 Symplectic geometry

A symplectic form is a closed, nondegenerate differential two-form $\omega$ on a smooth manifold. Nondegeneracy forces even dimension and converts the differential of a function into a Hamiltonian vector field. For example, with $\omega ={\sum }_{j}d{q}_j\wedge d{p}_j$ and the convention ${\iota }_{X_H}\omega =dH$, the Hamiltonian flow satisfiesThus the geometry of phase space encodes the equations of classical mechanics. The cotangent bundle supplies this structure intrinsically, without a choice of coordinates.^[ref]

The Symplectic Darboux theorem makes every such form locally standard; there is no local curvature invariant analogous to Riemann curvature tensor. Global geometry still imposes strong constraints. A symplectomorphism preserves symplectic volume, while the Gromov non-squeezing theorem obstructs squeezing a ball through a narrow conjugate coordinate plane even when the target has infinite volume. Lagrangian submanifolds, symplectic reduction and J-holomorphic curves describe different ways this global rigidity interacts with dynamics and topology.

Symplectic geometry is a branch of differential geometry and mathematics that deals with symplectic manifolds, which are even-dimensional manifolds equipped with a closed non-degenerate differential 2-form known as a symplectic form. This structure is pivotal in various areas of mathematics and physics, particularly in classical mechanics.