Principle of least action

In physics, the principle of least action or principle of stationary action is a variational principle by 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 action principle is preceded by earlier ideas in optics, dating back to Ancient Greece. 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.

Origins, statement, and dispute

The earliest precedents of the principle of least action can be found in studies of the specular reflection and refraction of light. Hero of Alexandria noted that the law of reflection $\theta_{i}=\theta_{r}$ follows from the assumption that light travels along the shortest distance between two given points. This was generalized to refraction by Pierre de Fermat, who, in the 17th century, refined the principle to "light travels between two given points along the path of shortest time"; now known as the principle of least time or Fermat's principle. Over the centuries, other natural philosophers, ranging from Socrates to Leibniz have suggested that Nature acts by the most economical means, although this concept was not defined mathematically.

Credit for the formulation of the principle of least action is commonly given to Pierre Louis Maupertuis, who wrote about it in 1744 and 1746, 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." 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". He begins the second paragraph :

"Sit massa corporis projecti

M, ejusque, dum spatiolum

ds emetitur, celeritas debita altitudini

v; erit quantitas motus corporis in hoc loco

M\sqrt{v}

; quae per ipsum spatiolum ds multiplicata, dabit

M\,ds\sqrt{v}

motum corporis collectivum per spatiolum ds. Iam dico lineam a corpore descriptam ita fore comparatam, ut, inter omnes alias lineas iisdem terminis contentas, sit

\int M ds \sqrt{v}

, seu, ob M constans,

\int ds \sqrt{v}

minimum."

A translation of this passage reads:

"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

M \sqrt{v}

that, when multiplied by the distance ds, will give

M ds \sqrt{v}

, 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

\int M ds \sqrt{v}

or, provided that M is constant,

\int ds \sqrt{v}

As Euler states, $\int M ds \sqrt{v}$ is the integral of the momentum over distance travelled which, in modern notation, equals the reduced action $\int p\,dq$ . 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, and even the King of Prussia entered the debate, defending Mauperius, 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.. The claims of forgery were re-examined 150 years later, and archival work by C.I. Gerhardt in 1898 and W. Kabitz in 1913 uncovered other copies of the letter, and three others cited by König, in the Bernoulli archives.

Further development

Euler continued to write on the topic; in his Reflexions sur quelques loix generales de la nature (1748), he call 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.

The full importance of the principle to mechanics was stated by Joseph Louis Lagrange in 1760 (need ref), although the variational principle was not used to derive the equations of motion until almost 75 years later, when William Rowan Hamilton in 1834 and 1835 applied the variational principle to the function $L=T-V$ to obtain what are now called the Lagrangian equations of motion.

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 two-dimensional surfaces . The first clear general statements were given by Marston Morse in the 1920's and 1930's, 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.

Apparent teleology?

Although equivalent mathematically, there is an important philosophical difference between the differential equations of motion and their integral counterpart. 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) 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.,

Given that the particle begins at position $x_{1}$ at time $t_{1}$ and ends at position $x_{2}$ at time $t_{2}$ , the physical trajectory that connects these two endpoints is an extremum of the action integral.

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.

References

P.L.N. de Maupertuis, Accord de différentes lois de la nature qui avaient jusqu'ici paru incompatibles. (1744) Mém. As. Sc. Paris p. 417.

P.L.N. de Maupertuis, Le lois de mouvement et du repos, déduites d'un principe de métaphysique. (1746) Mém. Ac. Berlin, p. 267.

Leonhard Euler, Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes. (1744) Bousquet, Lausanne & Geneva. 320 pages. Reprinted in Leonhardi Euleri Opera Omnia: Series I vol 24. (1952) C. Cartheodory (ed.) Orell Fuessli, Zurich. scanned copy of complete text at The Euler Archive, Dartmouth.

W.R. Hamilton, "On a General Method in Dynamics.", Philosophical Transaction of the Royal Society Part I (1834) p.247-308; Part II (1835) p. 95-144. (From the collection Sir William Rowan Hamilton (1805-1865): Mathematical Papers edited by David R. Wilkins, School of Mathematics, Trinity College, Dublin 2, Ireland. (2000); also reviewed as On a General Method in Dynamics)

G.C.J. Jacobi, Vorlesungen über Dynamik, gehalten an der Universität Königsberg im Wintersemester 1842-1843. A. Clebsch (ed.) (1866); Reimer; Berlin. 290 pages, available online Œuvres complètes volume 8 at Gallica-Math from the Gallica Bibliothèque nationale de France.

Gerhardt CI. (1898) "Über die vier Briefe von Leibniz, die Samuel König in dem Appel au public, Leide MDCCLIII, veröffentlicht hat", Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, I, 419-427.

Kabitz W. (1913) "Über eine in Gotha aufgefundene Abschrift des von S. König in seinem Streite mit Maupertuis und der Akademie veröffentlichten, seinerzeit für unecht erklärten Leibnizbriefes", Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, II, 632-638.

Marston Morse (1934). "The Calculus of Variations in the Large", American Mathematical Society Colloquium Publication 18; New York.

Chris Davis. Idle theory (1998)

Euler, Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes: Additamentum II, Ibid.

J J O'Connor and E F Robertson, "The Berlin Academy and forgery", (2003), at The MacTutor History of Mathematics archive.

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