Symplectic manifold

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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. 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.

Contents 1 Linear symplectic manifold 2 Volume form 3 Contact manifolds 4 Lagrangian submanifolds 5 See also 6 References

Linear symplectic manifold

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 and 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 (often normalized to be ωⁿ / n!).

Contact manifolds

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(e^t α)).

Lagrangian submanifolds

The two natural geometric notions of submanifolds of a symplectic manifold are symplectic submanifolds (potentially of any even dimension), where the symplectic form is required to induce a symplectic form on the submanifold, and a Lagrangian submanifold (of half the dimension) on whose tangent space the symplectic form restricts to zero. Lagrangian submanifolds arise naturally in many physical and geometric situations; for instance, the graph of a symplectomorphism in the product symplectic manifold (M × M, ω × −ω) is Lagrangian.

See also

• Kähler manifold
• Poisson bracket
• symplectic topology
• symplectic vector space
• almost complex manifold
• symplectic group, symplectic matrix
• tautological one-form

References

• Dusa McDuff and D. Salamon: Introduction to Symplectic Topology (1998) Oxford Mathematical Monographs, ISBN 0-198-50451-9.
• Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 3.2.de:Symplektische Geometrie

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