Principle of Least Action

This article discusses the history of the principle of least action. For the application, please refer to action (physics).

In physics, the principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system. The principle led to the development of the Lagrangian and Hamiltonian formulations of classical mechanics.

The principle remains central in modern physics and mathematics, being applied in the theory of relativity, quantum mechanics and quantum field theory, and a focus of modern mathematical investigation in Morse theory. This article deals primarily with the historical development of the idea; a treatment of the mathematical description and derivation can be found in the article on the action. The chief examples of the principle of stationary action are Maupertuis' principle and Hamilton's principle.

The action principle is preceded by earlier ideas in surveying and optics. The rope stretchers of ancient Egypt stretched corded ropes between two points to measure the path which minimized the distance of separation, and Claudius Ptolemy, in his Geographia (Bk 1, Ch 2), emphasized that one must correct for "deviations from a straight course"; in ancient Greece Euclid states in his Catoptrica that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection; and Hero of Alexandria later showed that this path was the shortest length and least time. But the credit for the formulation of the principle as it applies to the action is often given to Pierre-Louis Moreau de Maupertuis, who wrote about it in 1744 and 1746. However, scholarship indicates that this claim of priority is not so clear; Leonhard Euler discussed the principle in 1744, and there is evidence that Gottfried Leibniz preceded both by 39 years.