On the Origins of Hamilton’s Principle(s)

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1		Danilo Capecchi	--	1902	2024-09-29 07:10:17

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This is a peer-reviewed entry published in Encyclopedia Journal, full entry can be found at 10.3390/encyclopedia4040094

This entry first provides an overview of the historical, cultural and epistemological background that is key for Hamilton’s positions on mechanics. We consider the investigations on geometrical optics in the 17th and 18th centuries, Euler’s and Lagrange’s foundations of variational calculus in the 18th century to find extrema of physical quantities expressed as infinite sums of infinitesimals (today, we would say ‘definite integrals’), and Lagrange’s introduction of a revolutionary analytical mechanics, all of which are all fertile grounds for Hamilton’s steps—first, in what we could call analytical optics, then in an advanced form of analytical mechanics. Having provided such an overview, we run through some of Hamilton’s original papers to highlight how he posed his principle(s) in the wake of his forerunners and how his principles are linked with the search for a unitary view of physics.

Hamilton’s principle(s) analytical mechanics stationary action least action varying action

This entry aims to present some formulations of the principles of stationarity of action by William Rowan Hamilton (1805–1865). They are derived from two basic concepts, namely the purely mathematical concept of stationarity and the physical concept of action. Historically, the mathematical concept comes first and is generally presented as a minimum principle. Several examples, such as the isoperimetric problem and the law of reflection, date back to antiquity ^[1]. A problem of finding minima in physics, without the need to introduce the concept of action, is the minimum time problem formulated in 1657 by Pierre de Fermat (1601–1665) for the law of refraction ^[2].

Almost a century later, Pierre-Louis Moreau de Maupertuis (1698–1759) formulated the problem of finding minima in mechanics, introducing the concept of action; the field of application of this early work was still optics, which was treated as corpuscular theory. In the paper Accord de différents loix de la nature qui avoient jusqu’ici paru incompatibles of 1744 ^[3], which was devoted to the refraction of light rays seen as straight segments, Maupertuis formulated his principle of minimum by introducing the word action, which, by Maupertuis’ own admission, goes back to an analogous definition proposed by Gottfried Wilhelm Leibniz (1646–1716) and is based on the metaphysical principle according to which “nature for the production of its effects always operates with the simplest means” (^[4], p. 297). This presupposes the validity of final causes in physics.

Before applying the principle of minimum to the refraction of light, Maupertuis referred to Fermat, praising him for his brilliant idea but criticising him for using a wrong principle. Fermat assumed that a ray of light passing from a point (A) of a given medium, where light travels at speed a, to a point (B) of another medium, where light travels at speed b, with the two media separated by a plane, travels the path requiring the minimum time. If the first medium is more rarefied than the second, then

a > b

, and the angle of refraction is less than the angle of incidence, as experience shows ^[2]^[5]. According to Maupertuis, although the result is correct, the approach is wrong because he believed that the speed of light grows with the density of the medium. This position, which we know to be incorrect, was, however, assumed by both René Descartes (1596–1650) and Isaac Newton (1643–1727), and it was natural for Maupertuis to inherit it. Thus, Maupertuis proposed that the searched minimum is not time but the effort that nature makes, i.e., the action, which depends, according to Maupertuis, “on the velocity of the body and the space it passes through, but it is neither velocity nor space taken separately. It is rather proportional to the sum of the spaces multiplied by the speeds with which they are passed” (^[3], p. 423).

This definition of the physical quantity called action is, in no way, justified; thus, a skilful reader may suspect that Maupertuis chose an ad hoc expression on the basis of the result to be obtained, which was known in advance. If the speed of light is V in the less dense medium and W in the denser medium and the positions of points A and B are given, the action is defined by

V \times A R + W \times R B

, with R representing the point of incidence and refraction of the light ray. This action should be minimised by varying the position of the point of incidence and refraction (R); the obtained result is the correct one according to our standards. Maupertuis concluded his article by recalling the hostility of most mathematicians to the idea of resorting to final causes in physics, claiming that he, himself, partially agreed with this criticism, even considering the errors into which one can fall by using it, as Fermat and Leibniz did. But for him, “it is not the principle in itself that led them to error, but rather the hurry [with which they applied it]” (^[3], p. 423).

In the paper Les loix du mouvement et du repos, déduites d’un principe de métaphysique of 1746, Maupertuis extended his principle to mechanics, i.e., statics and dynamics (^[6], p. 425), defining action for a body as the product of its mass, its velocity and the length of the path it runs. However vague (the time lapse is unspecified, just to limit to one remark), this definition provided the right result for the test problem of the collision of two bodies, regardless of their rigidity ^[7]^[8]. An important part of the paper is the perspective with which the principle of least action is presented. Instead of simply referring to nature, which operates with the minimum effort, Maupertuis brought into play God himself and presented the principle of least action as a proof of the existence of God, to the extent that the original title of the paper was The laws of motion and rest derived from the attributes of God (^[9], p. 270). In fact, God had a dual role according to Maupertuis—on the one hand, the existence of God, which is certain, with the attribute of infinite wisdom making the principle of least action reliable and; on the other hand, the truthfulness of the principle of least action, deduced by experimental and theoretical results, is the proof of the existence of God.

The introduction of the principle of least action into mechanics is, however, much more complex than its application to refraction. Maupertuis was fully aware of this; thus, he came to consult Leonhard Euler (1707–1783), whose greater skill in mathematics he acknowledged and with whom he exchanged some letters on the matter ^[7]^[8]. Euler appreciated Maupertuis’ work, which provided him with some suggestions for applications to mechanics and to the development of the calculus of variations. The latter, which had found its very origin in the well-known problem of the brachistochrone posed by Johann Bernoulli (1667–1748) in 1696 ^[10], was becoming a trendy and challenging task for both pure and applied mathematicians.

Indeed, a powerful step towards the development of analytical mechanics and, thus, Hamilton’s principles is due to Euler’s masterpiece, Methodus inveniendi… of 1744 ^[11], where the solution of not only mathematical but also classical mechanical problems was reduced to the search of maxima and minima (maximi minimive) of certain definite integrals that could be the length of a curve or the surface of an area in geometry or the action in mechanics. As far as mechanics is concerned, Euler did not undertake the task of integrating Newton’s differential equations of motion directly, but he searched the ‘actual’ trajectory of the body as the one that makes action a minimum among the possible actions between fixed initial and final points. This poses the basis for variational calculus as an extension of differential calculus for functions of several variables ^[10]. The integral providing the action was made to depend on the values of the unknown minimising function in a finite number of sampling points between the initial and the final points so that the search of a stationary point was reduced to the ordinary vanishing of the action with respect to these unknown values. This let Euler find the trajectory of the body or the buckled shape of a compressed column (the Appendix Additamentum primum: de curvis elasticis of ^[11] is also quoted as the milestone for the mathematical theories on the bifurcation of static solutions).

Euler’s work ^[11] inspired Joseph-Louis Lagrange (1736–1813), who exchanged correspondence with Euler on resolution techniques for finding extremaof the definite integrals that we now call functionals. Thus, he contributed to the establishment of the basis of variational calculus and proposed an original technique still considered basic today, with small adjustments ^[10], i.e., that one should not consider the several possible values of the searched minimising (or maximising) function at sample points but at all points by introducing what we now call variations of the actual solution between fixed initial and final values. The variations are regular enough functions that have the same values at the initial and final points of the domain of integration for the functional; roughly speaking, one shall then evaluate the difference between the values attained by the action at any two ‘near’ variations and take the limit as a small variation, thus finding the so-called Euler-Lagrange equations for the stationarity of the functional.

Lagrange’s innovative application of such a mathematical approach to mechanics was presented in his masterpiece, Mécanique analytique (1st edition, 1788 ^[12]; 2nd edition, 1811 ^[13]). For what we now call conservative fields of forces, action is the accumulation, between the initial and final points of the trajectory, of a function (now dubbed ‘Lagrangian’) that expresses the excess of kinetic energy with respect to the potential energy. Then, Euler’s application of Maupertuis’ principle to the mathematical problems of the search of maxima and minima demands that this integral be stationary, so Lagrange could obtain what we call Euler–Lagrange equations of motion, which are still studied in every class of rational mechanics. Lagrange was well aware of this novelty, and in the preface of ^[12], he wrote,

I decided to reduce the theory of this Science [Mechanics] and the techniques to solve the relevant problems to general formulas, the simple development of which provides all the equations that are necessary to solve any problem. […] On the other hand, this work will have another usefulness: it will unite and present from the same viewpoint the different principles found until now to ease the resolution of problems in mechanics, and will let us be able to judge about their exactness and range of validity.

(^[12], p.v)

(Je me suis proposé de réduire la théorie de cette Science, & l’art de résoudre les problèmes qui s’y rapportent, à des formules générales, dont le simple développement donne toutes les équations nécessaires pour la solution de chaque problème. […] Cet Ouvrage aura d’ailleurs une autre utilité; il réunira & présentera sous un même point de vue, les differens Principes trouvés jusqu’ici pour faciliter la solution des questions de Mechanique, en montrera la liaison & la dépendance mutuelle, & mettra à portée de juger de leur justesse & de leur étendue.)

Therefore, it is undoubtable that, apart from his own admissions, Hamilton’s grounds are deeply rooted in these works and are indebted to these epistemological views, yet they were developed in a very personal way. Thus, in the following, we present Hamilton’s approach to the problem of the least action, where the action gradually leaves any physical meaning to become a function whose stationarity expresses the law of mechanics.

References

Rojo, A.; Bloch, A. The Principle of Least Action: History and Physics; The University Press: Cambridge, UK, 2018.
Fermat, P. Œuvres de Fermat, Publiées par les soins de MM. Paul Tannery et Charles Henry; Gauthier et Villars: Paris, France, 1891–1912. (In French)
Moreau de Maupertuis, P.-L. Accord de différentes loix de la nature qui avoient jusqu’ici paru incompatibles. MÉMoires L’AcadÉ Mie R. Des Sci. Paris 1744, 2, 417–426. (In French)
Capecchi, D. The Problem of the Motion of Bodies; Springer: Cham, Switzerland, 2014.
Andersen, K. The mathematical technique in Fermat’s deduction of the law of refraction. Hist. Math. 1983, 10, 48–62.
Moreau de Maupertuis, P.-L. Les loix du mouvement et du repos, déduites d’un principe de métaphysique. Hist. L’AcadÉMie R. Des Sci. Des Belles Lettres Berl. 1746, 2, 267–294. (In French)
Brunet, P. Étude Historique sur le Principe de la Moindre Action; Hermann: Paris, France, 1936. (In French)
Israel, G. Il principio di minima azione e il finalismo in meccanica. Scienze 1997, 58, 70–76. (In Italian)
Terrall, M. The Man Who Flattened the Earth: Maupertuis and the Sciences in the Enlightenment; Chicago University Press: Chicago, IL, USA, 2002.
Lanczos, C. The Variational Principles of Mechanics; The University of Toronto Press: Toronto, ON, Canada, 1949.
Euler, L. Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, Sive Solutio Problematis Isoperimetrici Latissimo Sensu Accepti; Bousquet & Socios: Lausanne & Genève, Switzerland, 1744. (In Latin)
Lagrange, J.-L. Mèchanique Analitique; Desaint: Paris, France, 1788. (In French)
Lagrange, J.-L. Mècanique Analytique; Courcier: Paris, France, 1811. (In French)

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Online Date: 29 Sep 2024

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