On the Origins of Hamilton’s Principle(s)

Definition

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.

1. Introduction

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 $V\times AR+W\times RB$, 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.

2. First Works on Optics

According to Hamilton, there are two methods in science, namely the inductive (or analytical) method and the deductive (or synthetic) method. Although (geometrical) optics was developed by means of an important mathematical apparatus, it remained essentially an inductive science, since “[…] it has benefited so little in proportion to the power of modern algebra” ([14], p. 5). Despite mentioning the original dates of publication of Hamilton’s works, we refer to their transcription and re-editing by David R. Wilkins, carried out around the 2000s and freely available at https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Papers.html (accessed on 24 July 2024). In Wilkins’ edition, pages are numbered starting from 1 for every paper. Therefore, to be a complete science, optics must develop its deductive side, and for such a deductive side to be satisfactory, it would be appropriate for it to be based on a single principle ([14], p. 5).

Hamilton presented his theory of optics in a lengthy paper with supplements called The theory of the system of rays. The main article was published in the Transactions of the Royal Irish Academy in 1828 [15]; three supplements appeared in the same Transactions in 1830, 1831 and 1837 [16,17,18]. In this paper, Hamilton derived what he called the principle of least action, in connection with the earlier use of this term, for the reflection and refraction of light, starting from the known principle of geometrical optics.

In [15], Hamilton showed how to pass from the inductive to the deductive phase, staring from the experimental law/principle of reflection of light on a uniform medium. He assumed the principle of equality of the angles of incidence and reflection, turning it into a vector version as follows:

When a ray of light is reflected on a mirror, we know from experience that the normal to the mirror at the point of incidence bisects the angle between the incident and reflected rays. If, therefore, two forces, each equal to unity, were to act at the point of incidence, in the directions of the two rays, their resultant would act in the direction of the normal, and would be equal to twice the cosine of the angle of incidence.

([15], p. 12)

The reference to force for rays of light is not uncommon (see, for, example Johann Bernoulli’s work [19], pp. 369–376.), but in this case, this apparently physical interpretation has only the function of introducing vector quantities into the analysis, which, for Hamilton, is fundamental to pursue an abstract and new procedure.

By indicating with $\rho l$, ${\rho }^{\prime }l$ and $nl$ the angles that the incident ray ($\rho$) and the reflected ray (${\rho }^{\prime }$) make with a generic line (l) and that line l makes with the normal (n) to reflecting surface at point O of incidence, the following relation holds:

$$cos\rho l+cos{\rho }^{\prime }l=2cosIcosnl$$

(1)

where I is the angle of incidence. Equation (1) is derived by assuming the rays to be vectors (indeed, the component of two unitary vectors forming an angle ($2I$) is exactly $2cosI$). When l coincides with the coordinate axes in succession, the following relations are obtained:

$$\begin{array}{c}\hfill \begin{array}{c}cos\rho x+cos{\rho }^{\prime }x=2cosIcosnx\\ cos\rho y+cos{\rho }^{\prime }y=2cosIcosny\\ cos\rho z+cos{\rho }^{\prime }z=2cosIcosnz\end{array}\end{array}$$

(2)

Hamilton imagined a variation ($\mathsf{\Delta }$) of the position of the point of incidence ($O\equiv (x,y,z)$), assuming that the incidence ray originates from a point ($P\equiv (X,Y,Z)$) and that the reflected ray converges on $Q\equiv (X^{\prime },Y^{\prime },Z^{\prime })$. As the variation of O is infinitesimal, $\mathrm{\Delta }$ is on the tangent plane of the surface in O and orthogonal to its normal (n); thus, one can write the following:

$$cosnxdx+cosnydy+cosnzdz=0$$

(3)

where $(dz,dy,dz)$ are the components of $\mathsf{\Delta }$ and $cosnx,cosnycosnz$ are the director cosines of n. By replacing the expressions of the director cosines from Equation (2) in Equation (3), one obtains the following:

$$cos\rho xdx+cos\rho ydy+cos\rho xdz+cos{\rho }^{\prime }xdx+cos{\rho }^{\prime }ydy+cos{\rho }^{\prime }zdz=0$$

(4)

According to ordinary geometry, the following relations can be derived:

$$\begin{array}{c}\hfill \begin{array}{c}\rho =\sqrt{{(X-x)}^2+{(Y-y)}^2+{(Z-z)}^2}\\ {\rho }^{\prime }=\sqrt{{(X^{\prime }-x)}^2+{(Y^{\prime }-y)}^2+{(Z^{\prime }-z)}^2}\end{array}\end{array}$$

(5)

where $\rho$ and ${\rho }^{\prime }$ indicate the length of the incident and reflexed rays, respectively, as measured from points O, P and Q. It is not difficult to prove that the cosines of the directions of the ray are the partial derivatives of $\rho$ and ${\rho }^{\prime }$ with respect to the Cartesian coordinates; thus, Equation (4) can be rewritten as follows:

$${\displaystyle \frac{\partial \rho }{\partial x}}dx+{\displaystyle \frac{\partial \rho }{\partial y}}dy+{\displaystyle \frac{\partial \rho }{\partial z}}dz+{\displaystyle \frac{\partial {\rho }^{\prime }}{\partial x}}dx+{\displaystyle \frac{\partial {\rho }^{\prime }}{\partial y}}dy+{\displaystyle \frac{\partial {\rho }^{\prime }}{\partial z}}dz=0,$$

(6)

that is:

$$\delta (\rho +{\rho }^{\prime })=0$$

(7)

In the third supplement to The theory of the system of rays ([16], p. 89), Hamilton treated refraction with a similar approach, coming to the following relation:

$$\delta (\rho +m{\rho }^{\prime })=0$$

(8)

where the symbols have the same meanings as above and m is the index of refraction. The relation can be further generalized to light passing through media with a varying index of refraction ([16], p. 104). The index of refraction (v) is proportional to the inverse of the speed of light in the medium.

$$\delta \int vd\rho =\delta V=0$$

(9)

where $d\rho$ is the elementary path and Hamilton dubs V a characteristic function ([16], p. 108).

Hamilton called his relations the principle of the least action, with reference to Pierre-Simon Laplace (1749–1827), who obtained Equations (7) and (8) with a mechanical approach, assuming that light consists of particles of matter moving with certain velocities that are subjected to forces that are insensible at a sensible distance,

The principle of least action then reduces to this: light arrives from a point outside to a point inside the crystal in such a way that if one adds the product of the straight [path] described outside by its original speed to the product of the straight [path] described inside by the relevant velocity, the sum is a minimum.

([20], p. 108)

(Le principe de la moindre action se réduit donc alors à ce que la lumiere parvient d’un point pris au-dehors, à un point pris dans l’intérieur du cristal; de manière que si l’on ajoute le produit de la droite qu’elle décrit au-dehors, par sa vitesse primitive, au produit de la droite qu’elle décrit au-dedans, par la vitesse correspondante, la somme soit un minimum.)

3. Later Works on Optics

We believe that a clear mark of Hamilton’s view if mechanics is evident in his work, “On a general Method of expressing the Paths of Lights, and of the Planets, by the Coefficients of a Characteristic Function” of 1833 [14]. Although less mentioned than the two papers that Hamilton dedicated to dynamics in the following two years [21,22], this work represents a crucial step in the evolutions of his ideas. Indeed, in the first place, it makes clear that the characteristic function that Hamilton introduces is derived from his studies in optics; in the second place, it shows that many of the most interesting applications of such functions are in dynamics, so that optics and dynamics can be expressed by means of a similar approach.

The foundational character of “On a general Method…” [14] is conspicuous from the beginning because Hamilton premises a brief historical introduction, where he synthetically expounds the evolution of optics and the novelty of his approach. Thus, he asserts that, from antiquity, one knows that light spreads in straight lines (the light rays of geometrical optics) if no obstacle is interposed between the source of light and one’s eye; however, if a mirror, a lens, or any reflecting or refracting medium is interposed between the source and our eye, then the path of light is not uniformly straight but broken into segments. Hamilton recalls that, while the reflection law was known from the Hellenistic period, the law of refraction is credited to Willebrord Snell (1580–1626). This version is not historically complete, since a refraction law was known to Arabic scholars, re-discovered in Western countries before Snell and popularized by Descartes. In the 17th century, three fundamental events happened in the investigation of light. Ole Rømer (1644–1710) proved that its speed is finite [23]; Newton proposed his corpuscular theory in papers beginning in 1675, then collected in his Opticks of 1704; and Christian Huygens (1629–1695) proposed his undulatory theory in the book, Traité de la lumière of 1690, which Hamilton described with the following poetic words:

So that great ocean of ether which bathes the furthest stars, is even newly stirred, by waves that spread and grow, from every source of light, till they move and agitate the whole with their mutual vibrations.

([14], p. 3)

According to Hamilton, Lagrange had the enormous merit to have established the whole of mechanics based on the principles of virtual velocity and those of of Jean-Baptiste Le Rond d’Alembert (1717–1783), who reduced dynamics to statics by introducing suitable inertia forces in his Traité de dynamique of 1743, which was published by David in Paris. Therefore, he naturally raised the question as to whether something similar and suitable for optics exists. Hamilton’s answer was positive—a principle according to which ‘analytical’ optics exists and corresponds to the law of least action or, more precisely, the law of stationary action. However, an even more general principle exists—that of varying action—for the discovery of which Hamilton claims the merit. What he wrote is a true research program, describing the law of stationary action as,

“[…] the last step in the ascending scale of induction, respecting linear paths of light, while the other law [of varying action] may be usefully made the first in the descending and deductive way”.

([14], p. 6)

This means that until Hamilton’s discovery of the law of varying action, deductive optics had not been founded.

Hamilton recalled that the merit of having introduced what we now would call a variational principle for a difficult optical problem must be attributed to Fermat, who assumed the principle of least time in refraction. This foresees that light travels quicker in a less dense medium that in a denser one, in contrast with Descartes’ ideas and, above all, Newton’s theory of light. Paradoxically, the variational principle that was generally accepted at that time derived from mechanics, at the beginning, was formulated in an imprecise manner, in contrast to Fermat’s perfect formulation. It was the famous Maupertuis principle of least action. Maupertuis thought that light respected his principle rather than Fermat’s (as, in fact, it does). Euler had the merit to clarify the concept of action and to show that the quantity is minimised in the actual motion of a material point of mass (m), urged by a central force and moving with velocity a of v along a curve with an arc length of $ds$, i.e., $m\int vds$. Lagrange extended this principle to systems of points and Laplace to refraction. Since, in fact, natural phenomena do not always correspond either to minimal or to maximal values of the action but always to a stationary value of this magnitude, it is appropriate to speak of the principle of stationary action rather than of least action. Hamilton aimed to show that a more general principle exists, namely that of varying action ([14], pp. 6–8).

The principle of stationary action is based on the idea of considering all possible trajectories between two fixed points and showing that, under appropriate conditions, the actual trajectory renders action stationary; on the other hand, the principle of varying action is based on the idea of considering the initial and final points of the searched trajectory as variable so that the first principle can be considered a particular case of the latter when no variation of the end points of the searched trajectory exists.

3.1. Principle of Stationary Action

First, Hamilton presented the easiest example to understand a stationary principle. Suppose that we have to find the shortest distance between two points in Euclidean ambient space, and let V be the length of a line expressed in Cartesian coordinates, i.e.,

$$V=\int dV=\int \sqrt{d{x}^2+d{y}^2+d{z}^2}$$

(10)

An increment is assigned to each coordinate by setting $x_{\epsilon }=x+\epsilon \xi , y_{\epsilon }=y+\epsilon \eta , z_{\epsilon }=z+\epsilon \zeta$, where $\epsilon$ is an arbitrary constant that is understood to be small and $\xi$, $\eta$ and $\zeta$ are arbitrary functions of x, y and z, respectively, that vanish at the extremes of integration. By substituting these into (10) one obtains the following:

$$V_{\epsilon }=\int \sqrt{d{x}_{\epsilon }^2+d{y}_{\epsilon }^2+d{z}_{\epsilon }^2}=\int \sqrt{{\left(dx+\epsilon d\xi \right)}^2+{\left(dy+\epsilon d\eta \right)}^2+{\left(dz+\epsilon d\zeta \right)}^2}$$

(11)

To find the stationary value of the functional in (10), we have to consider $\epsilon$ a variable parameter to calculate the following:

$$\underset{\epsilon \to 0}{lim}{\displaystyle \frac{V_{\epsilon }-V}{\epsilon }}=\int {\displaystyle \frac{dxd\xi +dyd\eta +dzd\zeta }{d{x}^2+d{y}^2+d{z}^2}}=-\int \left(\xi d{\displaystyle \frac{dx}{dV}}+\eta d{\displaystyle \frac{dy}{dV}}+\zeta d{\displaystyle \frac{dz}{dV}}\right)=0$$

(12)

The identity between the two integrals in (12) is obtained through integration by parts and considering that $\xi$, $\eta$ and $\zeta$ vanish at the extremes of integration. The last integral in (12) vanishes only if $d{\displaystyle \frac{dx}{dV}}=d{\displaystyle \frac{dy}{dV}}=d{\displaystyle \frac{dz}{dV}}=0$; hence, the searched curve is the segment of a straight line included between the two extremes ([14], pp. 8–9).

Let us now focus on the other principle, i.e., that of varying action, which is a distinctive feature of Hamilton’s approach, and come back to the example of the curve length. Suppose that two extremes (A and B) are not fixed but vary. In what follows, a prime denotes initial quantities so that $d’$ indicates an infinitesimal variation of the initial value of the quantity it refers to. Therefore, in the integration by parts performed in (12), the following additional term is considered:

$$\xi {\displaystyle \frac{dx}{dV}}+\eta {\displaystyle \frac{dy}{dV}}+\zeta {\displaystyle \frac{dz}{dV}}+{\xi }^{\prime }{\displaystyle \frac{d^{\prime }{x}^{\prime }}{d^{\prime }V}}+{\eta }^{\prime }{\displaystyle \frac{d^{\prime }{y}^{\prime }}{d^{\prime }V}}+{\zeta }^{\prime }{\displaystyle \frac{d^{\prime }{z}^{\prime }}{d^{\prime }V}}$$

(13)

where $d’V=-\sqrt{d^{\prime }{x}^{\prime 2}+d^{\prime }{y}^{\prime 2}+d^{\prime }{z}^{\prime 2}}$ is the initial element of length with a negative sign. Therefore, the law of varying action is expressed as follows:

$$\begin{array}{cc}{\displaystyle \underset{\epsilon \to 0}{lim}{\displaystyle \frac{V_{\epsilon }-V}{\epsilon }}}\hfill & =\xi {\displaystyle \frac{dx}{dV}}+\eta {\displaystyle \frac{dy}{dV}}+\zeta {\displaystyle \frac{dz}{dV}}+{\xi }^{\prime }{\displaystyle \frac{d^{\prime }{x}^{\prime }}{d^{\prime }V}}+{\eta }^{\prime }{\displaystyle \frac{d^{\prime }{y}^{\prime }}{d^{\prime }V}}+{\zeta }^{\prime }{\displaystyle \frac{d^{\prime }{z}^{\prime }}{d^{\prime }V}}=\hfill \\ {\displaystyle }\hfill & =\left(\xi -{\xi }^{\prime }\right){\displaystyle \frac{dx}{dV}}+\left(\eta -{\eta }^{\prime }\right){\displaystyle \frac{dy}{dV}}+\left(\zeta -{\zeta }^{\prime }\right){\displaystyle \frac{dz}{dV}}=0\hfill \end{array}$$

(14)

Introducing the symbol of variation ($\delta$), the last expression in (14) can be written as follows:

$$\delta V={\displaystyle \frac{dx}{dV}}\left(\delta x-\delta x^{\prime }\right)+{\displaystyle \frac{dy}{dV}}\left(\delta y-\delta y^{\prime }\right)+{\displaystyle \frac{dz}{dV}}\left(\delta z-\delta z^{\prime }\right)$$

(15)

Hamilton wrote the following:

[…] the length $V+dV$ of any other line which differs infinitely little from the straight ray in shape and in position, may be considered as equal to its own projection of the ray.

([14], p. 11)

After some mathematics, the following constraints on the characteristic functions are found, which are useful to its determination:

$$\left\{\begin{array}{c}{\displaystyle {\left({\displaystyle \frac{\partial V}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial V}{\partial y}}\right)}^2+{\left({\displaystyle \frac{\partial V}{\partial z}}\right)}^2=1}\hfill \\ {\displaystyle {\left({\displaystyle \frac{\partial V}{\partial x^{\prime }}}\right)}^2+{\left({\displaystyle \frac{\partial V}{\partial y^{\prime }}}\right)}^2+{\left({\displaystyle \frac{\partial V}{\partial z^{\prime }}}\right)}^2=1}\hfill \end{array}\right.$$

(16)

When one has a more complex function in the place of the length (V), such as action, the mathematical procedure is the same; obviously, there will be difficulties connected to the solution of integrals and differential equations. As an application, Hamilton compared the length of a chord with the length of a circular and elliptic arch ([14], pp. 9–11).

Hamilton then operated a series of passages of no particular difficulty, proving that the following identities hold:

$$\begin{array}{c}{\displaystyle {\displaystyle \frac{\partial V}{\partial x}}={\displaystyle \frac{\partial dV}{\partial dx}},{\displaystyle \frac{\partial V}{\partial y}}={\displaystyle \frac{\partial dV}{\partial dy}},{\displaystyle \frac{\partial V}{\partial z}}={\displaystyle \frac{\partial dV}{\partial dz}},}\\ {\displaystyle {\displaystyle \frac{\partial V}{\partial x^{\prime }}}={\displaystyle \frac{\partial dV}{\partial d{x}^{\prime }}},{\displaystyle \frac{\partial V}{\partial y^{\prime }}}={\displaystyle \frac{\partial dV}{\partial d{y}^{\prime }}},{\displaystyle \frac{\partial V}{\partial z^{\prime }}}={\displaystyle \frac{\partial dV}{\partial d{z}^{\prime }}}}\end{array}$$

(17)

Let us summarize how Hamilton applied the law of stationary action in optics. Consider a luminous path with $i+1$ points of sudden change of direction of luminous rays, e.g., in reflection or refraction. Then, action is expressed as $V={\sum }_{r=1}^{i+1}{V}^{\left(r\right)}$, where $V^{\left(r\right)}$ takes the following form:

$$V^{\left(r\right)}=\int d{V}^{\left(r\right)}=\int v^{\left(r\right)}\sqrt{d{x}^{\left(r\right)2}+d{y}^{\left(r\right)2}+d{z}^{\left(r\right)2}}$$

(18)

and the coefficients ($v^{\left(r\right)}$) depend on the optical properties of the medium and on the geometrical and optical properties of the ray (position, orientation and colour) ([14], p. 15). If we set

$${\sigma }^{\left(r\right)}={\displaystyle \frac{\partial d{V}^{\left(r\right)}}{\partial d{x}^{\left(r\right)}}},{\tau }^{\left(r\right)}={\displaystyle \frac{\partial d{V}^{\left(r\right)}}{\partial d{y}^{\left(r\right)}}},{\upsilon }^{\left(r\right)}={\displaystyle \frac{\partial d{V}^{\left(r\right)}}{\partial d{z}^{\left(r\right)}}},d{s}^{\left(r\right)}=\sqrt{d{x}^{\left(r\right)2}+d{y}^{\left(r\right)2}+d{z}^{\left(r\right)2}},$$

(19)

the homogeneity of $d{V}^{\left(r\right)}$ makes it is easy to prove that

$$d{V}^{\left(r\right)}={\sigma }^{\left(r\right)}d{x}^{\left(r\right)}+{\tau }^{\left(r\right)}d{y}^{\left(r\right)}+{\upsilon }^{\left(r\right)}d{z}^{\left(r\right)}$$

(20)

Following the principle of stationary action, Hamilton incremented the variables as follows:

$$x_{\epsilon }^{\left(r\right)}=x^{\left(r\right)}+\epsilon {\xi }^{\left(r\right)},y_{\epsilon }^{\left(r\right)}=y^{\left(r\right)}+\epsilon {\eta }^{\left(r\right)},z_{\epsilon }^{\left(r\right)}=z^{\left(r\right)}+\epsilon {\zeta }^{\left(r\right)}$$

(21)

Exactly as in the case of the shortest line between two points, it is necessary to calculate

$$\underset{\epsilon \to 0}{lim}{\displaystyle \frac{V_{\epsilon }-V}{\epsilon }}=\underset{\epsilon \to 0}{lim}\sum _{r=1}^{i+1}{\displaystyle \frac{V_{\epsilon }^{\left(r\right)}-V^{\left(r\right)}}{\epsilon }}=\underset{\epsilon \to 0}{lim}\sum _{r=1}^{i+1}\int {\displaystyle \frac{d{V}_{\epsilon }^{\left(r\right)}-d{V}^{\left(r\right)}}{\epsilon }}=\sum _{r=1}^{i+1}\int {\displaystyle \frac{\partial d{V}_{\epsilon }^{\left(r\right)}}{\partial \epsilon }}=0$$

(22)

Taking into account (21), it is possible to prove that

$${\displaystyle \frac{\partial d{V}_{\epsilon }^{\left(r\right)}}{\partial \epsilon }}={\sigma }^{\left(r\right)}d{\xi }^{\left(r\right)}+{\tau }^{\left(r\right)}d{\eta }^{\left(r\right)}+{\upsilon }^{\left(r\right)}d{\zeta }^{\left(r\right)}+{\xi }^{\left(r\right)}{\displaystyle \frac{\partial d{v}^{\left(r\right)}}{\partial d{x}^{\left(r\right)}}}+{\eta }^{\left(r\right)}{\displaystyle \frac{\partial d{v}^{\left(r\right)}}{\partial d{y}^{\left(r\right)}}}+{\zeta }^{\left(r\right)}{\displaystyle \frac{\partial d{v}^{\left(r\right)}}{\partial d{z}^{\left(r\right)}}},$$

(23)

so that the last term of (22), in view of (23), can be integrated by parts ([14], p. 16). Furthermore, the following conditions hold:

$$\begin{array}{c}{\displaystyle {\xi }^{\prime \left(1\right)}={\eta }^{\prime \left(1\right)}={\zeta }^{\prime \left(1\right)}={\xi }^{(i+1)}={\eta }^{(i+1)}={\zeta }^{(i+1)}=0,}\\ {\displaystyle {\xi }^{\left(r\right)}={\xi }^{(r+1)},{\eta }^{\left(r\right)}={\eta }^{(r+1)},{\zeta }^{\left(r\right)}={\zeta }^{(r+1)}}\end{array}$$

(24)

where symbols with primes denote values at the initial point of the interval (like above, where the initial point is $A\equiv (x^{\prime },y^{\prime },z^{\prime })$) and those without superscripts denote values at the end point. After some passages of no particular difficulty, Hamilton expressed the law of stationary action for optics through the following two groups of equations ([14], p. 17):

$$\begin{array}{c}\hfill {\displaystyle d{\sigma }^{\left(r\right)}={\displaystyle \frac{\partial v^{\left(r\right)}}{\partial x^{\left(r\right)}}}d{s}^{\left(r\right)},d{\tau }^{\left(r\right)}={\displaystyle \frac{\partial v^{\left(r\right)}}{\partial y^{\left(r\right)}}}d{s}^{\left(r\right)},d{\upsilon }^{\left(r\right)}={\displaystyle \frac{\partial v^{\left(r\right)}}{\partial z^{\left(r\right)}}}d{s}^{\left(r\right)},}\end{array}$$

(25)

$$\begin{array}{c}\hfill {\displaystyle {\sigma }^{\prime (r+1)}-{\sigma }^{\left(r\right)}={\lambda }^{\left(r\right)}{n}_x^{\left(r\right)},{\tau }^{\prime (r+1)}-{\tau }^{\left(r\right)}={\lambda }^{\left(r\right)}{n}_y^{\left(r\right)},{\upsilon }^{\prime (r+1)}-{\upsilon }^{\left(r\right)}={\lambda }^{\left(r\right)}{n}_z^{\left(r\right)}}\end{array}$$

(26)

In (25) and (26), $n^{\left(r\right)}$ indicates the semi-normal to the r-th reflecting or refracting surface at the r-th point of incidence, with $n_x^{\left(r\right)}$, $n_y^{\left(r\right)}$ and $n_z^{\left(r\right)}$ representing its director cosines with to the positive semi-axes x, y and z, respectively.

To provide an interpretation of Hamilton’s technique while considering refraction, we recall that the quantity of $v^{\left(r\right)}$ in the integral (18) is a function of the Cartesian coordinates and of the director cosines because the optical properties of the medium and the colour of light depend on them. If the r-th medium is isotropic, $v^{\left(r\right)}$ is its uniform refraction index and ${\displaystyle \frac{1}{v^{\left(r\right)}}}$ is the speed of the luminous ray in the medium if oriented along the director cosine. For an explanation of Hamilton’s optics, three old but good and clear papers are [24,25,26] (see also [27], pp. 59–87, 127–171). Therefore, in this case, the characteristic function (or action) is time, and Hamilton’s principle includes Fermat’s principleso that Snell’s law is directly deducible from his principle, expressed in its general form as Equations (25) and (26), which also lead to the properties of rectilinear propagation of light in a uniform medium and to the law of reflection.

3.2. Principle of Varying Action

Regarding the law of varying action applied to optics, the end points of a luminous path must be considered variable so that the initial values in (24) are nonzero ([14], p. 19). Using the usual notation of primes to denote the values at the starting point, with no superscripts to denote the final point, the principle of varying action is, thus, stated by the following equations:

$$\underset{\epsilon \to 0}{lim}{\displaystyle \frac{V_{\epsilon }-V}{\epsilon }}=\sum _{r=1}^{i+1}\int {\displaystyle \frac{\partial d{V}_{\epsilon }^{\left(r\right)}}{\partial \epsilon }}=\sigma \xi -{\sigma }^{\prime }{\xi }^{\prime }+\tau \eta -{\tau }^{\prime }{\eta }^{\prime }+\upsilon \zeta -{\upsilon }^{\prime }{\zeta }^{\prime }$$

(27)

Then, the variation of action is

$$\delta V=\sigma \delta x-{\sigma }^{\prime }\delta x^{\prime }+\tau \delta y-{\tau }^{\prime }\delta y^{\prime }+\upsilon \delta z-{\upsilon }^{\prime }\delta z^{\prime },\sigma ={\displaystyle \frac{\partial dV}{\partial dx}}={\displaystyle \frac{\partial vds}{\partial dx}},$$

(28)

and the expressions for $\tau$ and $\upsilon$ are analogous, changing x with y and z, respectively; on the other hand,

$${\sigma }^{\prime }={\left({\displaystyle \frac{\partial vds}{\partial dx}}\right)}^{\prime }=-{\displaystyle \frac{\partial d^{\prime }V}{\partial d^{\prime }{x}^{\prime }}},$$

(29)

and analogous expressions hold for ${\tau }^{\prime }$ and ${\upsilon }^{\prime }$, changing $x,x^{\prime }$ with $y,y^{\prime }$ and with $z,z^{\prime }$, respectively. Hamilton clearly explained the meaning of the variation of the initial point as follows:

[…] $d^{\prime }V$ being, according to the same analogy of notation, the infinitesimal change of the whole integral V, arising from the infinitesimal changes $d^{\prime }{x}^{\prime },d^{\prime }{y}^{\prime },d^{\prime }{z}^{\prime }$ of the initial coordinates, that is, from motion of the initial point $x^{\prime },y^{\prime },z^{\prime }$ along the initial element of the luminous path, so that $d’V$ is the initial element of the integral taken negatively

$$d^{\prime }V=-v^{\prime }\sqrt{{d^{\prime }{x}^{\prime }}^2+{d^{\prime }{y}^{\prime }}^2+{d^{\prime }{z}^{\prime }}^2}$$

([14], p. 19)

Hamilton proposes the following interesting explanation: since the action (V) is a function of the six coordinates of the initial and final points, (17) also holds with reference to the rectilinear propagation of a ray. If we know the initial coordinates of the ray, as well as its direction and colour, along with the initial properties of the first medium, we can restrict the initial quantities (${\displaystyle \frac{\partial d^{\prime }V}{\partial d^{\prime }{x}^{\prime }}},{\displaystyle \frac{\partial d^{\prime }V}{\partial d^{\prime }{y}^{\prime }}},{\displaystyle \frac{\partial d^{\prime }V}{\partial d^{\prime }{z}^{\prime }}}$), namely the right-hand side of (17)-2 to a finite variety. Thus, we can determine its left-hand side as well, namely components ${\displaystyle \frac{\partial V}{\partial x^{\prime }}}$, ${\displaystyle \frac{\partial V}{\partial y^{\prime }}}$ and ${\displaystyle \frac{\partial V}{\partial z^{\prime }}}$ of the gradient of the characteristic function (V) with respect to the initial coordinates. Therefore, if the form of V is known and the final coordinates ($x,y,z$) are considered variables, one finds that ${\displaystyle \frac{\partial V}{\partial x^{\prime }}}$, ${\displaystyle \frac{\partial V}{\partial y^{\prime }}}$ and ${\displaystyle \frac{\partial V}{\partial z^{\prime }}}$ are constant. Analogously, if V is considered given with respect to the final coordinates and the initial coordinates ($x^{\prime },y^{\prime },z^{\prime }$) are the variables, then the values of ${\displaystyle \frac{\partial V}{\partial x}}$, ${\displaystyle \frac{\partial V}{\partial y}}$ and ${\displaystyle \frac{\partial V}{\partial z}}$ are constant. Therefore, through his technique of varying action, Hamilton achieved the same result as that obtained by applying the principle of stationary action.

It is worth remarking that although the principle of stationary action can be seen as a limit case of that of variable action, as a matter of fact, from a conceptual standpoint, they are two different methods because in the principle of stationary action, one deals with a functional that varies between two extremes and to which the principles of the calculus of variations are applied. On the other hand, varying action is actually treated as a function—not as a functional—the initial and final values of which vary. One deduces, so to speak, what happens between these two values only when analysing their variation. The law of varying action is discussed, e.g., in [28,29,30,31,32] (see also [27], pp. 185–189). The paper we examined represents, in Hamilton’s view, the passage from optics to dynamics insofar as the principle of varying action is applicable to dynamics, as a hint with respect to the planetary theory at the end of this work shows. However, the application to dynamics is clearly explained in [21,22], allowing us to analyse these two long papers.

4. The Development of Hamilton’s Thought in Dynamics

Hamilton’s papers of 1834 and 1835 [21,22] are his most important contributions to dynamics, the former of which is foundational. He analysed all properties of the characteristic function, worked with Lagrangian coordinates in the configuration space, provided the theoretical bases of his method and offered two significant examples to which his procedure was applied. At the end of the contribution, he introduced his principal function, the features of which were better analysed in [22] in the context of what he called a perturbation theory, where the phase space was introduced, as well as the equations that we now dub Hamilton–Jacobi equations. On these two papers, there is a certain amount of good literature (see, e.g., [33] or [34], pp. 390–401, [1,27,35,36]). Therefore, we do not analyse their whole contents but restrict ourselves to stressing the aspects connected to the principle of varying action, which Hamilton considered one of his main discoveries, although he is more well-known for the principle of stationary action.

First, he considered what we call a weak formulation of the equations of motion for a system of n material points not subjected to external forces.

$$\sum m\left(\ddot{x}\delta x+\ddot{y}\delta y+\ddot{z}\delta z\right)=\delta U,$$

(30)

where $\delta U$ represents the infinitesimal variation of a force function that can be written as $U=m{m}^{\prime }f\left(r\right)$, with $f\left(r\right)$ being a function of the distance (r) between any two points with a mass of $m,m’$. Kinetic energy is classically written as $T=1/2\sum m\left({\dot{x}}^2+{\dot{y}}^2+{\dot{z}}^2\right)$, and because of the law of living force or, equivalently, the conservation of mechanical energy, it is $T=U+H$, H representing a special case of that function that was later named a Hamiltonian. This case is special, since here, energy—and not generalised energy—is considered.

In the case of infinitesimal variations, the equation of living force provides $\delta T=\delta U+\delta H$; if we multiply this by an infinitesimal time interval ($dt$) and integrate it, we obtain

$$\int \sum m\left(dx\delta \dot{x}+dy\delta \dot{y}+dz\delta \dot{z}\right)=\int \sum m\left(d\dot{x}\delta x+d\dot{y}\delta y+d\dot{z}\delta z\right)+\int \delta Hdt.$$

(31)

Then, Hamilton, after having introduced H, introduced another function that he named the characteristic function, which, today, is called “abbreviated action” and defined as follows:

$$V=\int \sum m\left(\dot{x}dx+\dot{y}dy+\dot{z}dz\right)={\int }_0^{t}2Tdt.$$

(32)

Hamilton spoke of “accumulated living force” ([21], p. 5), which provides an idea of how he saw the action as the sum (integral) of the kinetic energy of the system at every instant from the beginning to the end of its motion in the considered time interval. Hamilton calculated the variation of V, and according to calculus of variations and taking into account (31), he reached obtained following result. Hamilton was not detailed in describing all the mathematical passages, but they are well explained in [27] (pp. 184–186) as follows:

$$\delta V=\sum m\left(\dot{x}\delta x+\dot{y}\delta y+\dot{z}\delta z\right)-\sum m\left(\dot{a}\delta a+\dot{b}\delta b+\dot{c}\delta c\right)+t\delta H,$$

(33)

which is his law of varying action, where x, y and z denote the final values and a, b and c are the initial values. Hamilton explicitly claimed that the action (V) can be considered a function of the initial and final values and of function H so that the three following groups of equations are obtained when the index (i) assumes all values to lie between 1 and n:

$$\begin{array}{c}{\displaystyle {\displaystyle \frac{\partial V}{\partial x_i}}=m_i{\dot{x}}_i,{\displaystyle \frac{\partial V}{\partial y_i}}=m_i{\dot{y}}_i,{\displaystyle \frac{\partial V}{\partial z_i}}=m_i{\dot{z}}_i,}\\ {\displaystyle {\displaystyle \frac{\partial V}{\partial a_i}}=-m_i{\dot{a}}_i,{\displaystyle \frac{\partial V}{\partial b_i}}=-m_i{\dot{b}}_i,{\displaystyle \frac{\partial V}{\partial c_i}}=-m_i{\dot{c}}_i,}\\ {\displaystyle {\displaystyle \frac{\partial V}{\partial H}}=t.}\end{array}$$

(34)

Hamilton claimed that through his technique, the general problem of dynamics is reduced to the differentiation of the characteristic function (V) ([21], p. 5) so that, from his perspective, the main issue becomes the determination of V. According to Hamilton, the fundamental equation is (33), which expresses the law of varying action. It is appropriate to recall that such an action varies because the initial and final points are allowed to vary infinitesimally so that the values of $\delta x$, $\delta y$, $\delta z$, $\delta a$, $\delta b$ and $\delta c$ are nonzero. Therefore, it is clear that V is a function of the initial and final points, as well as of H. From a physical point of view, the remarkable property of the law of varying action is that momentum and time can be obtained simply by differentiating V.

The following comments of Hamilton are of extreme interest. He remarked that Lagrange imagined two fixed configurations for a system of bodies, distinguishing between geometrically possible movements and the real movement, fulfilling all the dynamical conditions. Hamilton continued by observing that Lagrange considered all geometrically possible but dynamically impossible motions, which differ infinitesimally minimally from the actual action; thus, for these geometrically possible motions, the action differs by an infinitely small quantity from its actual value, which justifies the law of least—or, more precisely, stationary—action. However, such a procedure is useful in determining the second-order equations of motion, not in solving them, whereas the law of varying action allows not only for the expression of the equations of motion but also for the determination of their integrals. Therefore, Hamilton considered his principle superior to that of Lagrange. He was very clear.

A different estimate, perhaps, will be formed of that other principle which has been introduced in the present paper under the name of the law of varying action, in which we pass from an actual motion to another motion dynamically possible, by varying the extreme positions of the system, and. (in general) the quantity H, and which serves to express, by means of a single function, not the mere differential equations of motion, but their intermediate and final integrals.

([21], p. 6. Italics in the text)

The following consideration seems paramount to us: the variation of the initial and final points (Jacobi showed that it is enough to consider the variation of only one of the two) (see [36], pp. 201–212). On the one hand, Jacobi fully recognised Hamilton’s merits but critically presented Hamilton’s theory. A fundamental text on the variational principles in dynamics is [10], where, regarding Hamilton’s ideas and Jacobi’s improvements, one can check Chapter V, “The Lagrangian equations of motion”; Chapter VII, “Canonical transformations”; and Chapter VIII, “The partial differential equation of Hamilton-Jacobi”. Among the numerous texts dealing with Lagrange’s and Hamilton’s approaches to dynamics, at least two classical works should be mentioned, namely that of Landau and Lifshitz [37] (first English edition, 1960; original Russian edition, 1957), where the authors introduced the principle of least action from the beginning of the text, and that of Arnold [38] (first English edition, 1978; original Russian edition, 1974), where Hamiltonian mechanics was introduced in the context of symplectic geometry. We also recall the papers by Capecchi and Drago [39], Capobianco et al. [40], Tomalin [41] and Van Weerden [42], allowed Hamiltonians to switch among dynamically possible motions, whereas, if the extremes are fixed, they can only pass from the true motion to other motions that are geometrically but not dynamically possible. This is a confirmation that the principle of stationary action and that of varying action rely upon different conceptual bases. The observation by Yourgrau and Mandelstam that the principle of stationary action “[…] suffers from the limitation that it applies only to virtual paths having the same energy as the real path” ([35], p. 46) is consistent with our remark on the more general view offered by the principle of varying action, where, in general, $\delta H\ne 0$. Obviously, if in (33), one sets all the variations to be equal to 0, one obtains the law of stationary action.

Hamilton showed that if $U_0$ is the opposite of the initial potential energy, the following equations are deductible from his principle:

$$\begin{array}{c}{\displaystyle {\displaystyle \frac{1}{2}}\sum {\displaystyle \frac{1}{m}}\left\{{\left({\displaystyle \frac{\partial V}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial V}{\partial y}}\right)}^2+{\left({\displaystyle \frac{\partial V}{\partial z}}\right)}^2\right\}=U+H}\\ {\displaystyle {\displaystyle \frac{1}{2}}\sum {\displaystyle \frac{1}{m}}\left\{{\left({\displaystyle \frac{\partial V}{\partial a}}\right)}^2+{\left({\displaystyle \frac{\partial V}{\partial b}}\right)}^2+{\left({\displaystyle \frac{\partial V}{\partial c}}\right)}^2\right\}=U_0+H}\end{array}$$

(35)

From (35), the laws of living forces, the motion of the centre of gravity and of the description of areas can be inferred ([21], pp. 6–10); therefore, the principle of varying action represents a crucial step in the foundation of dynamics. The next step consists of passing to Lagrangian coordinates and to the configuration space, with Hamilton showing that his principle is expressible in a form analogous to that where Cartesian coordinates are used ([21] p. 12), namely, for a system of n points.

$$\delta V=\sum _{i=1}^{3n}{\displaystyle \frac{\partial T}{\partial {\dot{\eta }}_i}}\delta {\eta }_i-\sum _{i=1}^{3n}{\displaystyle \frac{\partial T}{\partial {\dot{e}}_i}}\delta e_i+t\delta H,$$

(36)

where ${\eta }_i$ and $e_i$ represent the final and initial coordinates, respectively. In what followed, Hamilton offered further mathematical and dynamical justifications of his principle, showing how to separate the relative motion of a system from the motion of its gravity centre through the characteristic function and the principle of varying action ([21], pp. 19–22). Then, he applied his concepts to a system of two bodies ([21], pp. 26–34), described the motion of a planet or a comet around the Sun ([21], pp. 35–40) and analysed particular cases of the three-body problem ([21], pp. 40–48). A section concerning the way to find an approximate value to determine the characteristic function follows, where the explained technique should be applied when an exact value is impossible or very mathematically difficult to obtain. An application of this method to a multiple system is shown in [21] (pp. 48–56). The theory of approximation was further developed in the penultimate section of [21] (pp. 56–62). Finally, the principal function was introduced in [21] (pp. 62–63); however, since Hamilton’s following paper [22] dealt with the principal function, we analyse it later.

To see how Hamilton applied the law of varying action to solve a problem, we sketch his example about the system of two bodies. We do not expound upon the complete resolution, instead only focusing on the initial steps necessary for the reader to understand how varying action was introduced and used ([21], pp. 26–28). Hamilton considered a system of two mass points ($P_1=(x_1,y_1,z_1),P_2=(x_2,y_2,z_2)$) subjected only to their mutual attraction or repulsion. If their distance is indicated by r, the force function (U), as Hamilton said ([21], p. 26), namely the opposite of the potential energy of the system, takes the form of $U=m_1{m}_{2}f\left(r\right)$, with $f\left(r\right)$ being a function of distance such that its gradient expresses the force law. Hence, Newton’s equation can be written as

$$m_1\left({\ddot{x}}_1\delta x_1+{\ddot{y}}_1\delta y_1+{\ddot{z}}_1\delta z_1\right)+m_2\left({\ddot{x}}_2\delta x_2+{\ddot{y}}_2\delta y_2+{\ddot{z}}_2\delta z_2\right)=m_1{m}_{2}f\left(r\right).$$

(37)

Therefore, the following system of equations must hold:

$$\left\{\begin{array}{c}{\displaystyle {\ddot{x}}_1=m_2{\displaystyle \frac{\partial f\left(r\right)}{\partial x_1}},{\ddot{y}}_1=m_2{\displaystyle \frac{\partial f\left(r\right)}{\partial y_1}},{\ddot{z}}_1=m_2{\displaystyle \frac{\partial f\left(r\right)}{\partial z_1}},}\hfill \\ {\displaystyle {\ddot{x}}_2=m_1{\displaystyle \frac{\partial f\left(r\right)}{\partial x_2}},{\ddot{y}}_2=m_1{\displaystyle \frac{\partial f\left(r\right)}{\partial y_2}},{\ddot{z}}_2=m_1{\displaystyle \frac{\partial f\left(r\right)}{\partial z_2}}}\hfill \end{array}\right.$$

(38)

Hamilton pointed out that if the initial positions of the two mass points are $P_1=(a_1,b_1,c_1)$ and $P_2=(a_2,b_2,c_2)$, to integrate the previous system, it is necessary to assign six relations between the time (t), the masses ($m_1,m_2$), the variable coordinates ($x_1,y_1,z_1,x_2,y_2,z_2$), the initial coordinates and their velocities ($a_1,b_1,c_1,a_2,b_2,c_2,{\dot{a}}_1,{\dot{b}}_1,{\dot{c}}_1,{\dot{a}}_2,$${\dot{b}}_2,{\dot{c}}_2$) (he used the expression “rates of increase” to indicate velocities [21] (p. 27)). These six relations are assumed to be known and, combined with the law of living force ([21], p. 27) in the initial positions of the two points, are expressed as

$${\displaystyle \frac{1}{2}}{m}_1\left({\dot{a}}_1^2+{\dot{b}}_1^2+{\dot{c}}_1^2\right)+{\displaystyle \frac{1}{2}}{m}_2\left({\dot{a}}_2^2+{\dot{b}}_2^2+{\dot{c}}_2^2\right)=m_1{m}_{2}f\left(r_0\right)+H,$$

(39)

where $r_0=\sqrt{{(a_1-a_2)}^2+{(b_1-b_2)}^2+{(c_1-c_2)}^2}$ and H is the total energy, which Hamilton saw as a constant of integration. According to the seven relations supplied by the solutions of system (38) and Equation (39), it is possible to determine the time and the initial velocities (${\dot{a}}_1,{\dot{b}}_1,{\dot{c}}_1,{\dot{a}}_2,{\dot{b}}_2,{\dot{c}}_2$) as functions of the coordinates ($x_1,y_1,z_1,x_2,y_2,z_2,a_1,b_1,c_1,a_2,b_2,$$c_2$) and of H.

Furthermore, it is possible to calculate the (reduced) action or accumulated living force of the system as follows:

$$V=m_1{\int }_0^t\left({\dot{x}}_1^2+{\dot{y}}_1^2+{\dot{z}}_1^2\right)dt+m_2{\int }_0^t\left({\dot{x}}_2^2+{\dot{y}}_2^2+{\dot{z}}_2^2\right)dt$$

(40)

as a function of the thirteen quantities ($x_1,y_1,z_1,x_2,y_2,z_2,a_1,b_1,c_1,a_2,b_2,c_2,H$). It is, thus, possible to calculate the variation of V according to the traditional method.

In contrast to this, Hamilton wrote, “[…] the essence of our method is forming previously the expression of this variation by our law of varying action” ([21], p. 28; italics in the original). For the problem of two bodies, the variation of action is

$$\begin{array}{cc}\hfill \delta V& =m_1\left({\dot{x}}_1\delta x_1-{\dot{a}}_1\delta a_1+{\dot{y}}_1\delta y_1-{\dot{b}}_1\delta b_1+{\dot{z}}_1\delta z_1-{\dot{c}}_1\delta c_1\right)+\hfill \\ \hfill & +m_2\left({\dot{x}}_2\delta x_2-{\dot{a}}_2\delta a_2+{\dot{y}}_2\delta y_2-{\dot{b}}_2\delta b_2+{\dot{z}}_2\delta z_2-{\dot{c}}_2\delta c_2\right)+t\delta H.\hfill \end{array}$$

(41)

The function of V is considered a characteristic function of motion from which all the values of the final and initial momenta (Hamilton spoke of “all the intermediate and all the final integrals of all the known differential equations”, [21], p. 28) can be calculated by its simple derivation. Specifically, for the final values,

$$\left.\begin{array}{c}\hfill {\displaystyle {\displaystyle \frac{\partial V}{\partial x_1}}=m_1{\dot{x}}_1,{\displaystyle \frac{\partial V}{\partial y_1}}=m_1{\dot{y}}_1,{\displaystyle \frac{\partial V}{\partial z_1}}=m_1{\dot{z}}_1}\\ \hfill {\displaystyle {\displaystyle \frac{\partial V}{\partial x_2}}=m_2{\dot{x}}_2,{\displaystyle \frac{\partial V}{\partial y_2}}=m_2{\dot{y}}_2,{\displaystyle \frac{\partial V}{\partial z_2}}=m_2{\dot{z}}_2}\end{array}\right\},$$

(42)

while for the initial values,

$$\left.\begin{array}{c}\hfill {\displaystyle {\displaystyle \frac{\partial V}{\partial a_1}}=-m_1{\dot{a}}_1,{\displaystyle \frac{\partial V}{\partial b_1}}=-m_1{\dot{b}}_1,{\displaystyle \frac{\partial V}{\partial c_1}}=-m_1{\dot{c}}_1}\\ \hfill {\displaystyle {\displaystyle \frac{\partial V}{\partial a_2}}=-m_2{\dot{a}}_2,{\displaystyle \frac{\partial V}{\partial b_2}}=-m_2{\dot{b}}_2,{\displaystyle \frac{\partial V}{\partial c_2}}=-m_2{\dot{c}}_2}\end{array}\right\},$$

(43)

and for time,

$${\displaystyle \frac{\partial V}{\partial H}}=t.$$

(44)

Hamilton also claimed the following:

By this new method, the difficulty of integrating the six known equations of motion of the second order [here (38)] is reduced to the search and differentiation of a single function V.

([21], p. 28)

Although the determination of V can be a complicated mathematical task (and, in most cases, it is), for the determination of V in the problem of two bodies, see [21] (pp. 28–34). For an explanation of Hamilton’s mathematical procedures, see [36] (pp. 170–177). From a physical point of view, the principle of varying action allows for a very easy and clear method.

Further considerations of the principal function are as previously stated in [22]. Hamilton’s reasoning was developed as follows: given the configuration space and Lagrangian coordinates (${\eta }_i$), Lagrange’s equation can be written as follows: since U is the opposite of potential energy, if one sets $L=T+U$, (45) obviously coincides with the more common form of Lagrange’s equation, namely ${\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial L}{\partial {\dot{\eta }}_i}}-{\displaystyle \frac{\partial L}{\partial {\eta }_i}}=0$ (see, e.g., [27], pp. 189–190).

$${\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial L}{\partial {\dot{\eta }}_i}}-{\displaystyle \frac{\partial L}{\partial {\eta }_i}}={\displaystyle \frac{\partial U}{\partial {\eta }_i}}.$$

(45)

Since the kinetic energy (T) is homogeneous an of the second degree in the generalised velocities, Euler’s theorem on homogeneous functions can be applied so that

$$2T=\sum _{i=1}^{3n}{\dot{\eta }}_i{\displaystyle \frac{\partial T}{\partial {\dot{\eta }}_i}},$$

(46)

and, since T depends on Lagrangian coordinates as well, it is

$$\delta T=\sum _{i=1}^{3n}\left({\displaystyle \frac{\partial T}{\partial {\dot{\eta }}_i}}\delta {\dot{\eta }}_i+{\displaystyle \frac{\partial T}{\partial {\eta }_i}}\delta {\eta }_i\right),$$

(47)

so that one eventually obtains ([22], p. 4)

$$\delta T=\sum _{i=1}^{3n}\left[{\dot{\eta }}_i\delta \left({\displaystyle \frac{\partial T}{\partial {\dot{\eta }}_i}}\right)-{\displaystyle \frac{\partial T}{\partial {\eta }_i}}\delta {\eta }_i\right].$$

(48)

At this point, Hamilton introduced the quantity (${\varpi }_i={\displaystyle \frac{\partial T}{\partial {\dot{\eta }}_i}}$), which is now called the momentum conjugate to the Lagrangian velocity (${\dot{\eta }}_i$); this implies that, as a subtlety, the kinetic energy (T) can be considered not only a function of the Lagrangian coordinates and velocities (${\eta }_i,{\dot{\eta }}_i$) but also a function of the Lagrangian coordinates and conjugate momenta (${\eta }_i,{\varpi }_i$).

$$T({\eta }_i,{\dot{\eta }}_i)=F({\eta }_i,{\varpi }_i).$$

(49)

According to (48),

$$\delta F=\sum _{i=1}^{3n}\left({\dot{\eta }}_i\delta {\varpi }_i-{\displaystyle \frac{\partial T}{\partial {\eta }_i}}\delta {\eta }_i\right),$$

(50)

and (49) yields

$$\delta F=\sum _{i=1}^{3n}\left({\displaystyle \frac{\partial F}{\partial {\varpi }_i}}\delta {\varpi }_i+{\displaystyle \frac{\partial F}{\partial {\eta }_i}}\delta {\eta }_i\right).$$

(51)

Thus, comparing the expressions of (50) and (51) for the same quantity yields

$${\dot{\eta }}_i={\displaystyle \frac{\partial F}{\partial {\varpi }_i}},-{\displaystyle \frac{\partial T}{\partial {\eta }_i}}={\displaystyle \frac{\partial F}{\partial {\eta }_i}}$$

(52)

Since $H=F-U$, it is easy to derive Hamilton’s equations ([22], pp. 4–5). For an explanation of the passages leading to (53) with more details than those provided by Hamilton, see [27] (pp. 191–192).

$${\dot{\eta }}_i={\displaystyle \frac{\partial F}{\partial {\varpi }_i}},{\varpi }_i=-{\displaystyle \frac{\partial H}{\partial {\eta }_i}}.$$

(53)

Next, Hamilton introduced his principal function (S), which is called Hamiltonian action today.

$$S={\int }_0^t\left(\sum \varpi {\displaystyle \frac{\partial H}{\partial \varpi }}-H\right)dt={\int }_0^t{S}^{\prime }dt$$

(54)

If t and $dt$ do not vary, $\delta S={\int }_0^t\delta S^{\prime }dt$, and

$$\delta S^{\prime }=\sum \left[\varpi \delta \left({\displaystyle \frac{\partial H}{\partial \varpi }}\right)-{\displaystyle \frac{\partial H}{\partial \eta }}\delta \eta \right].$$

(55)

Taking into account (53), one immediately obtains

$$\delta S^{\prime }=\sum \left[\varpi \delta \left({\displaystyle \frac{\partial \eta }{\partial t}}\right)+{\displaystyle \frac{\partial \varpi }{\partial t}}\delta \eta \right]={\displaystyle \frac{d}{dt}}\sum \varpi \delta \eta$$

(56)

Therefore, Hamilton concluded ([22], p. 6) that

$$\delta S=\sum \left(\varpi \delta \eta -p\delta e\right)$$

(57)

where p and e are the initial values of $\varpi$ and $\eta$, respectively, so that ${\varpi }_i={\displaystyle \frac{\partial S}{\partial {\eta }_i}}$ and $p_i=-{\displaystyle \frac{\partial S}{\partial e_i}}$. Hamilton concluded this section of his work by claiming the following:

The difficulty of mathematical dynamics is therefore reduced to the search and the study of this one function S, which may for that reason be called Principal function of motion of a system,

([22], p. 6)

which is further evidence of the foundational character of his work.

One can remark that, while in [21], he had indicated the principal function is $S={\int }_0^t\left(T+U\right)dt$, it is easy to prove that this expression is equivalent to (54). Obviously, such a statement is what, today, we call Hamilton’s principle, namely that in order to find the true motion of a system, we must find its principal function (or Hamiltonian action; S) and impose $\delta {\int }_0^{t}Ldt=0$. Hamilton added a fundamental remark, namely that, given the latter considerations, i.e., the extremes of integration are fixed, the application of the principle of stationary action allows us to obtain Lagrange’s equation of motion. On the other hand, if the extremes are regarded as variable, it is possible to obtain the integrals of such equations. Hamilton wrote the following:

The variation of this definite integral S has therefore the double property, of giving the differential equations of motion for any transformed coordinates when the extreme positions are regarded as fixed, and of giving the integrals of those differential equations when the extreme positions are treated as varying.

([22], p. 6)

Hamilton was, hence, coherent in his thought in all the papers we have analysed. The principles of stationary action and of varying action have two different roles within physics. It is manifest that he found the latter to be more general and important and considered it his own invention and his most remarkable contribution to mathematical physics, although, today, the former is better known and more widely used.

5. Conclusions

Hamilton’s principles in optics and dynamics are of great interest for several reasons. They offer a new foundation for these disciplines, and they allow physical problems to be solved by means of systems of first-order differential equations (although the number of equations has to be doubled with respect to Lagrange’s approach).

From a historical point of view, they can be interpreted as the last step of ‘physics of principles’, although restricted to situations in which friction and heat are not considered. Such a story begins with Fermat’s principle of least time, includes Maupertuis’ principle of least action and passes through the more mature works by Euler and Lagrange before reaching Hamilton and Jacobi, who then improved upon Hamilton’s results.

At that time, Hamilton’s principles, especially that of stationary action, were a cornerstone of physics and became even more important when quantum mechanics was born.

From an epistemological and philosophical standpoint, one might wonder how these principles are included within Hamilton’s philosophy. He was a scientist who was profoundly interested in philosophy—for it is enough to recall that he was one of the first to introduce Kant in the English-speaking world and was involved in discussions with scientist–philosophers such as William Whewell (1794–1866) about the nature of physics.

However, we chose to follow another kind of approach. We analysed how Hamilton began to use variational principles in optics, his transition from optics to mechanics and the application of his principles to mechanics. In this respect, we attempted to remark on the importance of [14], a paper that is generally underestimated and that deservers great attention, since it is within this paper that the passage from optics to dynamics took place and the germs of Hamilton’s dynamical thought were expounded upon.

Our main goal was to show the relations between the stationary action principle and the varying action principle. We stressed the most relevant properties of the three main functions (which are actually not the only functions) introduced by Hamilton in dynamics, namely the characteristic function (V), the Hamiltonian function (H) and the principal function (S). The introduction of these functions and the exposition of Hamilton–Jacobi equations are useful to highlight the features of stationary and varying action.

Therefore, we have dealt with the mathematical developments of Hamilton’s theory insofar as they are necessary to understand the differences between the two principles. Thus, our aim was to provide the reader with an interpretative red line of Hamilton’s thought that passes through his two great principles.