12/08/11 22:13:45.98
Symplecticの由来
URLﾘﾝｸ(en.wikipedia.org)
Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds;
that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.
Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold.
Symplectic geometry has a number of similarities and differences with Riemannian geometry,
which is the study of differentiable manifolds equipped with nondegenerate, symmetric 2-tensors (called metric tensors).
Unlike in the Riemannian case, symplectic manifolds have no local invariants such as curvature.
This is a consequence of Darboux's theorem which states that a neighborhood of any point of a 2n-dimensional symplectic manifold is isomorphic to the standard symplectic structure on an open set of R2n.
Another difference with Riemannian geometry is that not every differentiable manifold need admit a symplectic form; there are certain topological restrictions. For example, every symplectic manifold is even-dimensional and orientable.
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