A symplectic manifold is a pair (M,ω) of a smooth manifold M together with a closed non degenerate differential 2-form ω, the symplectic form. "Non-degenerate" means that for every vector u in the tangent space at a point, there is a vector v such that the skew product
Fundamental examples of symplectic manifolds are given by the cotangent bundles of manifolds; these arise in classical mechanics, where 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. Kähler manifolds[?] are also symplectic manifolds. Many exotic examples exist, for example those constructed in dimension 4 by Gompf and others.
Directly from the definition, one can show that M is of even dimension 2n and that ωn is a nowhere vanishing form, the symplectic volume form. It follows that a symplectic manifold is canonically oriented and comes with a canonical measure, the Liouville measure[?].
On a symplectic manifold, every differentiable function, H, defines a unique Hamiltonian vectorfield[?] XH. It is defined such that for every vectorfield Y on M the identity
holds. The Hamiltonian vectorfields give the functions on M the structure of a Lie algebra with bracket the Poisson bracket[?]
(other sign conventions are also in use)
The flow of a Hamiltonian vectorfield is a symplectomorphism i.e. a diffeomorphism that preserves the symplectic form. This follows from the closedness of the symplectic form and the expression of the Lie derivative[?] in terms of the exterior derivative. As a direct consequence we have Liouville's theorem: the symplectic volume is invariant under the Hamiltionan flow. Since {H,H} = XH(H) = 0 the flow of a Hamiltonian vectorfield 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.
Unlike Riemannian manifolds, symplectic manifolds are extremely non-rigid: they have many symplectomorphisms coming from Hamiltonian vectorfields. 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
Finite-dimensional subgroups of the group of symplectomorphisms are Lie groups. Representations of these Lie groups 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 Lie operator from the Lie algebra to the Lie algebra of continuous linear operators is also sometimes called the quantization, and is a more common way of looking at it among physicists.
In symplectic topology pseudo holomorphic curves and symplectic capcities should be mentioned.
Examples would be nice. Why are these things studied? I suspect because of physics?
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