In differential geometry, a symplectic manifold is a smooth manifold, <math> M </math>, equipped with a closed nondegenerate differential 2-form <math> \omega </math>, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent 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 <math>dH</math> of a Hamiltonian function <math>H</math>. So we require a linear map <math>TM \rightarrow T^*M </math> from the tangent manifold <math>TM</math> to the cotangent manifold <math> T^* M </math>, or equivalently, an element of <math>T^*M \otimes T^*M</math>. Letting <math>\omega</math> denote a section of <math>T^*M \otimes T^* M</math>, the requirement that <math>\omega</math> be non-degenerate ensures that for every differential <math>dH</math> there is a unique corresponding vector field <math>V_H</math> such that <math>dH = \omega (V_H, \cdot)</math>. Since one desires the Hamiltonian to be constant along flow lines, one should have <math>\omega(V_H, V_H) = dH(V_H) = 0</math>, which implies that <math>\omega</math> is alternating and hence a 2-form. Finally, one makes the requirement that <math>\omega</math> should not change under flow lines, i.e. that the Lie derivative of <math>\omega</math> along <math>V_H</math> vanishes. Applying Cartan's formula, this amounts to (here <math> \iota_X</math> is the interior product):
:<math>\mathcal{L}_{V_H}(\omega) = 0\;\Leftrightarrow\;\mathrm d (\iota_{V_H} \omega) + \iota_{V_H} \mathrm d\omega= \mathrm d (\mathrm d\,H) + \mathrm d\omega(V_H) = \mathrm d\omega(V_H)=0</math>
so that, on repeating this argument for different smooth functions <math>H</math> such that the corresponding <math>V_H</math> span the tangent space at each point the argument is applied at, we see that the requirement for the vanishing Lie derivative along flows of <math>V_H</math> corresponding to arbitrary smooth <math>H</math> is equivalent to the requirement that ω should be closed.
Definition
Let <math> M </math> be a smooth manifold. A symplectic form on <math> M </math> is a closed non-degenerate differential 2-form <math> \omega </math>. Here, non-degenerate means that for every point <math> p \in M </math>, the skew-symmetric pairing on the tangent space <math> T_p M </math> defined by <math> \omega </math> is non-degenerate. That is to say, if there exists an <math> X \in T_p M </math> such that <math> \omega( X, Y ) = 0 </math> for all <math> Y \in T_p M </math>, then <math> X = 0 </math>. The closed condition means that the exterior derivative of <math> \omega </math> vanishes.
By nondegeneracy, <math> \omega </math> can be used to define a pair of musical isomorphisms <math> \omega^\flat: T M \rightarrow T^* M, \omega^\sharp : T^* M \rightarrow T M </math>, such that <math> \omega(X, Y) = \omega^\flat(X) (Y) </math> for any two vector fields <math> X, Y </math>, and <math> \omega^\sharp \circ\omega^\flat = \operatorname{Id} </math>.
A symplectic manifold <math>(M, \omega)</math> is exact iff the symplectic form <math>\omega</math> is exact, i.e. equal to <math>\omega = -d\theta</math> for some 1-form <math>\theta</math>. The symplectic form on any compact symplectic manifold without boundary is inexact, by Stokes' theorem.
By Darboux's theorem, around any point <math>p</math> there exists a local coordinate system, in which <math>\omega = \Sigma_i dp_i \wedge dq^i</math>, 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 <math>T^*\R^n</math> and generated by the corresponding tautological 1-form <math>\theta = \Sigma_i p_i dq^i, \;\omega = d\theta</math>.
A (local) Liouville form is any (locally defined) <math>\lambda</math> such that <math>\omega = d\lambda</math>. A vector field <math> X</math> is (locally) Liouville iff <math> \mathcal L_X \omega = \omega</math>. By Cartan's magic formula, this is equivalent to <math> d(\omega(X, \cdot)) = \omega</math>. 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 <math>H:M\to\mathbb R</math> determines a Hamiltonian vector field <math>X_H</math> by <math>\iota_{X_H}\omega=dH</math>, up to sign convention. The integral curves of <math>X_H</math> are the Hamiltonian flow of <math>H</math>. In classical mechanics, <math>H</math> 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 <math>(\operatorname{Ham}(M), [\cdot, \cdot])</math> where <math>[\cdot, \cdot]</math> is the Lie bracket.
Given any two smooth functions <math> f, g : M \to \R </math>, their Poisson bracket is defined by <math> \{f,g\} = \omega (X_g,X_f) </math>. This makes any symplectic manifold into a Poisson manifold. The Poisson bivector is a bivector field <math> \pi </math> defined by <math> \{ f,g \} = \pi(df \wedge dg) </math>, or equivalently, by <math> \pi := \omega^{-1} </math>. The Poisson bracket and Lie bracket are related by <math display="inline"> X_{\{f,g\ = [X_f,X_g]</math>.
Basic properties
If <math>(M,\omega)</math> is a symplectic manifold of dimension <math>2n</math>, then <math>\omega^n</math> is a nowhere-vanishing top-degree form. Thus every symplectic manifold is orientable and has a natural volume form, called the symplectic volume form.
A Lagrangian fibration of a symplectic manifold M is a fibration where all of the fibers are Lagrangian submanifolds.
Given a submanifold <math>N \subset M</math> of codimension 1, the characteristic line distribution on it is the duals to its tangent spaces: <math>T_p N^\omega </math>. If there also exists a Liouville vector field <math>X</math> in a neighborhood of it that is transverse to it. In this case, let <math>\alpha := \omega(X, \cdot)|_N</math>, then <math>(N, \alpha)</math> 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 <math>u: \R^n \to M</math>. It is a Lagrangian submanifold if <math>\omega(\partial_i , \partial_j) = 0</math> for all <math>i, j \in 1:n</math>. If locally there is a canonical coordinate system <math>(q, p)</math>, then the condition is equivalent to <math display="block">
[ 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
</math>where <math>[\cdot, \cdot]_{p, q}</math> is the Lagrange bracket in this coordinate system.
The graph of a closed 1-form on <math>M</math> is a Lagrangian submanifold of <math>T^*M</math>. In particular, the graph of <math>df</math> is Lagrangian. Conversely, if a Lagrangian submanifold <math>L\subset T^*M</math> projects diffeomorphically to <math>M</math>, then it is the graph of a closed 1-form.
Two Lagrangian maps and 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.
Coadjoint orbits
Coadjoint orbits of Lie groups carry natural symplectic forms. If <math>\mathcal O\subset\mathfrak g^*</math> is the coadjoint orbit through <math>\xi</math>, then tangent vectors at <math>\xi</math> have the form <math>\operatorname{ad}^*_X\xi</math>, and the symplectic form is given, up to sign convention, by
:<math>\omega_\xi(\operatorname{ad}^*_X\xi,\operatorname{ad}^*_Y\xi)=\langle \xi,[X,Y]\rangle.</math>
Coadjoint orbits also arise naturally in moment map theory and symplectic reduction.
Lagrangian correspondences
A symplectomorphism can be described as a Lagrangian submanifold. If <math>\phi:(M,\omega_M)\to (N,\omega_N)</math> is a symplectomorphism, then its graph is a Lagrangian submanifold of <math>\overline{M}\times N</math>, where <math>\overline{M}</math> denotes <math>M</math> equipped with the symplectic form <math>-\omega_M</math>.
More generally, a Lagrangian correspondence from <math>M</math> to <math>N</math> is a Lagrangian submanifold of <math>\overline{M}\times N</math>. Lagrangian correspondences are used in formulations of the symplectic category and in Floer homology.
Generalizations
- Presymplectic manifolds generalize the symplectic manifolds by only requiring <math>\omega</math> 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 <math>(n+2)</math>-form; it is utilized in Hamiltonian field theory.
See also
- —an odd-dimensional counterpart of the symplectic manifold.