, a symplectic manifold
is a smooth manifold M
equipped with a closed
ω 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, 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.
Any real-valued differentiable function H on a symplectic manifold can serve as an energy function or Hamiltonian. Associated to any Hamiltonian is a Hamiltonian vector field; the integral curves of the Hamiltonian vector field are solutions to the Hamilton–Jacobi equations. The Hamiltonian vector field defines a flow on the symplectic manifold, called a Hamiltonian flow or symplectomorphism. By Liouville's theorem, Hamiltonian flows preserve the volume form on the phase space.
A symplectic form on a manifold M
is a nondegenerate closed two-form ω. Explicitly, nondegeneracy of the form means that, relative to any given basis Xi of the tangent space of M at a point, the matrix
is nonsingular (meaning that its determinant is non-zero). Note that Ω, being a skew-symmetric non-singular matrix, must have an even number of rows and columns. Thus the dimension of M is necessarily an even number 2n. In intrinsic terms, ω is nondegenerate if and only if its n-th exterior power is non-zero:
Furthermore, ω is required to be closed, meaning that
where d is the exterior derivative.
Linear symplectic manifold
There is a standard 'local' model, namely R2n with
- and for i = 1, ..., 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.
- for all j, k = 1, ..., n with j ≠ k − n and j ≠ k + n.
Directly from the definition, one can show that every symplectic manifold M is of even dimension 2n and 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 (often normalized to be ωn / n!).
Closely related to symplectic manifolds are the odd-dimensional manifolds known as contact manifolds. Any 2n+1-dimensional contact manifold (M, α) gives rise to a 2n+2-dimensional symplectic manifold (M × R, d(et α)).
Lagrangian and other submanifolds
There are several natural geometric notions of submanifold of a symplectic manifold. There are symplectic submanifolds (potentially of any even dimension), where the symplectic form is required to induce a symplectic form on the submanifold. On isotropic submanifolds, the symplectic form restricts to zero, i.e. each tangent space is an isotropic subspace of the ambient manifold's tangent space. Similarly, if each tangent subspace to a submanifold is coisotropic (the dual of an isotropic subspace), the submanifold is called coisotropic.
The most important case of the above is that of Lagrangian submanifolds, which are isotropic submanifolds of maximal dimension, namely half the dimension of the ambient manifold. Lagrangian submanifolds arise naturally in many physical and geometric situations. One major example is that the graph of a symplectomorphism in the product symplectic manifold (M × M, ω × −ω) is Lagrangian. Their intersections display rigidity properties not possessed by smooth manifolds; the Arnold conjecture gives the sum of the submanifold's Betti numbers as a lower bound for the number of self intersections of a smooth Lagrangian submanifold, rather than the Euler characteristic in the smooth case.
Special cases and generalizations
A symplectic manifold endowed with a metric that is compatible with the symplectic form is an almost Kähler manifold in the sense that the tangent bundle has an almost complex structure, but this need not be integrable. Symplectic manifolds are special cases of a Poisson manifold. The definition of a symplectic manifold requires that the symplectic form be non-degenerate everywhere, but if this condition is violated, the manifold may still be a Poisson manifold.
A multisymplectic manifold of degree k is a manifold equipped with a closed nondegenerate k-form. See F. Cantrijn, L. A. Ibort and M. de León, J. Austral. Math. Soc. Ser. A 66 (1999), no. 3, 303-330.
- Dusa McDuff and D. Salamon: Introduction to Symplectic Topology (1998) Oxford Mathematical Monographs, ISBN 0-19-850451-9.
- Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 3.2.
- Alan Weinstein, "Symplectic manifolds and their Lagrangian submanifolds", Adv. Math. 6 (1971), 329–346.