Symplectic manifold

<p>In <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>, a subject of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a symplectic manifold is a <a href="/facts/Differentiable_manifold/aSnbXKcV">smooth manifold</a>, 
  
    
      
        M
      
    
    {\displaystyle M}
  
, equipped with a <a href="/facts/Closed_and_exact_differential_forms/uvEUbzUK">closed</a> <a href="/facts/Nondegenerate_form/9TLGBvSH">nondegenerate</a> <a href="/facts/Differential_form/T9gyHtMv">differential 2-form</a> 
  
    
      
        ω
      
    
    {\displaystyle \omega }
  
, called the symplectic form. The study of symplectic manifolds is called <a href="/facts/Symplectic_geometry/runXFCea">symplectic geometry</a> or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of <a href="/facts/Classical_mechanics/ykG8upsN">classical mechanics</a> and <a href="/facts/Analytical_mechanics/aotB0n4t">analytical mechanics</a> as the <a href="/facts/Cotangent_bundle/dewBXXxr">cotangent bundles</a> of manifolds. For example, in the <a href="/facts/Hamiltonian_mechanics/2ReerlNA">Hamiltonian formulation</a> 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 <a href="/facts/Phase_space/qPe4Fq57">phase space</a> of the system.
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Symplectic manifold open-in-new

Symplectic manifold