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. Often the term vectorial mechanics is applied to the form based on Newton's work, to contrast it with analytical mechanics. This distinction makes sense because analytical mechanics uses two scalar properties of motion, the kinetic and potential energies, instead of vector forces, to analyze the motion.^{[1]}
The subject has two parts: Lagrangian mechanics and Hamiltonian mechanics. The Lagrangian formulation identifies the actual path followed by the motion as a selection of the path over which the time integral of kinetic energy is least, assuming the total energy to be fixed, and imposing no conditions on the time of transit. The Hamiltonian formulation is more general, allowing timevarying energy, identifying the path followed to be the one with least action (the integral over the path of the difference between kinetic and potential energies), holding the departure and arrival times fixed.^{[1]}^{[2]} These approaches underlie the path integral formulation of quantum mechanics.
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 HamiltonJacobi equations leads naturally to the study of symplectic manifolds and symplectic topology.^{[3]}^{[4]} In this formulation, the solutions of the Hamilton–Jacobi equations are the integral curves of Hamiltonian vector fields.
