Abstract
Gibbs and Helmholtz provide the strongest scientific influences on Duhem’s works in what is now called mathematical physics. With the help of examples exhibiting this influence in thermo-mechanics and electrodynamics, it is shown that this conduced Duhem and his followers to a definite style and practice of physical science marked by abstraction and mathematical rigor. This has practically become the rule while helping to classify the numerous, linear or non linear, effects and giving rise to fruitful developments, in continuum physics.
Unpublished contribution to the Wissenschaftliche Veranstaltungen aus Anlass des 100. Todestages von Hermann von Helmholtz. Fachkolloquien zu Themen Helmholtzscher Traditionen, 10 September 1994, Humboldt-Universtät zu Berlin (Thermodynamik: Von den Berliner Anfängen zu Modernen Entwicklungen). Most of the bibliographical material needed in this study was gathered while the author was a member of the Wissenschaftskolleg zu Berlin (1991–92). He received there the help of a formidable library service.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
In shock waves for one-dimensional motions in fluids the Hugoniot jump relation reads
$$H := \left[ {e + \left\langle p \right\rangle \tau } \right]$$(a)where \(e\left( {\tau ,\eta } \right)\) is the internal energy per unit mass, a function of specific volume τ and specific entropy η; p is the thermodynamical pressure, \(\theta = \partial e/\partial \eta \, > 0\) is the thermodynamic temperature, <..> is the mean value of a quantity at the shock, and [..] its jump. The best known disciple of Duhem was E. Jouguet, a specialist of shock and detonation waves, and explosives. For one-dimensional phase-transition fronts (this dimensionality is chosen for illustrative purpose only) in solids, the driving force acting on the front reads:
$$F = - \left[ {W\left( {\varepsilon ,\theta } \right) - \left\langle \sigma \right\rangle :\varepsilon } \right]$$(b)where W is the free energy per unit volume, a function of strain ε and temperatureθ, the entropy per unit volume is given by \(S = - \partial W/\partial \theta\) , and σ is the stress. H = 0 at shocks whereas F is in general not zero at irreversibly progressing phase-transition fronts. Both F and the nonzero propagation velocity V of the front satisfy jointly at the front the second law of thermodynamics in the form F.V > 0 or = 0 (for the exact three-dimensional theory in conductors of heat see [39]).
- 3.
Much more on rational thermodynamics is to be found in Truesdell [46].
- 4.
This is rightly emphasized by Miller [42], p. 229.
- 5.
This is exemplified by the author’s course that deals with strongly nonlinear dissipative processes Maugin [36]. This style of thermodynamical exposition is to be found in the Journal of Non-Equilibrium Thermodynamics, de Gruyter, Berlin. The book by Bridgman [3] was instrumental in this development, especially in influencing Joseph Kestin from whom we all more or less learned our “thermodynamics”. The points of view of Duhem, Bridgman and Kestin are examined in parallel and comparison in the book [37].
- 6.
Duhem [15, English translation, p. 79] claims that Maxwell justifies the introduction of the displacement current by means of two lines:“ The variation of the electric displacement should be added to the current in order to obtain the total movement of the electricity”.
- 7.
This is mainly exposed in Helmholtz [24]—also (Helmholtz [26], posthumous). In modern times, this theory has been discussed several times, e.g. by Hirosize [29] and Buchwald [5, see Chap. 21]. Strangely enough, none of the modern commentaries cites Duhem’s thorough analysis, perhaps because Duhem went through some purgatory period and the original work in French was never reprinted or translated. For the information of the reader, Helmholtz's equations using potentials read (in modern notation).
$$\begin{aligned} \nabla^{2} {\mathbf{U}} &= \left( {1 - k} \right)\nabla \left( {\partial \phi /\partial t} \right) - 4\pi {\mathbf{J}} ,\\ \nabla .{\mathbf{U}} &= - k\,\partial \phi /\partial \,t\; ,\\ \nabla^{2} \phi &= - 4\pi \rho_{f} ,\quad \left( {\partial \rho_{f} /\partial t} \right) + \nabla .{\mathbf{J}} ,\\ \end{aligned}$$where U and ϕ are a vector potential and a scalar potential, and k is a constant to be found
by means of experiments conducted on an open circuit. It is to be noted that the time rate
of change of ϕ affects U by virtue of the continuity equation. This, in fact, is a hindrance
in the reduction of Helmholtz’ to Maxwell’s equations. We recommend Buchwald’s
discussion as very enlightening, especially in so far as the “Maxwell limit” is concerned. Duhem’s analysis is also briefly given in his Duhem [17, pp. 147–150].
- 8.
- 9.
We remind the reader that this “principle” recommends to enter the whole set of independent field variables as possible arguments in all constitutive equations.
- 10.
See Chap. 4 in Glandsdorff and Prigogine [23] for the stability according to Gibbs and Duhem. The general theory of the stability of thermodynamic equilibrium makes use of the Gibbs-Duhem approach and the balance of entropy. Chapters 6 and 7 deal with systems out of equilibrium. The minimum property of the dissipation function has been established by Helmholtz for a linear viscous fluid. The relationship between the Le Châtelier-Braun principle and Duhem's work on the displacement out of equilibrium is reported in Manville [32, pp. 259–260].
- 11.
The original works of P. Duhem on hysteretic systems are published in 1901 in the Zeitschrift für physikalisch Chemie and in the Mémoires présentés à la Classe de Sciences de l’Académie de Belgique. The most relevant equations are best expressed by Manville [32]—apparently the finest and sharpest analyst of Duhem’s scientific works—e.g. his Eq. (9) in p. 310, dA.da > 0, and his un-numbered equation in p. 313: Integral of A da > 0 for an isothermal closed cycle, are identical to the expressions of Drucker's and Ilyushin’s local and global stability conditions of modern plasticity with hardening—compare to Eqs. (5.75) and (5.88) in Maugin [36], pp. 108 and 111, respectively, where the proof relies on the convexity of the free energy with respect to a, and the convexity of the homogeneous positive dissipation potential in A, the thermodynamical force associated to a. This applies to so-called generalized standard (thermodynamic) materials whose two basic potentials (free energy and dissipation) exhibit these properties. Duhem did not possess the last concept but he had a rather clear view of incremental laws exhibiting hysteresis as shown by Manville’s [32] equations in pp. 307–310.
- 12.
- 13.
To Manville [32], p. 197 Gibbs and Helmholtz are not dissociable in Duhem’s vision.
- 14.
References
Boltzmann L (1893) Vorlesungen über Maxwells Theorie der Elektrizität und des Lichtes, 2 volumes. Leipzig
Born M (1921) Kritische Betrachtungen zur traditionellen Darstellung der Thermodynamik. Phys Zeit 22:218–224, 249–254, 282–286
Bridgman P (1943) The nature of thermodynamics. Harvard University Press, Cambridge
Brouzeng P (1987) Duhem: science et providence. Belin, Paris
Buchwald JZ (1985) From Maxwell to Microphysics (Aspects of Electromagnetic Theory in the Last Quarter of the Nineteenth Century). The University of Chicago Press, Chicago [see Chap. 21]
Caratheorory C (1909) Untersuchungen über die Grundlagen der Thermodynamik. Math Ann 67:355–386
Coleman BD (1973) The energy criterion for stability in continuum thermodynamics. Rend Semin Mat Fis Milano 43:85–99
Coleman BD, Noll W (1964) Thermodynamics of materials with memory. Arch Rat Mech Anal 17:1–46
Duhem P (1886) Le potentiel thermodynamique et ses applications à la mécanique physique et à la théorie des phénomènes électriques. A. Hermann, Paris 248 pp
Duhem P (1893) Commentaires aux principes de la thermodynamique, Seconde partie: Le principe de Sadi Carnot et de R. Clausius. Journ Math Pures et Appl 9:293–359
Duhem P (1902) Sur les conditions nécessaires pour la stabilité de l’équilibre d un système visqueux. C R Acad Sci Paris 135:1290–1294
Duhem P (1903) Considérations sur la stabilité, et plus particulièrement la stabilité des corps élastiques. In: Procès verbaux des séances de la Société des Sciences Physiques et Naturelles de Bordeaux, 25 juin 1903 [also in: Recherches sur l’élasticité, Third part: La stabilité des milieux élastiques, 218 pages. Gauthier-Villars, Paris, 1906]
Duhem P (1903) L’évolution de la mécanique, 348 pages. A. Joanin, Paris [Collection of seven papers published in 1903 in the « Revue Générale des sciences Pures et Appliquées »]
Duhem P (1906) Les théories électriques de J. Clerk Maxwell - Etude historique et critique, 228 pages. A. Hermann, Paris [This is a collection of papers published in 1900–1901 in the Annales de la Société Scientifique de Bruxelles]
Duhem P (1906) La théorie physique, son objet et sa structure, 450 pages. Chevalier et Rivière, Paris (This is a collection of papers published in 1904 and 1905 in the Revue de Philosophie). English translation: The Aim and Structure of Physical Theory, Princeton University Press, 1954; paperback edition 1991; precise references are to this edition; German translation: Autorisierte übersetzung: Ziel und Struktur der Physikalischen Theörien, mit einem Vorwort von Ernst Mach, 367 pages. Johan Ambrosius Barth, Leipzig (1908)]
Duhem P (1911) Traité d’énergétique ou de thermodynamique générale, Two volumes, 528 and 504 pages. Gauthier-Villars, Paris
Duhem P (1913) Notice sur les Titres et Travaux Scientifiques de Pierre Duhem, rédigée par lui-même lors de sa candidature à l’Académie des Sciences [Reprinted in: Mémoires de la Société des Sciences Physiques et Naturelles de Bordeaux, 7ème série, T. 1, pp 40–169 (1917)]
Ericksen JL (1966) Thermoelastic stability. In: Proceedings. 5th National Congress Applied Mechanics, pp 187–193
Ericksen JL (1977) Special Topics in Elastostatics. In: Advances in Applied Mechanics, Ed. C.-S. Yih, 17:189–244, Academic Press, New York
Ericksen JL (1991) Introduction to the thermodynamics of solids. Chapman and Hall, London
Eringen AC (Editor 1971–1976) Continuum Physics, Four volumes. Academic Press, New York
Eringen AC, Maugin GA (1990) Electrodynamics of Continua, Two volumes. Springer, New York [Soft-cover reprint, Springer, New York, 2012]
Glandsdorff P, Prigogine I (1971) Structure, stabilité et fluctuations. Masson, Paris
Helmholtz H von (1870) Über die Bewegungsgleichungen der Elektrizität für ruhende leitende Kôrper. (Crelle) J reine angewandte Math 72:57–129
Helmholtz H von (1882) Die thermodynamik chemischer Vorgänge. Sitz Deutsch Akad Wiss Berlin 1:22–39
Helmholtz H von (1897, posthumous) Vorlesungen über die Theorie des Lichtes. Unter Rucksicht auf. d. elastische und der elektromagnetische Ausschauung, J.A. Barth, Leipzig (Lectures)
Hertz H (1894) Die Prinzipien der Mechanik, in neuem Zuzammenhange dargestellt, Leipzig [English translation: The Principles of Mechanics, New York 1899; Dover reprint, New York, 1956]
Hertz H (1894–5) Untersuchungen über eine Ausbreitung der elektrischen Kraft. In: Vol. 2 of Gesammelthe Werke, Ed. P. Lenard, Leipzig
Hirosize T (1969) Origins of Lorentz Theory of Electrons and the Concept of the Electromagnetic Field. In: Historical Studies in the Physical Sciences, Ed. McCormmach, Vol 1, pp 151–209, Princeton University Press, New Jersey
Hunt BR (1991) The maxwellians. Cornell University Press, Ithaca
Jaki SL (1984) Uneasy genius: the life and the work of Pierre Duhem. Martinus Nijjhoff, The Hague
Manville O (1927) La physique de Pierre Duhem. In: Pierre Duhem, Sa vie - Ses Oeuvres, Mém Soc Sci Phys Nat Bordeaux T.1, 7ème Série, pp 171–636
Manville O (1928) L’oeuvre scientifique de Pierre Duhem, Vol l, Bordeaux (see especially pp 71–76 for the scientific heritage of Helmholtz)
Massieu F (1869) Sur les fonctions caractéristiques des divers fluides. C R Acad Sci Paris 69(858–864):1057–1061
Maugin GA (1988) Continuum mechanics of electromagnetic solids. North-Holland, Amsterdam
Maugin GA (1992) Thermomechanics of plasticity and fracture. Cambridge University Press, U.K.
Maugin GA (1999) Thermodynamics of nonlinear irreversible behaviors. World Scientific, Singapore
Maugin GA, Muschik W (1994) Thermodynamics with internal variables. Part 1: General concepts, Part II. Applications. J Non-Equilibr Thermodynam 19: 217–249, 250–289
Maugin GA, Trimarco C (1994) Material Forces and Irreversible Progress of Thermoelastic Phase-transition Fronts. Euromech Colloquium on Phase Transitions in Solids in the Honour of J.L. Ericksen, Udine, May 1994 [Cf. Meccanica (Italy) 30: 605–619, 1995]
Maxwell JC (1873) A Treatise on Electricity and Magnetism. Clarendon Press, Oxford (Dover reprint, two volumes, New York, 1954)
Mayergoz I (1991) Mathematical models of hysteresis. Springer, New York
Miller DG (1970) Entry: “Pierre-Maurice-Marie Duhem”. In: Dictionary of Scientific Biographies. Ed. C.C. Gillispie, Vol 3, pp 225–233, Charles Scribner’s Sons, New York
O’Rahilly A (1931) Electromagnetics. London, (reprinted as Electromagnetic Theory, Two volumes, Dover, New York, 1965)
Poincaré H (1890) Electricité et optique, Two volumes. Paris [Lecture notes of the Physics Course at the Sorbonne in 1888, the year electromagnetic waves were experimentally discovered and their speed of propagation shown to be equal to that of light by H. Hertz]
Roy L (1923) L’électrodynamique des milieux isotropes en repos d’après Helmholtz et Duhem. Paris
Truesdell CA (1986) Rational thermodynamics (revised and enlarged edition). Springer, New York (original edition, McGraw Hill), New York, p 1969
Truesdell CA, Noll W (1965) The Nonlinear Field Theories of Mechanics. In: Handbuch der Physik, Bd.III/3, Ed. S. Flügge, Springer, Berlin
Visintin A (1991) Bibliography of hysteresis. Preprint, University of Trento, New Jersey
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Maugin, G.A. (2014). Helmholtz Interpreted and Applied by Duhem. In: Continuum Mechanics Through the Eighteenth and Nineteenth Centuries. Solid Mechanics and Its Applications, vol 214. Springer, Cham. https://doi.org/10.1007/978-3-319-05374-5_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-05374-5_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-05373-8
Online ISBN: 978-3-319-05374-5
eBook Packages: EngineeringEngineering (R0)