Symplectic geometry

Symplectic geometry is a branch of differential geometry and differential topology that 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.Weyl (1939, p. 165) Symplectic geometry is a branch of differential geometry and differential topology that 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. A symplectic geometry is defined on a smooth even-dimensional space that is a differentiable manifold. On this space is defined a geometric object, the symplectic form, that allows for the measurement of sizes of two-dimensional objects in the space. The symplectic form in symplectic geometry plays a role analogous to that of the metric tensor in Riemannian geometry. Where the metric tensor measures lengths and angles, the symplectic form measures oriented areas. Symplectic geometry arose from the study of classical mechanics and an example of a symplectic structure is the motion of an object in one dimension. To specify the trajectory of the object, one requires both the position q and the momentum p, which form a point (p,q) in the Euclidean plane ℝ2. In this case, the symplectic form is and is an area form that measures the area A of a region S in the plane through integration: The area is important because as conservative dynamical systems evolve in time, this area is invariant. Higher dimensional symplectic geometries are defined analogously. A 2n-dimensional symplectic geometry is formed of pairs of directions in a 2n-dimensional manifold along with a symplectic form This symplectic form yields the size of a 2n-dimensional region V in the space as the sum of the areas of the projections of V onto each of the planes formed by the pairs of directions Symplectic geometry has a number of similarities with and differences from 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. Additionally, if M is a closed symplectic manifold, then the 2nd de Rham cohomology group H2(M) is nontrivial; this implies, for example, that the only n-sphere that admits a symplectic form is the 2-sphere. A parallel that one can draw between the two subjects is the analogy between geodesics in Riemannian geometry and pseudoholomorphic curves in symplectic geometry: Geodesics are curves of shortest length (locally), while pseudoholomorphic curves are surfaces of minimal area. Both concepts play a fundamental role in their respective disciplines.