Symplectic manifold

A smooth manifold equipped with a closed, nondegenerate differential two-form. This structure provides a geometric framework for classical mechanics and Hamiltonian dynamics.

Differential form: A smooth assignment of alternating multilinear functions to tangent spaces, used to integrate and differentiate geometric quantities. A symplectic structure is a differential two-form.
Hamiltonian vector field: The vector field determined by a function and a symplectic form through contraction with that form. It translates a Hamiltonian function into instantaneous motion.
Phase space: A space whose points encode the complete states of a physical system. A classical system's phase space typically carries a natural symplectic form.
Riemannian manifold: A smooth manifold equipped with a smoothly varying positive-definite inner product on tangent spaces. It measures lengths and angles, unlike a symplectic form's pairings.
Symplectic capacity: A numerical invariant that measures symplectic size and restricts symplectic embeddings. Capacities expose global rigidity that local coordinate equivalence cannot detect.
Exterior derivative: An operator that maps differential forms to forms of one higher degree and satisfies a graded product rule. Closedness means the exterior derivative of the symplectic form vanishes.
Hamiltonian mechanics: A formulation of classical mechanics in which states and their evolution are described using Hamiltonian functions. Symplectic geometry supplies the structure behind its equations of motion.
Cotangent bundle: The bundle of covectors over a manifold, naturally equipped with a canonical symplectic structure. It provides the standard phase space for mechanical systems.
Contact manifold: An odd-dimensional manifold equipped with a maximally nonintegrable field of hyperplanes. Contact geometry is the odd-dimensional counterpart often linked to symplectic geometry.
Gromov non-squeezing theorem: A theorem stating that a symplectic ball cannot be embedded into a narrower cylinder of smaller radius. It demonstrates a global constraint on symplectic embeddings.

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