In physics, the principle of least action or more accurately principle of stationary action is a variational principle which, 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 distance which minimized the path 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.^{[1]} The credit for the formulation of the principle as it applies to the action is often given to PierreLouis Moreau de Maupertuis, who wrote about it in 1744^{[2]} and 1746^{[3]}. However, scholarship indicates that this claim of priority is not so clear; Leonhard Euler discussed the principle in 1744^{[4]}, and there is evidence that Gottfried Leibniz preceded both by 39 years^{[5]}^{[6]}^{[7]}.
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In the 17th century Pierre de Fermat postulated that "light travels between two given points along the path of shortest time" which is known as the principle of least time or Fermat's principle.
Credit for the formulation of the principle of least action is commonly given to Pierre Louis Maupertuis, who wrote about it in 1744^{[2]} and 1746^{[3]}, although the true priority is less clear, as discussed below.
Maupertuis felt that "Nature is thrifty in all its actions", and applied the principle broadly:
The laws of movement and of rest deduced from this principle being precisely the same as those observed in nature, we can admire the application of it to all phenomena. The movement of animals, the vegetative growth of plants ... are only its consequences; and the spectacle of the universe becomes so much the grander, so much more beautiful, the worthier of its Author, when one knows that a small number of laws, most wisely established, suffice for all movements.^{[8]}
This notion of Maupertuis, although somewhat deterministic today, does capture much of the essence of mechanics.
In application to physics, Maupertuis suggested that the quantity to be minimized was the product of the duration (time) of movement within a system by the "vis viva", twice what we now call the kinetic energy of the system.
Leonhard Euler gave a formulation of the action principle in 1744, in very recognizable terms, in the Additamentum 2 to his Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes^{[4]}. He begins the second paragraph ^{[9]}:
Let the mass of the projectile be M, and let its squared velocity resulting from its height be v while being moved over a distance ds. The body will have a momentum Mv that, when multiplied by the distance ds, will give Mv ds, the momentum of the body integrated over the distance ds. Now I assert that the curve thus described by the body to be the curve (from among all other curves connecting the same endpoints) that minimizes or, provided that M is constant, .
As Euler states, is the integral of the momentum over distance traveled which, in modern notation, equals the reduced action . Thus, Euler made an equivalent and (apparently) independent statement of the variational principle in the same year as Maupertuis, albeit slightly later. Curiously, Euler did not claim any priority, as the following episode shows.
Maupertuis' priority was disputed in 1751 by the mathematician Samuel König, who claimed that it had been invented by Gottfried Leibniz in 1707. Although similar to many of Leibniz's arguments, the principle itself has not been documented in Leibniz's works. König himself showed a copy of a 1707 letter from Leibniz to Jacob Hermann with the principle, but the original letter has been lost. In contentious proceedings, König was accused of forgery^{[5]}, and even the King of Prussia entered the debate, defending Maupertuis, while Voltaire defended König.
Euler, rather than claiming priority, was a staunch defender of Maupertuis, and Euler himself prosecuted König for forgery before the Berlin Academy on 13 April 1752.^{[5]} The claims of forgery were reexamined 150 years later, and archival work by C.I. Gerhardt in 1898^{[6]} and W. Kabitz in 1913^{[7]} uncovered other copies of the letter, and three others cited by König, in the Bernoulli archives.
Euler continued to write on the topic; in his Reflexions sur quelques loix generales de la nature (1748), he called the quantity "effort". His expression corresponds to what we would now call potential energy, so that his statement of least action in statics is equivalent to the principle that a system of bodies at rest will adopt a configuration that minimizes total potential energy.
Much of the calculus of variations was stated by Joseph Louis Lagrange in 1760^{[10]}^{[11]} and he proceeded to apply this to problems in dynamics. In Méchanique Analytique (1788) Lagrange derived the general equations of motion of a mechanical body^{[12]}. William Rowan Hamilton in 1834 and 1835^{[13]} applied the variational principle to the function L = T − V to obtain the Lagrangian equations in its present form.
In 1842, Carl Gustav Jacobi tackled the problem of whether the variational principle found minima or other extrema (e.g. a saddle point); most of his work focused on geodesics on twodimensional surfaces^{[14]}. The first clear general statements were given by Marston Morse in the 1920s and 1930s^{[15]}, leading to what is now known as Morse theory. For example, Morse showed that the number of conjugate points in a trajectory equalled the number of negative eigenvalues in the second variation of the Lagrangian.
Other extremal principles of classical mechanics have been formulated, such as Gauss' principle of least constraint and its corollary, Hertz's principle of least curvature.
The mathematical equivalence of the differential equations of motion and their integral counterpart has important philosophical implications. The differential equations are statements about quantities localized to a single point in space or single moment of time. For example, Newton's second law F = ma states that the instantaneous force F applied to a mass m produces an acceleration a at the same instant. By contrast, the action principle is not localized to a point; rather, it involves integrals over an interval of time and (for fields) an extended region of space. Moreover, in the usual formulation of classical action principles, the initial and final states of the system are fixed, e.g.,
In particular, the fixing of the final state appears to give the action principle a teleological character which has been controversial historically. This apparent teleology is eliminated in the quantum mechanical version of the action principle.
