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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 Curve s 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. LINEAR SYMPLECTIC MANIFOLD There is a standard 'local' model, namely R2''n'' with ''ωi,n+i'' = 1; ''ωn+i,i'' = -1; ''ωj,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 2''n''; this follows because ω''n'' 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 ω''n'' / ''n''!). CONTACT MANIFOLDS Closely related to symplectic manifolds are the odd-dimensional manifolds known as Contact Manifold s. Any ''2n+1''-dimensional contact manifold (''M'', α) gives rise to a ''2n+2''-dimensional symplectic manifold (''M'' × R, d(e''t'' α)). LAGRANGIAN AND OTHER SUBMANIFOLDS There are several natural geometric notions of Submanifold of a symplectic manifold. There are symplectic submanifolds (potentially of any even dimension), where the symplectic form is required to induce a symplectic form on the submanifold. On '''isotropic submanifolds''', the symplectic form restricts to zero, i.e. each tangent space is an isotropic subspace of the ambient manifold's tangent space. Similarly, if each tangent subspace to a submanifold is coisotropic (the dual of an isotropic subspace), the submanifold is called '''coisotropic'''. The most important case of the above is that of Lagrangian submanifold, which are isotropic submanifolds of maximal dimension, namely half the dimension of the ambient manifold. Lagrangian submanifolds arise naturally in many physical and geometric situations. One major example is that the graph of a symplectomorphism in the product symplectic manifold (''M'' × ''M'', ω × −ω) is Lagrangian. Their intersections display rigidity properties not possessed by smooth manifolds; the Arnold Conjecture gives the sum of the submanifold's Betti numbers as a lower bound for the number of self intersections of a smooth Lagrangian submanifold, rather than the Euler characteristic in the smooth case. SPECIAL CASES AND GENERALIZATIONS A symplectic manifold endowed with a Metric that is compatible with the symplectic form is a Kähler Manifold . Symplectic manifolds are special cases of a Poisson Manifold ; the definition of a symplectic manifold requires that the symplectic form be non-degenerate everywhere. If this condition is violated, the manifold may still be a Poisson manifold. SEE ALSO
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