Symplectic manifold

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In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, 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.

Contents

Definition

A symplectic form on a manifold M is a closed non-degenerate differential 2-form ω[1]. The non-degeneracy condition means that for all pM we have the property that there does not exist non-zero XTpM such that ω(X,Y) = 0 for all YTpM. The skew-symmetric condition means that for all pM we have ω(X,Y) = −ω(Y,X) for all X,YTpM. Recall that in odd dimensions antisymmetric matrices aren't invertible. Since ω is a differential two-form the skew-symmetric condition implies that M has even dimension[1]. The closed condition means that the exterior derivative of ω, namely dω, is identically zero. A symplectic manifold consists a pair (M,ω), of a manifold M and a symplectic form ω. Assigning a symplectic form ω to a manifold M is referred to as giving M a symplectic structure.

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