Symplectic Geometry

Symplectic geometry is a branch of mathematics that studies spaces equipped with a specific structure, called a symplectic form, which captures the notion of area in a way that is preserved under certain transformations. In Hamiltonian mechanics, this structure allows us to describe the phase space of a physical system—comprising positions and momenta—so that the evolution of the system can be analyzed. The symplectic form ensures that the laws of physics, such as conservation of energy, emerge naturally from the geometric properties of the space, facilitating our understanding of dynamics in a geometric framework.