On the necessary truth of the laws of classical mechanics

https://doi.org/10.1016/j.shpsb.2006.11.004Get rights and content

Abstract

The idealization of primitive mechanical experience is shown to lead to four mutually related formulations of classical mechanics based on connections, action at a distance, stresses, and collisions. For a given structure of spacetime and a given characterization of mechanical systems, fundamental laws (including Newton's law of acceleration and d’Alembert's principle) are derived from a few general principles regarding the comprehensibility of motion. Special emphasis is placed on the “secular principle,” according to which the evolution of a system at the relevant time scale should not depend on finer details of the applied forces.

Introduction

In the introduction to his famous Traité de dynamique of 1743, Jean le Rond d’Alembert couched his conception of the laws of mechanics in terms of the Leibnizian distinction between contingent and necessary truth (d’Alembert, 1743, pp. xxviii–xxix):

The laws of statics and mechanics…are the laws that result from the existence of matter and motion. Now, experience teaches us that these laws obtain in the bodies that surround us. Therefore, the laws of equilibrium and motion revealed by observation have necessary truth.

d’Alembert meant that the principle of virtual work (for equilibrium) and his principle of dynamics followed from the ideal rigidity of the parts of mechanical systems. His reasoning involved a now obsolete concept of perfect hardness (according to which a hard body falling normally on a fixed hard plane is completely stopped) and abundant use of the principle of sufficient reason.1

Although later writers on mechanics adopted d’Alembert's principle for the dynamics of connected systems, they usually rejected the necessary character of the laws of mechanics. In their opinion, other theories of motion were conceivable; only experience could teach us which of the possible theories applied to observed motions; d’Alembert's concept of hardness was unrealistic; and his recourse to the principle of sufficient reason smelled of bad metaphysics. At the turn of the century, Fourier (1798), Lagrange (1798), and Ampère (1806), Ampère (1826) nonetheless attempted to derive the principle of virtual work from more evident premises. Later in the century, Violle's treatise on mechanics (1883–1892) included a derivation of Newton's law of acceleration from the (Galilean) principle of relativity.2

These derivations have disappeared from modern treatises. Although only one (non-relativistic) mechanics of macroscopic bodies is ever taught, most physicists seem to believe in the contingency of the laws of mechanics. The present essay challenges this view by showing that for a given spacetime structure (Galilean or Minkowskian) the known laws of mechanics are the only ones for which motion is measurable and comprehensible.3

An analogy with the case of geometry may help to understand my purpose. Hermann Helmholtz (1868), showed that the measurability of space by freely mobile rigid bodies implied its locally Euclidean character. We may similarly wonder whether the ideal measurability of space, time, and force constrain the form that the laws of mechanics can take. In addition, we may wonder whether the very comprehensibility of macroscopic motion as resulting from macroscopically identifiable causes has an effect on the form of its laws. This is analogous to Helmholtz's early endeavor, in his memoir of 1847 on the conservation of force, to derive a general concept of interaction from the comprehensibility of nature. Although I do not intend to follow the details of Helmholtz's considerations (much of which later proved to be flawed) nor the precise form of empiricized Kantianism he was advocating, I retain the general idea that considerations of measurability and comprehensibility may by themselves determine the laws of a theory.4

My starting point is an ideal characterization of the class of systems under study, together with the description of ideal measurement procedures for basic mechanical quantities. These are inspired by primitive mechanical experiments and the desire to make them quantitative. Nonetheless, I regard both the systems and the measurement procedures as pure mathematical constructs, in conformity with the semantic view of theories. The laws of mechanics apply strictly to these ideal systems and measurements. The ideal measurement procedures should not be regarded as concrete operational definitions à la Percy Bridgman; nor do they imply correspondence rules in the sense of logical positivism. Their ultimate concrete realization depends on the whole theory and may strongly depart from the primitive devices that historically inspired them.

There are several definitions of the basic class of mechanical systems, corresponding to four basic intuitions of mechanical interaction: articulated connections, direct action at a distance, permanent contact, collisions. The first intuition leads to the mechanics of connected systems, the second to molecular mechanics, the third to the stress-based dynamics of continua, the fourth to the conservation laws of collision processes. The intended applications of these theories are different, and so are too the ways in which they are connected to experiment.

In each case, the laws of mechanics appear to be uniquely determined by very broad assumptions about the predictability or comprehensibility of motion. More precisely, they are found to result from the following elements: the definition of mechanical systems, the structure of ideal measurements, the spacetime structure (assumed to be Galilean in most of this essay), and a comprehensibility requirement that includes the impossibility of perpetual motion, a causality principle, and what I call the secular principle. According to the latter principle, two systems of applied forces yield the same secular motion if their secular average is the same.5 The intuitive justification for this property is that the evolution of a system at the relevant time scale should not depend on details of the applied forces at a much finer time scale. Newton (1687) implicitly assumed so much in his mechanics. The secular principle fills the holes in Violle's derivation of the relation between force and acceleration and in earlier derivations of d’Alembert's principle.

The kind of necessity here claimed for the laws of mechanics does not imply their empirical truth. Rather, my aim is to prove the following conditional proposition: if motion is comprehensible in a certain sense, and if there are concrete systems corresponding to the ideal systems defined in this paper, then the usual laws of classical mechanics hold. My criteria for the comprehensibility of motion are not necessarily met: for instance, motion may not be directly describable in spacetime, as happens for quantum objects. Or concrete realization of ideal systems may not be possible: for instance, the perfectly rigid bodies of connected systems do not exist in relativistic physics. At best one can hope that approximate realizations of ideal systems exist and that their motion is approximately comprehensible in causal and spatial terms.

Moreover, the necessity claimed in this paper does not concern force laws: the choice of the external forces applied to connected systems, the expression of the forces acting between two molecules, or the expression of the stresses occurring in continuous bodies remain largely free. What is severely constrained by comprehensibility is the relation between forces and motion.6

2 The statics of connected systems, 3 The dynamics of connected systems of this paper are devoted to the case of connected systems. 4 Molecular mechanics, 5 Continuum mechanics, 6 Collisions deal with the cases of molecular mechanics, continuum mechanics, and collisions. Section 7 provides the following comments and conclusions. Firstly, the various assumptions made in the previous sections are shown to derive from the comprehensibility of motion. Secondly, the various forms of mechanics are shown to be multiply interrelated (this feature deflects the criticism that arbitrariness in the definition of mechanical systems could compromise the necessity of mechanical laws). Thirdly, the principle of least action is shown to be intimately related to the more natural principles adopted in this essay. Lastly, the logic of application of mechanics is shown to be far more subtle than a plausible misinterpretation of the present paper's emphasis on ideal measurements would suggest, in conformity with earlier philosophical analysis by Sneed (1977), Torretti (1980), and others.

Section snippets

The statics of connected systems

The simplest mathematical physics is applied geometry and chronometry. The next simple one is the science of the motion of bodies, namely, mechanics. Bodies in our environment change place, and we want to predict when and how they do so. Within terrestrial physics, the most conspicuous cases of motion are the fall of bodies, collisions of various kinds, the deformation of bodies, the flow of liquids, and the working of connected mechanisms. The only obvious regularities are qualitative: every

The relation between force and acceleration

The simplest case of motion is that of a point-like body subjected to a constant force. Judging from statics, the free fall of a small body would seem to belong to this category. Yet common observation does not lead to any simple regularity of free fall. It rather favors the Aristotelian intuition that fall is a means to reach a given end, proximity to the center of the earth, in ways that complexly depend on the shape and the weight of the body. It took a Galileo to suspect that the fall of a

Saint-Venant's foundations

The mechanics so far developed could be that of a blind man, who only experiments on objects that can be reached by touch. Vision, optical instruments and the laws of geometrical optics greatly open the range of observation. The motion of stars and planets thus become a natural object of study. As is well known, Newton managed to explain most celestial regularities by assuming the attractive force Gmm/r2 between any two mass points m and m’ with the mutual distance r and selecting an inertial

Concepts of pressure

Is it possible to emulate the large scope of molecular mechanics without leaving the macroscopic realm of continuous matter? Euler (1755) and Cauchy (1828) gave a positive answer to this question. Their success depended on the introduction of the concept of internal pressure. So far, the basic concepts of force we have used are the force applied to a material point of a connected system, and the force applied to a point atom. For continuous distributions of matter, we admitted two derived

Collisions

We have so far addressed three ways of founding mechanics: through connected systems, through point-like atoms and central forces, and through stresses. There is still a third way, through collisions, which played a significant role in history. Descartes, Huygens, Newton, Leibniz, and d’Alembert all gave a central role to collisions in their quest for the laws of dynamics. Johann and Daniel Bernoulli tried to reduce every force, including gas pressure, to molecular impacts. Later in the

The necessity and unity of mechanics

We may now judge to which extent and in which sense the laws of classical mechanics can be regarded as necessary truths. The form of the invariants in collision processes result directly from the spacetime structure. The derivation of the laws of the full mechanics of connected systems, molecular systems, and continua requires additional assumptions. The first is a causality principle according to which forces (at least impulses) alter the motion in a well-defined manner. The second is the

Acknowledgments

I am grateful to two anonymous referees for useful suggestions.

References (56)

  • E. McMullin

    Galilean idealization

    Studies in History and philosophy of science

    (1985)
  • J.M. Ampère

    Démonstration générale du principe des vitesses virtuelles, dégagée de la considération des infiniment petits

    Journal de l’École Polytechnique

    (1806)
  • J.M. Ampère

    Nouvelle demonstration du principe des vitesses virtuelles

    Correspondances de mathématique et de physique

    (1826)
  • W. Balzer et al.

    An architectonic for science: The structuralist program

    (1987)
  • F. Bevilacqua

    Helmholtz's Über die Erhaltung der Kraft: The emergence of a theoretical physicist

  • Boltzmann L. (1897–1904). Vorlesungen über die Principe der Mechanik (2 vols.). Leipzig:...
  • J. Buchwald

    From Maxwell to microphysics: Aspects of electromagnetic theory in the last quarter of the nineteenth century

    (1985)
  • L. Carnot

    Principes fondamentaux de l’équilibre et du mouvement

    (1803)
  • A. Cauchy

    Sur les équations qui expriment les conditions d’équilibre ou les lois du mouvement d’un corps solide, élastique ou non élastique

    Exercices de mathématiques

    (1828)
  • H. Chabot

    Nombre et approximations dans la théorie de la gravitation de Lesage

    Sciences et techniques en perspective

    (2004)
  • Chatzis, K. (2004). Les conceptions de Barré de Saint-Venant en matière de théorie de la connaissance. In L’art de...
  • C. Comte

    Leibniz aurait-il pu découvrir la théorie de la relativité?

    European Journal of Physics

    (1986)
  • Cornu A. (1875–1879). Cours de physique (3 vols.). Paris: Ecole...
  • d’Alembert, J. le Rond (1743). Traité de dynamique (2nd ed., 1758). Paris:...
  • O. Darrigol

    God, waterwheels, and molecules: Saint-Venant's anticipation of energy conservation

    Historical Studies in the Physical and Biological Sciences

    (2001)
  • O. Darrigol

    Between hydrodynamics and elasticity theory: The first five births of the Navier–Stokes equation

    Archive for the History of Exact Sciences

    (2002)
  • O. Darrigol

    Worlds of flow: A history of hydrodynamics from the Bernoullis to Prandlt

    (2005)
  • de Courtenay, N. (1999). Science et philosophie chez Ludwig Boltzmann: La liberté des images par les signes. Thèse de...
  • J. Dhombres et al.

    Joseph Fourier 1768–1830. Créateur de la physique

    (1998)
  • P. Dugas

    Histoire de la mécanique

    (1950)
  • L. Euler

    Principes généraux de l’état d’équilibre d’un fluide. Principes généraux du mouvement des fluides

  • A. Flamant

    Mécanique générale

    (1888)
  • J. Fourier

    Mémoire sur la statique, contenant la démonstration du principe des vitesses virtuelles, et la théorie des momens

    Journal de l’École Polytechnique

    (1798)
  • C. Fraser

    Lagrange's early contributions to the principles and methods of mechanics

    Archive for the History of Exact Sciences

    (1983)
  • C. Fraser

    d’Alembert's principle: The original formulation and application in Jean d’Alembert's ‘Traité de dynamique’ (1743)

    Centaurus

    (1985)
  • T. Hankins

    Introduction

  • T. Hankins

    Sir William Rowan Hamilton

    (1980)
  • R. Giere

    Explaining science: A cognitive approach

    (1988)
  • Cited by (0)

    View full text