Darrigol O., Frisch U. «From Newton's mechanics to Euler's equations» (1123933)

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Physica D 237 (2008) 1855–1869www.elsevier.com/locate/physdFrom Newton’s mechanics to Euler’s equationsIO. Darrigol a , U. Frisch b,∗a CNRS:Rehseis, 83, rue Broca, 75013 Paris, Franceb Labor. Cassiopée, UNSA, CNRS, OCA, BP 4229, 06304 Nice Cedex 4, FranceAvailable online 17 August 2007AbstractThe Euler equations of hydrodynamics, which appeared in their present form in the 1750s, did not emerge in the middle of a desert. We shallsee in particular how the Bernoullis contributed much to the transmutation of hydrostatics into hydrodynamics, how d’Alembert was the firstto describe fluid motion using partial differential equations and a general principle linking statics and dynamics, and how Euler developed themodern concept of internal pressure field which allowed him to apply Newton’s second law to infinitesimal elements of the fluid.c 2007 Elsevier B.V.

All rights reserved.Keywords: History of science; Fluid dynamics; Euler equationsQuelques sublimes que soient les recherches sur les fluides, dont nous sommesredevables à Mrs. Bernoullis, Clairaut, & d’Alembert, elles découlent sinaturellement de mes deux formules générales : qu’on ne scauroit assés admirercet accord de leurs profondes méditations avec la simplicité des principes, d’oùj’ai tiré mes deux équations, & auxquels j’ai été conduit immédiatement par lespremiers axiomes de la Mécanique.1(Leonhard Euler, 1755)1. IntroductionLeonhard Euler had a strong interest in fluid dynamics andrelated subjects during all his adult life.

In 1827, at age twenty,he published an important paper on the theory of sound. Inthat paper, he gave a quantitative theory of the oscillationsof the column of air in a flute or similar instruments. Ona slate found after his death on 7 September 1783 he hadI The present article includes large sections of Chapter 1 of Darrigol, 2005,thanks to the kind permission of Oxford University Press.

We mention that oneof the authors (OD) is a theoretical physicist by early training who became ahistorian of science some twenty years ago, while the other one (UF) is a fluiddynamicist interested in Euler’s equations since the seventies.∗ Corresponding author. Tel.: +33 4 92003035; fax: +33 4 92003058.E-mail address: uriel@obs-nice.fr (U. Frisch).1 Euler, 1755c: 316[original publication page]/92[omnia page]: Howeversublime the researches on fluids that we owe to Messrs Bernoullis, Clairaut,and d’Alembert may be, they derive so naturally from my two general formulasthat one could not cease to admire this agreement of their profound meditationswith the simplicity of the principles from which I have drawn my two equationsand to which I have been immediately driven by the first axioms of Mechanics.c 2007 Elsevier B.V.

All rights reserved.0167-2789/$ - see front matter doi:10.1016/j.physd.2007.08.003developed a theory of aerostatic balloons, having just learnedabout the first manned ascent of a balloon designed by theMontgolfier brothers. Altogether, he published more than fortypapers or books devoted to fluid dynamics and applications.After his arrival in Saint-Petersburg in 1727, and perhapsbefore, Euler was planning a treatise on fluid mechanics basedon the principle of live forces.

He recognized the similarityof his project with Daniel Bernoulli’s and left the field opento this elder friend. During the fourteen years of his firstPetersburg stay, Euler was actively involved in establishing thetheoretical foundations of naval science, thereby contributingto the ongoing effort of the Russian state in developing amodern and powerful fleet. His Sciencia Navalis, completed by1738 and published in 1749, contained a clear formulation ofhydrostatic laws and their application to the problem of shipstability. It also involved a few Newtonian considerations onship resistance. Soon after his move to Berlin in 1741, he editedthe German translation of Benjamin Robins’s New Principles ofGunnery, as a consequence of Frederick II’s strong interest inthe science of artillery. Published in 1745, this edition includedmuch innovative commentary on the problem of the resistanceof the air to the motion of projectiles, especially regarding theeffects of high speed and cavitation.22 Euler, 1727, [1784] (balloons), 1745, 1749.

For general biography, cf.Youschkevitch, 1971; Knobloch, 2008 and references therein. On Euler and1856O. Darrigol, U. Frisch / Physica D 237 (2008) 1855–1869Today’s fluid dynamics cannot be conceived without thefundamental basis of Euler’s equations, as they appear in“Principes généraux du mouvement des fluides”, presented tothe Académie Royale des Sciences et Belles-Lettres (Berlin)on 4 September 1755 and published in 1757. In Euler’s ownnotation, they read: d.qud.qvd.qwdq+++=0dtdxdydz 1 dpP−q dx dudududu+u+v+w=dtdxdydz 1 dpQ−(1)q dy dvdvdvdv+u+v+w=dtdxdydz 1 dpR−q dz dwdwdwdw=+u+v+w.

dtdxdydzHere, P, Q, and R are the components of an external force, suchas gravity. The modern reader with no special training in thehistory of science will nevertheless recognize these equationsand be barely distracted by the use of q instead of ρ for density,duof dxinstead of ∂∂ux and of d.qu instead of ∂(qu).3Euler’s three memoirs on fluid dynamics written in 1755contain, of course, much more than these equations. They areimmediately intelligible to the modern reader, the argumentsbeing strikingly close to those given in modern treatises. Theymark the emergence of a new style of mathematical physicsin which fundamental equations take the place of fundamentalprinciples formulated in ordinary or geometrical language.Euler’s equations are also the first instance of a nonlinear fieldtheory and remain to this day shrouded in mystery, contrary forexample to the heat equation introduced by Fourier in 1807 andthe Maxwell equations discovered in 1862.Our main goal is to trace the development and maturationof the physical and mathematical concepts, such as internalpressure, which eventually enabled Euler to produce hismemoirs of the 1750s.4 The emergence of Euler’s equationswas the result of several decades of intense work involvingsuch great figures as Isaac Newton, Alexis Clairaut, Johannand Daniel Bernoulli, Jean le Rond d’Alembert .

. . and Eulerhimself. It is thus also our goal to help the reader to see howsuch early work, which is frequently difficult because it is notcouched in modern scientific language, connects with Euler’shydraulics, cf. Mikhailov, 1983. On sound, cf. Truesdell, 1955: XXIV–XXIX.On the early treatise on fluids, cf. Mikhailov, 1999, and pp. 61–62, 80 in Euler,1998. On naval science, cf. Nowacki, 2006; Truesdell, 1954: XVII–XVIII,1983. On gunnery, cf.

Truesdell, 1954: XXVIII–XLI.3 Euler, 1755b.4 Detailed presentations of these may be found in Truesdell’s 1954 landmarkwork on Euler and fluid dynamics.maturing views on continuum mechanics and his papers of the1750s.Section 2 is devoted to the first applications of Newtonianmechanics to fluid flow, from Newton to the Bernoullis.Whereas Isaac Newton treated a few particular problemswith heteroclite and ad hoc methods, Daniel and JohannBernoulli managed to solve a large class of problems througha uniform dynamical method. Section 3 shows how Jean leRond d’Alembert’s own dynamical method and mathematicalcreativity permitted a great extension of the investigated classof flows. Despite its now antiquated formulation, his theoryhad many of the key concepts of the modern theory ofincompressible flows.

In Section 4 we discuss Euler’s memoirsof the 1750s. Finally, a few conclusions are presented inSection 5. Another paper in these Proceedings focuses onEuler’s 1745 third remark (Theorem 1) à propos Robins’sGunnery. This remark, which actually constitutes a standalonepaper of eleven pages on the problem of steady flow arounda solid body, is at the crossroads of eighteenth-century fluiddynamics: it uses many ideas of the Bernoullis to write theequations in local coordinates and has been viewed, correctlyor not, as a precursor of d’Alembert’s derivation of the paradoxof vanishing resistance (drag) for ideal flow.52.

From Newton to the Bernoullis2.1. Newton’s principiaThrough the eighteenth century, the main contexts forstudies of fluid motion were water supply, water-wheels,navigation, wind-mills, artillery, sound propagation, andDescartes’s vortex theory. The most discussed questions werethe efflux of water through the short outlet of a vessel, theimpact of a water vein over a solid plane, and fluid resistancefor ships and bullets. Because of its practical importance andof its analogy with Galilean free-fall, the problem of efflux gotspecial attention from a few pioneers of Galilean mechanics.

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