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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. 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. 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 Hamilton's 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. == Motivation == Symplectic manifolds arise from classical mechanics, in particular, they are a generalization of the phase space of a closed system.〔Ben Webster: ''What is a symplectic manifold, really?'' http://sbseminar.wordpress.com/2012/01/09/what-is-a-symplectic-manifold-really/〕 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 ''dH'' of a Hamiltonian function ''H''. As Newton's laws of motion are linear differential equations, such a map should be linear as well.〔Henry Cohn: ''Why symplectic geometry is the natural setting for classical mechanics'' http://research.microsoft.com/en-us/um/people/cohn/thoughts/symplectic.html〕 So we require a linear map ''TM'' → ''T'' *''M'', or equivalently, an element of ''T'' *''M'' ⊗ ''T'' *''M''. Letting ''ω'' denote a section of ''T'' *''M'' ⊗ ''T'' *''M'', the requirement that ''ω'' be non-degenerate ensures that for every differential ''dH'' there is a unique corresponding vector field ''VH'' such that ''dH'' = ''ω''(''VH'', · ). Since one desires the Hamiltonian to be constant along flow lines, one should have ''dH''(''VH'') = ''ω''(''VH'', ''VH'') = 0, which implies that ''ω'' is alternating and hence a 2-form. Finally, one makes the requirement that ''ω'' should not change under flow lines, i.e. that the Lie derivative of ''ω'' along ''VH'' vanishes. Applying Cartan's formula, this amounts to : which is equivalent to the requirement that ''ω'' should be closed. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Symplectic manifold」の詳細全文を読む スポンサード リンク
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