The Principle of Least Action

Selected references.

Maupertuis

[In 1746 Maupertuis formulated] the Principle of Least Action, which is all too commonly credited to one of the three great mathematicians, Euler, Lagrange, and Hamilton, who further developed it. This Principle is indeed one of the greatest generalizations in all physical science, although not fully appreciated until the advent of quantum mechanics in the present century. Maupertuis arrived at this principle from a feeling that the very perfection of the universe demands a certain economy in nature and is opposed to any needless expenditure of energy. Natural motions must be such as to make some quantity a minimum. It was only necessary to find that quantity, and this he proceeded to do. It was the product of the duration (time) of movement within a system by the "vis viva" or twice what we now call the kinetic energy of the system. Having found the quantity that tends to a minimum, Maupertuis regarded the principle as all-inclusive: "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 sort of talk aroused vigorous opposition. Translated into recent biological terminology, however, what is it other than Claude Bernard's principle of the maintenance of the internal environment, W.B. Cannon's principle of homeostasis, or the principle of Le Chatelier: "In a system in equilibrium, when one of the factors which determine the equlibrium is made to vary, the system reacts in such a way as to oppose the variation of the factor, and partially to annul it.¹

"Le Grand principe que la nature, dans la production de ses effets, agit toujours par les voies les plus simples." ²

Leibniz

Leibniz in Tentamen Anagogicum, "On ne regarde pas au plus grand ou au plus petit, mais generalement au plus determine ou au plus simple."³

Euler

In his 1748 paper, Euler in "Reflexions sur quelques loix generales de la nature.." starts by declaring his commitment to the least-action principle. 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. Euler called this quantity "effort".³

Fermat, Hamilton, Thomson, Feynman

Geodesy, as the term is used in physics, is the tendency of physical changes and processes to take the easiest or minimum path. Almost the whole of physics can be represented in geodetic form. Water running downhill seeks the steepest descent, the quickest way down, and water running into a basin, even one with irregular shape and bottom, distributes itself so that its surface is as low as possible, the water then has the minimum potential energy in the earth's gravitational field. Light finds the quickest trajectory through an optical system (Fermat's principle of Least Time). The path of a body in a gravitational field (i.e. free fall in space time) is a geodesic. Feynman's formulation of quantum mechanics is based on a least-action principle, using path integrals. Maxwell's equations can be derived as conditions of least action. Newton's mechanics is contained in Hamilton's principle of least action, and also Gauss's principle of least constraint. Thomson's theorem states that electrically charged particles arrange themselves so as to have the least energy. The Second Law of Thermodynamics requires that thermal systems change along a sequence of configurations, each having a higher probability of occurrence than the preceding configuation. ⁴

When light goes through optical systems, it finds the path of least time, taking short cuts in glass and water, where light travels slower, and longer paths through air... The problem for the physicist is like that of figuring out the quickest way of reaching a person drowning down the beach from you... You run down the beach to almost the point where you are nearest the drowning person, then plunge in and swim. ⁴

Pierre Louis Moreau de Maupertuis, b. 1698, member of the Royal Society, seems to have been an interesting character. He became a disciple of Newton and converted Voltaire and others to the Newtonian view. In 1736 he went on an expedition to Lapland to test the flattening of the earth at the poles. He attempted a calculus of pain and pleasure, studied hexadactylism in family trees, developing a theory of heredity. And he anticipated the survival of the fittest:

"May we not say that, in the fortuitous combination of the productions of Nature, since only those creatures could survive in whose organization a certain degree of adaptation was present, there is nothing extraordinary in the fact that such adaptation is actually found in all those species which now exist? Chance, one might say, turned out a vast number of individuals; a small proportion of these were organized in such a manner that the animal's organs could satisfy their needs. A much greater number showed neither adaptation nor order; these last have perished... Thus the species which we see today are but a small part of all those that a blind destiny has produced. (Maupertuis. Essaie de cosmologie, 1750) ¹

References.

1. Forerunners of Darwin. Glass, Temkin, Strauss. Johns Hopkins. 1959

2. Maupertuis. Oeuvres.

3. Maupertuis: an intellectual biography. David Beeson. The Voltaire Foundation. 1992.

4. Six Roads from Newton. Edward Speyer. Wiley. 1994.