Hamiltonian Systems and Noether’s Theorem

This paper uses the machinery of symplectic geometry to make rigorous the mathematical framework of Hamiltonian mechanics. This framework is then shown to imply Newton’s laws and conservation of energy, thus validating it as a physical theory. We look at symmetries of physical systems in the form of Lie groups, and show that the Hamiltonian framework grants us the insight that the existence of a symmetry corresponds to the conservation of a physical quantity, i.e. Noether’s Theorem. Throughout the paper we pay heed to the correspondence between mathematical definitions and physical concepts, and supplement these definitions with examples.