Hertz's Mechanics and a unitary notion of force

https://doi.org/10.1016/j.shpsa.2021.10.004Get rights and content

Abstract

Heinrich Hertz dedicated the last four years of his life to a systematic reformulation of mechanics. One of the main issues that troubled Hertz in the customary formulation of mechanics was a ‘logical obscurity’ in the notion of force. However, it is unclear what this logical obscurity was, hence it is unclear how Hertz took himself to have avoided it.

In this paper, I argue that a subtle ambiguity in Newton's original laws of motion lay at the basis of Hertz's concerns; an ambiguity which led to the development of two slightly different notions of force. I then show how Hertz avoided this ambiguity by deriving a unitary notion of force, thus dispelling the obscurity that lurked in the customary representation of mechanics.

Introduction

In 1890 Heinrich Hertz began a grand project in the foundations of physics, aiming to systematically reformulate the theory of mechanics. Tragically, however, the next four years of labor were to prove to be the last four years of Hertz's short life. On New Years Day of 1894, Hertz died just thirty-six years old while the manuscript of his treatise was still in press. The broader physics community greeted the publication of The Principles of Mechanics Presented in a New Form with a mixture of sadness and curiosity, wondering what it was, exactly, that Hertz had poured his last energies into.

In his introduction, Hertz explains that his motivation to reformulate mechanics stemmed from a sense of discomfort concerning a lack of clarity in the foundations of the theory:

I have not attempted this task because mechanics has shown signs of inappropriateness in its applications, nor because it in any way conflicts with experience, but solely in order to rid myself of the oppressive feeling that to me its elements were not free from things obscure and unintelligible. (Hertz, 1899, p. 33)

In seeking to convey this issue to his readers, Hertz first discusses the elementary problem of swinging a stone in a circle, aiming to illustrate how easy it is ‘to attach to the fundamental laws considerations which are quite in accordance with the usual modes of expression in mechanics and which yet are an undoubted hindrance to clear thinking’ (Hertz, 1899, p. 5). He then offers three ‘general observations’ as further evidence for this lack of clarity. The first is the dissatisfaction felt in introducing the basic concepts and definitions of mechanics, and the desire ‘to move rapidly over the introductory material on to examples which speak for themselves’ (Hertz, 1899, p. 7). The second is the existence of disagreements concerning the rigor of supposedly elementary theorems in mechanics, disagreements which ‘in a logically complete science, such as pure mathematics … [are] utterly inconceivable’ (ibid). The final observation is the concern felt in the physics community over the ‘essence’ (Wesen) of force:

Weighty evidence seems to be furnished by the statements which one hears with wearisome frequency, that the essence of force is still a mystery, that one of the chief problems of physics is the investigation of the essence of force, and so on. In the same way electricians are continually attacked as to the essence of electricity. Now, why is it that people never in this way ask what is the essence of gold, or what is the essence of velocity? Is the essence of gold better known to us than that of electricity, or the essence of velocity better than that of force? Can we by our conceptions, by our words, completely represent the essence of any thing? Certainly not. (Hertz, 1899, pp. 7–8)

For Hertz, the fact that the essence of something is regarded as mysterious implies the existence of an underlying state of confusion. Hence he prompts his readers to consider why we might be drawn to ask such questions with regard to certain concepts but not others:

I fancy the difference must lie in this. With the terms “velocity” and “gold” we connect a large number of relations to other terms; and between all these relations we find no contradictions which offend us. We are therefore satisfied and ask no further questions. But we have accumulated around the terms “force” and “electricity” more relations than can be completely reconciled amongst themselves. We have an obscure feeling of this and want to have things cleared up. Our confused wish finds expression in the confused question as to the essence of force and electricity.

One of the aims of Principles is thus to alleviate the confusion surrounding the notion of force so as to avoid confused questions as to its essence from arising. As Hertz famously put the matter:

the answer which we want is not really an answer to this question [what is the essence of force?]. It is not by finding out more and fresh relations and connections that it can be answered; but by removing the contradictions existing between those already known, and thus perhaps by reducing their number. When these painful contradictions are removed, the question as to essences will not have been answered; but our minds, no longer vexed, will cease to ask illegitimate questions. (Hertz, 1899, p. 8)

The present paper has two related goals. First, I aim to give an account of the state of confusion that surrounded the notion of force as Hertz understood it. Second, I aim to show how Hertz derived a perspicuous notion of force in his own reformulation of mechanics so that our minds, no longer vexed, would cease to ask illegitimate questions. I begin in section 2 with Hertz's discussion of the ‘hindrance to clear thinking’ encountered in the case of swinging a stone in a circle—a difficulty whose roots, Hertz suggests, can be traced back to Newton's laws of motion. In seeking to understand the basis of Hertz's concerns, I turn in section 3 to consider the way in which the historical development of mechanics was understood in this period, focusing in particular on Thomson and Tait (1879). With this in hand, I argue in section 4 that there was in fact an ambiguity in the notion of force contained in the received interpretation of Newton's laws, and that Hertz was therefore well motivated in seeking a unitary notion of force which avoided such ambiguity. In section 5, I provide an overview of Hertz's reformulation of mechanics, leading to a discussion in section 6 of the notion of force that Hertz's framework makes available. Section 7 concludes.

Section snippets

Swinging a stone in a circle

Hertz's first attempt to convey the lack of clarity in the foundations of mechanics involves a discussion of a simple example: ‘We swing in a circle a stone tied to a string, and in so doing we are conscious of exerting a force upon the stone’ (Hertz, 1899, p. 5). As Hertz notes, it is easy to confirm that, in relation to the stone's circular motion, the magnitude of this force is consistent with Newton's second law. Newton's third law then instructs us that an equal and opposite reaction force

‘The customary representation of mechanics’

In order to make progress in identifying Hertz's logical obscurity, we need to appreciate Hertz's understanding of the domain in which it lies—the customary representation of mechanics:

By this we mean the representation, varying in detail but identical in essence, contained in almost all text-books which deal with the whole of mechanics, and in almost all courses of lectures which cover the whole of this science. This is the path by which the great army of students travel and are inducted into

A logical obscurity

According to d’Alembert's principle, all the forces acting on a system are balanced by equal and opposite ‘reactions against acceleration.’ These reactions are therefore naturally regarded as themselves forces. Returning to Lanczos, we can say that d’Alembert's principle ‘introduces a new force, the force of inertia, defined as the negative of the product of mass and acceleration’ (Lanczos, 1962, p. 92). Hence d’Alembert's principle itself can be understood as the claim that ‘any system of

Hertz's alternative approach

In his own presentation of mechanics, Hertz begins with just three primitive notions: space, time, and mass. Furthermore, in contrast to Newton's three laws of motion, Hertz distils the core empirical content of classical mechanics into a single ‘fundamental law’:

Fundamental Law. Every free system persists in its state of rest or of uniform motion in a straightest path. (Hertz, 1899, §309)14

Hertz's notion of force

Hertz's notion of force adds nothing beyond the application of the fundamental law to a system of connected material points. Even so, Hertz is able to show that his notion is in accord with the notion of force in the customary representation of mechanics to a remarkable degree.

Most immediately, Hertz's forces are vector quantities (of various dimensions), both with regard to the systems that exert them and with regard to the systems that they are exerted on. All the expected compositional

Conclusion

The goals of this paper have been two-fold. First, I hope to have unearthed the source of the logical obscurity in the customary notion of force that troubled Hertz. Second, I hope to have shown how Hertz derived a unitary—hence unambiguous—notion of force. With regard to my first goal, I have argued that a subtle ambiguity in Newton's original laws of motion led to the emergence of two slightly different notions of force—Newtonian and Lagrangian—the relationship between which was obscured in

Acknowledgements

My thanks to Gal Ben Porath, Janet Folina, Doreen Fraser, Lydia Patton, and Sheldon Smith for various comments that have helped to improve this paper. Special thanks to Katherine Brading and Mark Wilson.

References (16)

  • Ludwig Boltzmann

    Theoretical physics and philosophical problems: Selected writings

    (1974)
  • Jeremy Butterfield

    Between laws and models: Some philosophical morals of Lagrangian mechanics

  • Joshua Eisenthal

    Mechanics without mechanisms

  • Eisenthal, Joshua. Models and Multiplicities. Journal of the History of Philosophy,...
  • Craig Fraser

    D'Alembert's principle: The original formulation and application in Jean d'Alembert's Traité de Dynamique (1743)

    Centaurus

    (1985)
  • Heinrich Hertz

    The principles of mechanics presented in a new form

    (1899)
  • Heinrich Hertz et al.

    Heinrich Rudolf Hertz (1857–1894): A collection of articles and addresses

    (1994)
  • Cornelius Lanczos

    The variational principles of mechanics

    (1962)
There are more references available in the full text version of this article.

Cited by (0)

Einstein Papers Project, Caltech M/C 20-7, 1200 East California Blvd., Pasadena, CA 91125, USA.

View full text