Essays in the History of Mechanics pp 334-366 | Cite as

# Recent Advances in Rational Mechanics (1956)

Chapter

## Abstract

I begin by answering the question most of you asked upon reading the title: What is rational mechanics? It is difficult to define rational mechanics, but no more so than to define chemistry or physics or mathematics. However, a chemist coming before you would not be expected to begin by defining chemistry. The reason most of you wish me to define rational mechanics is that in the United States, at least, it is not a recognized science. Indeed, there are some who disbelieve in its existence^{1}.

## Keywords

Phase Average Vortex Line Vortex Tube Rational Mechanic Bibliographical Note
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## Literatur

- 3.I. Newton,
*Philosophiae Naturalis Principia Mathematica*,London, 1687; 2nd ed., Cambridge, 1713; 3rd ed., London, 1726. The only available English translation is unreliable, especially in the allegedly modernized reprint. Some of the relevant passages, newly translated, are quoted in Essay III, above. A new edition of the*Principia*,collating the three original ones, and presumably also to be provided with a correct translation, has been undertaken by I. B. Cohen and the late A. Koyre ; its appearance is eagerly awaited.Google Scholar - 4.D. Hilbert, “Mathematische Probleme,”
*Nachr. Ges. Wiss. Göttingen*1900, 253–297. An English translation of an amplified version is printed in*Bull. Am. Math. Soc*. (2) 8, 437–479 (1902).*Note added for the reprinting*. Only two significant attempts to solve the part of Hilbert’s sixth problem that concerns mechanics have been published: that of Hamel, Über die Grundlagen der Mechanik,*Math. Annalen*66, 350–397 (1908), and that of Noll, mentioned in Essay VII, above, and presented in full in the notes of his lectures at the C.I.M.E. Course on*Recent Developments in the Mechanics of Continua*at Bressanone in 1965, Edizioni Cremonese, Rome, 1966, and issued also as a report of the Mathematics Department of the Carnegie Institute of Technology, 1966.Google Scholar - 5.J. W. Gibbs,
*Elementary Principles in Statistical Mechanics*, 1902, reprinted in Volume 2 of Gibbs’*Collected Papers*.MATHGoogle Scholar - 6.H. v. Helmholtz, “Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen,”J.
*Reine Angew. Math*. 55, 25–55 (1858)=*Ges. Abb*. 1, 101–134. English translation,*Phil. Mag*. (4) 33, 485–512 (1867).CrossRefMATHGoogle Scholar - 7.W. Thomson, “On vortex motion,”
*Trans. Roy. Soc. Edinburgh*25, 217–260 (1869) =*Collected Papers*4, 13–66.Google Scholar - 8.H. Ertel, “ Ein Theorem über asynchron-periodische Wirbelbewegungen kompressibler Flüssigkeiten,”
*Misc. Acad. Berol*. 1, 62–68 (1950).MathSciNetGoogle Scholar - 9.E. Reissner, “On a variational theorem in elasticity,”J.
*Math. Phys*. 29, 90–95 (1950); “On a variational theorem for finite elastic deformations,” J.*Math. Phys*. 32, 129–135 (1953).*Note added for the reprinting*. In fact, the variational principle seems to be due to M. Born, who gave it for two-dimensional problems, with boundary conditions neglected, in § IV of the Anhang to his Habilitationsschrift,*Untersuchungen über die Stabilität der elastischen Linie in Ebene und Raum, unter verschiedenen Grenzbedingungen*,Göttingen, 1906. In three dimensions, again without boundary conditions, it was given by E. Hellinger in § 7e of his article, “Die allgemeinen Ansätze der Mechanik der Kontinua,”*Enz. Math. Wiss*. 44, 602–694 (1914). The new contribution of Reissner lies in the formulation of a boundary integral such as to yield appropriate boundary conditions for all the standard problems of linear elasticity, and in applications to cases.MathSciNetMATHGoogle Scholar - 10.A. J. C. B. de St. Venant, “Memoire sur la torsion des prismes ...,”
*Mem. Divers Savants Acad. Sci. Paris*14, 233–560 (1855); also issued separately,*“De la Torsion des Prismes*...,” Paris, 1855.Google Scholar - 11.J. Boussinesq,
*Application des Potentiels à l ’Étude de l ’Équilibre et du Mouvement des Solides É lastiques*,Paris, Gauthier-Villars, 1885.Google Scholar - 12.R. V. Mises, “On Saint-Venant’s Principle,”
*Bull. Am. Math. Soc*. 51, 555–562 (1945).CrossRefMATHGoogle Scholar - 13.E. Sternberg, “On Saint-Venant’s Principle,”
*Quart. Appl. Math*. 11, 393–402 (1954).MathSciNetMATHGoogle Scholar - 16.P. G. Bordoni, “ Sopra le trasformazioni termoelastiche finite di certi solidi omogenei ed isotropi,”
*Rend. mat. e delle sue Applic*. (Roma) (5) 12, 237–266 (1953). See §5.*Note added for the reprinting*. The thermodynamic basis of this work does not now seem clear to me.MathSciNetGoogle Scholar - 18.M.Reiner, “A mathematical theory of dilatancy,”
*Am. J. Math*. 67, 350–362 (1945). Reprinted in*Rational Mechanics of Materials*•, Gordon &Breach, 1965.*Cf*. also W. Prager, “ Strain hardening under combined stresses,” J.*Appl. Phys*. 16, 837–840 (1945), where the same algebraic theorem is inferred in a different context.MathSciNetADSCrossRefMATHGoogle Scholar - 19.R. S. Rivlin, “The hydrodynamics of Non-Newtonian fluids, I,”
*Proc. R. Soc. London*193, 260–281 (1948); reprinted in*Rational Mechanics of Materials*, Gordon &Breach, 1965.MathSciNetADSCrossRefMATHGoogle Scholar - 23.A.-L. Cauchy, Sur l’équilibre et le mouvement intérieur des corps considérés comme des masses continues,
*Ex. Math*. 4, 293–319 (1829)=*(Euvres*(2) 9, 342–369.Google Scholar - 24.S. Zaremba, “ Sur une forme perfectionnée de la théorie de la relaxation,”
*Bull, inter. Acad. Sci. Cracovie*1903, 594–614.Google Scholar - 25.J. G. Oldroyd, “On the formulation of rheological equations of state,”
*Proc. R. Soc. London*A 200, 523–541 (1950). Reprinted in*Rational Mechanics of Materials*,Gordon &Breach, 1965.MathSciNetADSGoogle Scholar - 26.W. Noll, “On the continuity of the solid and fluid states,” J.
*Rational Mech. Anal*. 4, 3–81 (1955). Reprinted in*Rational Mechanics of Materials*,Gordon &Breach, 1965.MathSciNetMATHGoogle Scholar - 28.J. C. Maxwell, “On the dynamical theory of gases,”
*Phil. Trans. R. Soc. London*A157, 49_88 (1867)=*Phil. Mag*. (4) 35, 129–145, 187–217 (1868)=*Papers*2, 26–78.Google Scholar - 32.J. Irving &J. G. Kirkwood, The statistical mechanical theory of transport processes, IV, The equations of hydrodynamics, J.
*Chem. Phys*. 18, 817–829 (1950).MathSciNetADSCrossRefGoogle Scholar - 33.W. Noll, “ Die Herleitung der Grundgleichungen der Thermomechanik aus der statistischen Mechanik,” J.
*Rational Mech. Anal*. 4, 627–646 (1955).MathSciNetMATHGoogle Scholar

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