Analytical mechanics is a term used for a refined, highly mathematical form of classical mechanics, constructed from the eighteenth century onwards as a formulation of the subject as founded by Isaac Newton. It began with d'Alembert's principle. By analogy with Fermat's principle, which is the variational principle in geometric optics, Maupertuis' principle was discovered in classical mechanics. Using generalized coordinates, we obtain Lagrange's equations. Using the Legendre transformation, we obtain generalized momentum and the Hamiltonian. Hamilton's canonical equations provides integral, while Lagrange's equation provides differential equations. Finally we may derive the Hamilton-Jacobi equation. The study of the solutions of the Hamilton-Jacobi equations leads naturally to the study of symplectic manifolds and symplectic topology. In this formulation, the solutions of the Hamilton-Jacobi equations are the integral curves of Hamiltonian vector fields.
