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Concept
Symplectic manifold
Summary
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , 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. For example, 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 modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.
Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the symplectic form should allow one to obtain a vector field describing the flow of the system from the differential of a Hamiltonian function . So we require a linear map from the tangent manifold to the cotangent manifold , or equivalently, an element of . Letting denote a section of , the requirement that be non-degenerate ensures that for every differential there is a unique corresponding vector field such that . Since one desires the Hamiltonian to be constant along flow lines, one should have , which implies that is alternating and hence a 2-form. Finally, one makes the requirement that should not change under flow lines, i.e. that the Lie derivative of along vanishes. Applying Cartan's formula, this amounts to (here is the interior product):
so that, on repeating this argument for different smooth functions such that the corresponding span the tangent space at each point the argument is applied at, we see that the requirement for the vanishing Lie derivative along flows of corresponding to arbitrary smooth is equivalent to the requirement that ω should be closed.
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