ABSTRACT
This chapter explores how Hamiltonian’s equations of motion motivate the notion of symplectic manifolds. In special relativity, Einstein’s perspective of viewing spacetime as a single unit, equipped with a modified notion of metric, is properly modeled by Minkowski spaces. The chapter discusses this, along with its natural generalization to pseudo-Riemannian manifolds. The classical study of dynamics relies almost exclusively on Sir Isaac Newton’s laws of motion, in particular, his second law. The chapter illustrates how the theory of manifolds, equipped with some additional structure, is ideally suited for the area of mathematical physics. The definition of a Riemannian manifold arose from assuming a smooth manifold came equipped with an inner product on every tangent space, that varied smoothly across the manifold. Most textbooks on general relativity take a considerable amount of time to develop the techniques of analysis on manifolds, in particular, pseudo-Riemannian manifolds.
