The Lagrangian formulation of mechanics is popular in practically all modern treatments of the subject. The ideas date back to the pioneering work of Euler, Lagrange and Hamilton, who showed how the equations of Newtonian dynamics could be derived from variational principles. In these, the evolution of a system is viewed as a path in some parameter space. The path the system follows is one which extremises a quantity called the action, which is the integral of the Lagrangian with respect to the evolution parameter (usually time). The mathematics behind this approach was clear from the outset, but a thorough physical understanding had to wait until the arrival of quantum theory. In the path-integral formulation of quantum mechanics a particle is viewed as simultaneously following all possible paths. By assigning a phase factor to the action for each path and summing these, one obtains the amplitude for a quantum process. The classical limit can then be understood as resulting from trajectories that reinforce the amplitude. In this manner classical trajectories emerge as those which make the action stationary.

A closely related idea is the Hamiltonian formulation of dynamics. The advantage of this approach is that it produces a set of first-order equations, making it well suited to numerical methods. The Hamiltonian approach also exposes the appropriate geometry for classical dynamical systems, which is a symplectic manifold.