   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. == introduction == 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 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, 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 :\omega = dp \wedge dq and is an area form that measures the area a of a region s in the plane through integration: :a = \int_s \omega. 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 : ((x_1,x_2), (x_3,x_4),\ldots(x_{2n-1},x_{2n})) in a 2n-dimensional manifold along with a symplectic form :\omega = dx_1 \wedge dx_2 + dx_3 \wedge dx_4 + \cdots + dx_{2n-1} \wedge dx_{2n}. 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 :a = \int_v \omega = \int_v dx_1 \wedge dx_2 + \int_v dx_3 \wedge dx_4 + \cdots + \int_v dx_{2n-1} \wedge dx_{2n}. ==comparison with riemannian geometry== symplectic geometry has a number of simil
Symplectic_geometry