Abstract
James Clerk Maxwell’s 1865 paper, “A Dynamical Theory of the Electromagnetic Field,” is usually remembered as replacing the mechanical model that underpins his 1862 publication with abstract mathematics. Up to this point historians have considered Maxwell’s usage of Lagrangian dynamics as the sole important feature that guides Maxwell’s analysis of electromagnetic phenomena in his 1865 publication. This paper offers an account of the often ignored mechanical analogy that Maxwell used to guide him and his readers in the construction of his new electromagnetic equations. The mechanical system consists of a weighted flywheel geared into two independently driven crank wheels in what amounts to a mechanical differential. I will demonstrate how Maxwell made use of the analogy between his flywheel system and electromagnetic induction to ground his study of electromagnetism in clear mechanical conceptions and to structure the derivation of the equations that together are now recognized as Maxwell’s equations for electrodynamics. By reconceiving specific components of his model in electromagnetic terms, while at the same time retaining many of the relations between concepts in the mechanical case, Maxwell gradually assembled increasingly generalized equations for electromotive force. Maxwell thus realized a much sought after balance between physical analogy and abstract mathematics in this, the last of his three seminal papers on electromagnetism.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The picture of Maxwell’s 1865 paper “A Dynamical Theory of the Electromagnetic Field” as founded exclusively on the abstract analysis of Lagrangian dynamics is commonly accompanied by quotations from two of Maxwell’s letters, one before the paper’s publication and one after. In the first, a letter from Maxwell to William Thomson dated October 15, 1864, Maxwell states: “I can find the velocity of transmission of electromagnetic disturbances independently of any hypothesis now and it is equal to v [the speed of light]” (Maxwell 1995, p. 180). As both Martin Goldman and Harman dutifully point out, by 1864 Maxwell had absolved himself of his early reliance on hypotheses about the electromagnetic medium as well as his impious inclusion of terms that emerge directly from his mechanical analogy (Goldman 1983, p. 155; Harman 1998, p. 113). The second quotation, from a late December 1867 letter from Maxwell to Peter Guthrie Tait, compares the 1862 and 1865 papers: “The former is built up to show that the phenomena are such as can be explained by mechanism...The latter is built on Lagrange’s Dynamical Equations and is not wise about vortices” (Maxwell 1995, p. 337). Harman and Goldman both accept this statement as evidence of the purely mathematical nature of Maxwell’s methodology in the 1865 paper (Goldman 1983, p. 155; Harman 1998, p. 118). Nevertheless, they both admit that despite reforming his methods, Maxwell’s new conception of the electromagnetic field as a “complicated mechanism” is still explicitly based upon a physical/mechanical analogy (Maxwell 1865, p. 533). Each letter makes clear (as does the content of the paper itself) that Maxwell is no longer relying on a hypothetical model, but the second is supposed to show that Maxwell believes that the analogy in the 1865 paper is to nothing but Lagrangian dynamics.
Olivier Darrigol gestures to the presence of the flywheel in the 1865 paper, although he assigns it little significance (Darrigol 2000, p. 156). Francis Everitt takes the time to explain how the analogy is constructed although he does not delve much into its uses (Everitt 1975, pp. 103–105). Daniel Siegel acknowledges the status of the flywheel as a mechanical analogue for inductive circuits in his notes, citing Everitt, but given his book’s focus on Maxwell’s 1862 paper, he understandably does not push any deeper (Siegel 1991, p. 199). Harman and Goldman ignore its existence entirely. In a guided study of Maxwell’s electromagnetic papers, Thomas K. Simpson suggests that the flywheel might be of some significance: it need “not necessarily be an inferior but quite possibly a more insightful way of grasping the principles of a connected mechanical system” (Simpson 1997, p. 367). I will advocate for something very close to this view.
This is not to say that Maxwell viewed this clarifying role as unimportant. Maxwell still regarded this application of analogy as crucial when giving his “Introductory Lecture on Experimental Physics” at Cambridge in October 1871: illustrative analogy lends “vividness and relief to ideas which, when presented as mere abstract terms, are apt to fade entirely from the memory” (Maxwell 1871, p. 242).
What Maxwell calls electromagnetic momentum is what we would now call the vector potential \(\mathbf{A}\). It should not be confused with the modern definition of electromagnetic momentum, \(\varepsilon _0(\mathbf{E}\times \mathbf{B})\), where \(\varepsilon _0\) is the dielectric constant in vacuo and \(\mathbf{E}\) and \(\mathbf{B}\) are the electric and the magnetic field, respectively.
Unless noted otherwise, all page references refer to the reprint of this paper in The Scientific Papers of James Clerk Maxwell, Volume 1. Equations marked M(x), where x is an integer or capital letter, refer to the equation number or letter within Maxwell’s 1865 paper.
In his 1862 cogwheel model Maxwell had felt it necessary to insert idle wheels to overcome a similar mechanical problem, namely “coupling neighboring vortices” (Siegel 1991, p. 66). That Maxwell appeared unconcerned with the potential mechanical difficulties of coupling an arbitrary number of these systems together demonstrates his move away from modeling electromagnetic phenomena and the ether itself, instead offering only a “dynamical illustration”.
Maxwell had previously described the relevance of William Siemens’ differentially geared governor to electromagnetic phenomena in Part II of “On Physical Lines of Force,” where he noted that elements of this governor are capable of motions similar to those of the idle wheels in his cogwheel model (Maxwell 1862, pp. 468–469).
Section titles closely resemble Maxwell’s own titles and proceed in the order given in Maxwell’s “Dynamical Theory”.
Maxwell’s comment on p. 542 that this mechanical force acts to maximize L, M, and N remains puzzling.
Here we are looking at the electromagnetic momentum of a circuit (not a single conductor as before) from a perspective such that the circuit is defined as the boundary of the area \(\hbox {d}y\hbox {d}x\). As the circuit lengthens in the y-direction, the area increases per unit length in x. As such, the number of lines of magnetic force that pass through the circuit will also increase per unit length in x. If we remember that the number of these lines is a measure of electromagnetic momentum, the fact that the expansion is in the y-direction entails a change in G, the y-component of electromagnetic momentum, per unit length in x (cf. p. 15).
For discussion of Maxwell’s use of Faraday’s “mutually embracing curves” see (Wise 1979).
References
Boltzmann, Ludwig. 1891. Vorlesungen über Maxwells Theorie der Elektricität und des Lichtes, vol. 1. Leipzig: Johann Ambrosius Barth.
Campbell, Lewis, and William Garnett. 1882. The life of James Clerk Maxwell: With a selection from his correspondence and occasional writings and a sketch of his contributions to science. London: Macmillan.
Darrigol, Olivier. 2000. Electrodynamics from Ampere to Einstein. Oxford: Oxford University Press.
Eckert, Michael. 2001. Sommerfeld und das Boltzmannsche Bizykel. (read 2001) Personal Communication by Michael Eckert.
Everitt, C.W.F. 1975. James Clerk Maxwell: Physicist and natural philosopher. New York: Scribners.
Goldman, Martin. 1983. The demon in the aether: The life of James Clerk Maxwell. Edinburgh: Paul Harris.
Gooday, Graeme J.N. 2004. The morals of measurement: Accuracy, irony, and trust in late victorian electrical practice. Cambridge: Cambridge University Press.
Harman, Peter M. 1998. The natural philosophy of James Clerk Maxwell. Cambridge: Cambridge University Press.
Hopkinson, John. 1883. On some points in electric lighting. (read 5 April 1883) Original papers by the late John Hopkinson, D.Sc, F.R.S. Vol. 1, 57–83. Cambridge: Cambridge University Press, 1901.
Jeans, James. 1931. James Clerk Maxwell’s method. In James Clerk Maxwell: A commemoration volume, 1831–1931, 91–108. Cambridge: Cambridge University Press.
Lagrange, Joseph Louis. 1997. Analytical mechanics, Trans. and ed. Auguste Boissonnade and Victor N. Vagliente. Boston: Kluwer Academic.
Maxwell, James Clerk 1855. On Faraday’s lines of force. (read 10 Dec. 1855 and 11 Feb. 1856) Transactions of the Cambridge philosophical society. 10. 1864, 27–65. All page numbers refer to the reprint on Faraday’s lines of force. The scientific papers of James Clerk Maxwell, ed. W.D. Niven, Vol. 1, 155–229. Cambridge: Cambridge University Press, 1890.
Maxwell, James Clerk. 1862. On physical lines of force, part I: The theory of molecular vortices applied to magnetic phenomena. Philosophical Magazine 21(139): 161–175 (Mar 1861); On physical lines of force, part II: The theory of molecular vortices applied to electric currents. Philosophical Magazine 21(140): 281–291 (April 1861); On physical lines of force, part II: The theory of molecular vortices applied to electric currents. Philosophical Magazine 21(141): 338–345 (May 1861); On physical lines of force, part III: The theory of molecular vortices applied to statical electricity. Philosophical Magazine 23(151): 12–24 (Jan 1862); On physical lines of force, part IV: The theory of molecular vortices applied to the action of magnetism on polarized light. Philosophical Magazine 23(152): 85–95 (Feb 1862). All page numbers refer to the collected reprint on physical lines of force. The scientific papers of James Clerk Maxwell, ed. W.D. Niven, Vol. 1, 451–513. Cambridge: Cambridge University Press.
Maxwell, James Clerk. 1864. A dynamical theory of the electromagnetic field. (received Oct. 27. 1864). In A dynamical theory. London: Libraries of the Royal Society.
Maxwell, James Clerk. 1865. A dynamical theory of the electromagnetic field. (read Dec. 8 1864) Philosophical Transactions of the Royal Society of London. 155. Jan. 1865. 459–512. All page numbers refer to the reprint a dynamical theory of the electromagnetic field. The scientific papers of James Clerk Maxwell, ed. W.D. Niven, Vol. 1, 526–597. Cambridge: Cambridge University Press, 1890.
Maxwell, James Clerk. 1868. On governors. Proceedings of the Royal Society of London. 16. March 5, 270–283. All page numbers refer to the reprint on governors. The scientific papers of James Clerk Maxwell, ed. W.D. Niven, Vol. 2, 105–120. Cambridge: Cambridge University Press, 1890.
Maxwell, James Clerk. 1871. Introductory lecture on experimental physics. (read Oct. 1871) The scientific papers of James Clerk Maxwell, ed. W.D. Niven. Vol. 2, 241–255. Cambridge: Cambridge University Press, 1890.
Maxwell, James Clerk. 1873. A treatise on electricity and magnetism, 2 volumes, 1st ed. Cambridge: Cambridge University Press.
Maxwell, James Clerk. 1879. Thomson and Tait’s Natural Philosophy. Rev. of Natural Philosophy, by William Thomson and Peter Guthrie Tait. Nature. 20. 3 July 1879. 213–216. All page numbers refer to the reprint Thomson and Tait’s Natural Philosophy. The scientific papers of James Clerk Maxwell, ed. W.D. Niven. Vol. 2, 776–785. Cambridge: Cambridge University Press, 1890.
Maxwell, James Clerk. 1892. A treatise on electricity and magnetism, vol. 2, 3rd ed, ed. J.J. Thomson. New York: Dover.
Maxwell, James Clerk. 1990. The scientific letters and papers of James Clerk Maxwell, vol. 1, ed. Peter M. Harman. Cambridge: Cambridge University Press.
Maxwell, James Clerk. 1995. The scientific letters and papers of James Clerk Maxwell, vol. 2, ed. Peter M. Harman. Cambridge: Cambridge University Press.
Maxwell, James Clerk. 2002. The scientific letters and papers of James Clerk Maxwell, vol. 3, ed. Peter M. Harman. Cambridge: Cambridge University Press.
Mayr, Otto. 1971. Victorian physicists and speed regulation: An encounter between science and technology. Notes and Records of the Royal Society of London 26(2): 205–228.
Rayleigh, John William Strutt, Baron. 1890. On Huygens’s gearing in illustration of the induction of electric currents. (Read May 16 1890). Proceedings of the Physical Society of London. 10: 434–437.
Siegel, Daniel M. 1991. Innovation in Maxwell’s electromagnetic theory: Molecular vortices, displacement current, and light. Cambridge: Cambridge University Press.
Simpson, Thomas K. 1997. Maxwell on the electromagnetic field: A Guided Study. New Jersey: Rutgers University Press.
Sommerfeld, Arnold. 1952. Lectures on theoretical physics: Mechanics. Trans. Martin O. Stern. Vol. 1. New York: Academic Press.
Turner, Joseph. 1956. Maxwell on the logic of dynamical explanation. Philosophy of Science 23: 36–47.
Wise, Matthew Norton. 1977. The flow analogy to electricity and magnetism: Kelvin and Maxwell. PhD dissertation, Princeton University. Ann Arbor: ProQuest/UMI. (No. 77-14252).
Wise, Matthew Norton. 1979. The mutual embrace of electricity and magnetism. Science. New Series 203(4387): 1310–1318.
Acknowledgments
I am grateful to Michel Janssen for his help in the development of the ideas and reconstructions presented in this paper. Additionally, I would like to thank the members of the University of Minnesota’s “Physics Interest Group” for their thoughtful comments when this paper was delivered as a talk, Michael Eckert for pointing out that the flywheel analogy had at least two German speaking fans, and Ekaterina Titova for her encouragement and for always flipping through the pictures. Additionally, I find myself greatly indebted to the late Martin Goldman in whose book I first encounted the crucial quote from “On Faraday’s Lines of Force” describing the balance between analogical and mathematical approaches. Finally, I want to thank Jed Buchwald for his help in editing this paper and pushing me to more critically examine my own understanding of Maxwell and his methodology.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Jed Buchwald.
Rights and permissions
About this article
Cite this article
Lazaroff-Puck, C. Gearing up for Lagrangian dynamics. Arch. Hist. Exact Sci. 69, 455–490 (2015). https://doi.org/10.1007/s00407-015-0157-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00407-015-0157-9