Symplectic geometry

Symplectic geometry studies smooth manifolds equipped with a closed, nondegenerate differential two-form. This structure underlies Hamiltonian mechanics and differs from a metric because it measures oriented area rather than length.

Differential form: A differential form is a smooth field of alternating multilinear functions on tangent spaces, integrated over chains. A symplectic form is a differential two-form.
Hamiltonian mechanics: Hamiltonian mechanics describes classical motion using a state space, an energy function, and equations for its evolution. Its equations of motion are encoded by the symplectic form and a Hamiltonian.
Phase space: Phase space is the space of all possible states of a physical system, with positions and momenta represented together. A classical system's phase space commonly carries its canonical symplectic form.
Riemannian geometry: Riemannian geometry studies smooth manifolds equipped with smoothly varying positive-definite inner products on tangent spaces. A Riemannian metric measures lengths and angles, unlike a symplectic form.
Joseph-Louis Lagrange: Joseph-Louis Lagrange was an Italian-born mathematician and astronomer whose work shaped analytical mechanics. His variational mechanics helped establish the mathematical setting later expressed geometrically.
Exterior derivative: The exterior derivative maps differential forms to forms of one higher degree and satisfies a graded product rule. Closedness means the symplectic form has zero exterior derivative.
Hamiltonian vector field: A Hamiltonian vector field is the vector field determined by a function and a symplectic form through contraction. It converts a function on a symplectic manifold into a flow.
Cotangent bundle: The cotangent bundle of a manifold collects all covectors based at its points. Cotangent bundles provide the standard symplectic phase spaces of mechanics.
Contact geometry: Contact geometry studies odd-dimensional manifolds equipped with maximally nonintegrable hyperplane fields. Contact structures are the odd-dimensional counterpart to symplectic structures.
William Rowan Hamilton: William Rowan Hamilton was an Irish mathematician who formulated a influential reformulation of classical mechanics. Hamiltonian mechanics supplied the central physical structure behind symplectic geometry.

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