Inculcation: Symplectic geometry

motivating Hamiltonian vector fields

Given a smooth , how can we produce a vector field that “preserves” this function? Well Hamilton’s equations on

produces a vector field perpendicular to the gradient of which does the job! But in doing so, this “Hamiltonian vector field” of produces nice geometrical properties. For example it has zero divergence, implying its flow preserves area (in )!

Symplectic geometry helps study the geometry of such vector fields in a general setting on “Symplectic manifolds” which are smooth manifolds with the minimum structure needed to define Hamiltonian vector fields.