanifolds

Lagrangian submanifolds are the most important submanifolds. 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.

A Lagrangian fibration of a symplectic manifold M is a fibration where all of the fibers are Lagrangian submanifolds.

Given a submanifold Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \subset M} of codimension 1, the characteristic line distribution on it is the duals to its tangent spaces: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_p N^\omega } . If there also exists a Liouville vector field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} in a neighborhood of it that is transverse to it. In this case, let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha := \omega(X, \cdot)|_N} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (N, \alpha)} is a contact manifold, and we say it is a contact type submanifold. In this case, the Reeb vector field is tangent to the characteristic line distribution.

An n-submanifold is locally specified by a smooth function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u: \R^n \to M} . It is a Lagrangian submanifold if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(\partial_i , \partial_j) = 0} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i, j \in 1:n} . If locally there is a canonical coordinate system Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (q, p)} , then the condition is equivalent to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\cdot, \cdot]_{p, q}} is the Lagrange bracket in this coordinate system.

The graph of a closed 1-form on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is a Lagrangian submanifold of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^*M} . In particular, the graph of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df} is Lagrangian. Conversely, if a Lagrangian submanifold Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L\subset T^*M} projects diffeomorphically to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , then it is the graph of a closed 1-form. It is globally the graph of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df} only when that closed 1-form is exact.

Lagrangian mapping

File:TIKZ PICT FBN.svg

See also: symplectic category

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. Symbolically:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.

Symmetries

Main article: Symplectomorphism

A map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: (M, \omega) \to (M', \omega')} between symplectic manifolds is a symplectomorphism when it preserves the symplectic structure, i.e. the pullback is the same Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^* \omega' = \omega} . The most important symplectomorphisms are symplectic flows, i.e. ones generated by integrating a vector field on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, \omega)} .

Given a vector field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, \omega)} , it generates a symplectic flow iff Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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, symplectic geometry is the study of symplectic invariants.

Examples

The standard symplectic structure

Main article: Symplectic vector space

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{v_1, \ldots, v_{2n}\}} be a basis for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^{2n}.} We define our symplectic form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} on this basis as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_n} denotes the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\times n} identity matrix then the matrix, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} , of this bilinear form is given by the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2n\times 2n} block matrix:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}. }

That is,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} , i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\R^n \times \{y\} : y \in \R^n\}} .

A Liouville form for this is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lambda=\frac{1}{2} \sum_i\left(x_i d y_i-y_i d x_i\right)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \omega=d \lambda} , the Liouville vector field isFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma_i x_i dy_i} , with Liouville vector field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} is a smooth manifold, its cotangent bundle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^*Q} carries a canonical 1-form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} , also called the tautological or Liouville 1-form. The exterior derivative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega=d\lambda} , up to sign convention, is the canonical symplectic form on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^*Q} , also called the Poincaré two-form.

The canonical 1-form is defined by the property that, for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v\in T_{x,\alpha}T^*Q} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda(v)=\alpha(\pi_*v)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi:T^*Q\to Q} is the bundle projection. In local coordinates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^i} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} , the canonical 1-form is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = \sum_{i=1}^n p_idq^i} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_i} are fiber coordinates on the cotangent bundle such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = \sum_{i=1}^n p_i(\alpha)dq^i} . In these coordinates, the canonical symplectic form is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = \sum_{i=1}^n dp_i \wedge dq^i }

The tautological 1-form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = \sum_i p_i dq^i} has Liouville vector field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y = \sum_i p_i \partial_{p_i} } , the fiberwise radial field. Its flow dilates covectors: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle (q, p) \mapsto\left(q, e^t p\right)} .

The zero section of the cotangent bundle is Lagrangian.

Kähler manifolds

A Kähler manifold is a symplectic manifold equipped with a compatible integrable complex structure. They form a particular class of complex manifolds. A large class of examples come from complex algebraic geometry. Any smooth complex projective variety Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V \subset \mathbb{CP}^n} has a symplectic form which is the restriction of the Fubini—Study form on the projective space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{CP}^n} .

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. A compatible almost-complex structure is an endomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} of the tangent space such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J^2=-I} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(X,JY)=-\omega(JX,Y)} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(X,JX) \ge 0} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . For such a compatible almost complex structure, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(X,Y)=\omega(X,JY)} defines a Riemannian metric. When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} is integrable, the resulting symplectic manifold is Kähler.

Coadjoint orbits

Coadjoint orbits of Lie groups carry natural symplectic forms. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal O\subset\mathfrak g^*} is the coadjoint orbit through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \xi} , then tangent vectors at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \xi} have the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{ad}^*_X\xi} , and the symplectic form is given, up to sign convention, by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_\xi(\operatorname{ad}^*_X\xi,\operatorname{ad}^*_Y\xi)=\langle \xi,[X,Y]\rangle.}

Coadjoint orbits also arise naturally in moment map theory and symplectic reduction.

Lagrangian correspondences

A symplectomorphism can be described as a Lagrangian submanifold. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi:(M,\omega_M)\to (N,\omega_N)} is a symplectomorphism, then its graph is a Lagrangian submanifold of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{M}\times N} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{M}} denotes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} equipped with the symplectic form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\omega_M} .

More generally, a Lagrangian correspondence from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is a Lagrangian submanifold of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{M}\times N} . Lagrangian correspondences are used in formulations of the symplectic category and in Floer homology.

Generalizations

• Presymplectic manifolds generalize the symplectic manifolds by only requiring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} to be closed, but possibly degenerate. Any submanifold of a symplectic manifold inherits a presymplectic structure.
• Poisson manifolds generalize the symplectic manifolds by preserving only the differential-algebraic structures of a symplectic manifold.
• Dirac manifolds 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.
• A polysymplectic manifold is a Legendre bundle provided with a polysymplectic tangent-valued Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n+2)} -form; it is utilized in Hamiltonian field theory.

See also

• Almost symplectic manifold
• Contact manifold—an odd-dimensional counterpart of the symplectic manifold.
• Covariant Hamiltonian field theory
• Fedosov manifold
• Poisson bracket
• Symplectic group
• Symplectic matrix
• Symplectic topology
• Symplectic vector space
• Symplectomorphism
• Tautological one-form
• Wirtinger inequality (2-forms)

General and cited references

• McDuff, Dusa; Salamon, D. (1998). Introduction to Symplectic Topology. Oxford Mathematical Monographs. ISBN 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 978-3-0348-0104-1.
• Auroux, Denis. "Seminar on Mirror Symmetry".
• Meinrenken, Eckhard. "Symplectic Geometry" (PDF).
• Abraham, Ralph; Marsden, Jerrold E. (1978). Foundations of Mechanics. London: Benjamin-Cummings. See Section 3.2. ISBN 0-8053-0102-X.
• de Gosson, Maurice A. (2006). Symplectic Geometry and Quantum Mechanics. Basel: Birkhäuser Verlag. 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.
• Arnold, V. I. (1990). "Ch.1, Symplectic geometry". Singularities of Caustics and Wave Fronts. Mathematics and Its Applications. 62. Dordrecht: Springer Netherlands. doi:10.1007/978-94-011-3330-2. ISBN 978-1-4020-0333-2. OCLC 22509804.

Further reading

• Dunin-Barkowski, Petr (2024). "Symplectic duality for topological recursion". Transactions of the American Mathematical Society. arXiv:2206.14792. doi:10.1090/tran/9352.
• "How to find Lagrangian Submanifolds". Stack Exchange. December 17, 2014.
• Template:Springer
• Sardanashvily, G. (2009). "Fibre bundles, jet manifolds and Lagrangian theory". Lectures for Theoreticians. arXiv:0908.1886.
• McDuff, D. (November 1998). "Symplectic Structures—A New Approach to Geometry" (PDF). Notices of the AMS.
• Hitchin, Nigel (1999). "Lectures on Special Lagrangian Submanifolds". arXiv:math/9907034.