Symplectic manifold
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In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. (A 2-form ω on a manifold M is called nondegenerate if for every nonzero vector u in the tangent space at a point, there is a vector v such that ω(u,v) ≠ 0.) The form is called the symplectic form. The contemporary name for the study of symplectic manifolds is symplectic topology.
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Origins in theoretical mechanics
Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds, e.g. 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 modelled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.
Linear case
There is a standard 'local' model, namely R2n with ωi,n+i = 1; ωn+i,i = -1; ωj,k = 0 for all i = 0,...,n-1; j,k=0,...,2n-1 (k ≠ j+n or j ≠ k+n). This is an example of a linear symplectic space. See symplectic vector space.
Volume form
Directly from the definition, one can show that every symplectic manifold M is of even dimension 2n; this follows because aωn is a nowhere vanishing form, the symplectic volume form. It follows that every symplectic manifold is canonically oriented and comes with a canonical measure, the Liouville measure.
Hamiltonian vector fields
Since the symplectic form on a symplectic manifold is nondegenerate, it sets up an isomorphism between the tangent bundle and the cotangent bundle, thus establishing a one-to-one correspondence between tangent vectors and one-forms. As a special case, every differentiable function, H, on a symplectic manifold M defines a unique vector field, XH, called a Hamiltonian vector field. It is defined such that for every vector field Y on M the identity
- dH(Y) = ω(XH,Y)
holds. The Hamiltonian vector fields give the functions on M the structure of a Lie algebra with bracket the Poisson bracket
- {f,g} = ω(Xf,Xg) = Xg(f)
(Warning: other sign conventions are in use).
Symplectomorphisms
The flow of a Hamiltonian vector field (and more generally that of any vector field which corresponds to a closed one-form via the correspondence mentioned above) is a symplectomorphism, i.e., a diffeomorphism that preserves the symplectic form. This follows from the closedness of the symplectic form and Cartan's formula for the Lie derivative in terms of the exterior derivative.
As a direct consequence we have Liouville's theorem: the symplectic volume is invariant under a Hamiltionan flow. Since
- {H,H} = XH(H) = 0
the flow of a Hamiltonian vector field also preserves H. In physics this is interpreted as the law of conservation of energy. Liouville's theorem is interpreted as the conservation of phase volume in Hamiltonian systems, which is the basis for classical statistical mechanics.
We have shown that there is a one-to-one correspondence between infinitesimal symplectomorphisms and closed one-forms on a symplectic manifold. If the first Betti number of the manifold is zero, and it is connected, the latter set is the same as the space of smooth functions modulo addition of constants.
Unlike Riemannian manifolds, symplectic manifolds are extremely non-rigid: they have many symplectomorphisms coming from Hamiltonian vector fields. The fundamental difference between Riemannian and symplectic geometry is that a symplectic manifold has no local invariants: according to Darboux's theorem for every point x in a symplectic manifold there is a local coordinate system called action-angle with coordinates
- p1,...,pn, q1,...,qn,
such that
- ω = ∑ dpi ∧ dqi
Finite-dimensional subgroups of the group of symplectomorphisms are Lie groups. Representations of these Lie groups (after <math>\hbar<math>-deformations, in general) on Hilbert spaces are called quantizations. When the Lie group is the one defined by a Hamiltonian, it is called a "quantization by energy". The corresponding operator from the Lie algebra to the Lie algebra of continuous linear operators is also sometimes called the quantization; this is a more common way of looking at it in physics.
Locally, symplectomorphisms can be generated by a generating function over a (local) Darboux coordinates. See Hamilton-Jacobi equation.
Relationship with Kähler manifolds
Every Kähler manifold is also a symplectic manifold. Well into the 1970s, symplectic experts were unsure of whether any compact non-Kähler symplectic manifolds existed, but since then many examples have been constructed; in particular, Robert Gompf has shown that every finitely presented group occurs as the fundamental group of some symplectic 4-manifold, in marked contrast with the Kähler case.
Pseudoholomorphic curves
Most symplectic manifolds, one can say, are not Kähler; and so do not have an integrable complex structure compatible with the symplectic form. Mikhail Gromov made, however, the important observation that symplectic manifolds do admit an abundance of compatible almost complex structures, so that they satisfy all the axioms for a complex manifold except the requirement that the transition functions be holomorphic.
A Riemann surface mapped into a symplectic manifold compatibly with the almost complex structure is called a pseudoholomorphic curve, and Gromov proved a compactness theorem for such curves; this result has led to the development of a fairly large subdiscipline of symplectic topology. Results arising from Gromov's theory include Gromov's nonsqueezing theorem concerning symplectic embeddings of spheres into cylinders, and also a proof of a conjecture of Vladimir Arnol'd concerning the number of fixed points of Hamiltonian flows. This was proven (in increasing generality) by several researchers beginning with Andreas Floer, who introduced what is now known as Floer homology using Gromov's methods.
Pseudoholomorphic curves are also a source of symplectic invariants, known as Gromov-Witten invariants, by which two different symplectic manifolds could in principle be distinguished.
Related topics
References
- McDuff, D. and Salamon, D.: Introduction to Symplectic Topology (Oxford Mathematical Monographs)
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