Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold M equipped with a closed, nondegenerate, 2-form ω called the symplectic form. The study of symplectic manifolds is called symplectic topology.

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 ω_i,n+i = 1; ω_n+i,i = -1; ω_j,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. A proposition known as Darboux's theorem says that locally any symplectic manifold resembles this simple one.

Volume form

Directly from the definition, one can show that every symplectic manifold M is of even dimension 2n; this follows because ωⁿ 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, X_H, called a Hamiltonian vector field. It is defined such that for every vector field Y on M the identity
:dH(Y) = ω(X_H,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(X_f,X_g) = X_g(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} = X_H(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,...,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 (after -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.