Symplectic topology

Symplectic topology is that part of mathematics concerned with the study of symplectic manifolds. These manifolds arise in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field.

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

ω(u,v) ≠ 0

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[?] X_H. It is defined such that for every vectorfield Y on M the identity

dH(Y) = ω(X_H,Y)

holds. The Hamiltonian vectorfields give the functions on M the structure of a Lie algebra with bracket the Poisson bracket[?]

{f,g} = ω(X_f,X_g) = X_g(f)

(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} = X_H(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,...,p_n, q₁,...,q_n, such that

ω = ∑ dp_i ∧ dq_i

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.

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REMARK as is, this section should be called symplectic geometry The relation between symplectic geometry and Hamiltonian mechanics should explained in more detail.

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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References

• Mc Duff D., Salomon D.: Introduction to Symplectic Topology (Oxford Mathematical Monographs)