Darrigol O., Frisch U. «From Newton's mechanics to Euler's equations» (1123933), страница 8
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Cf. Truesdell, 1954: LXXXV–C.56 As observed by Truesdell, 1954: XC–XCI), in Section 66 Euler reverts tothe assumption of non-vortical flow, a possible leftover of an earlier version ofthe paper.57 Euler, 1755c: 345/117.58 Euler, 1755b: 315/91.1868O. Darrigol, U. Frisch / Physica D 237 (2008) 1855–18695. ConclusionsReferencesIn retrospect, Euler was right in judging that his “twoequations” were the definitive basis of the hydrodynamics ofperfect fluids. He reached them at the end of a long historicalprocess of applying dynamical principles to fluid motion. Anessential element of this evolution was the recurrent analogybetween the efflux from a narrow vase and the fall of acompound pendulum. Any dynamical principle that solvedthe latter problem also solved the former.
Daniel Bernoulliappealed to the conservation of live forces; Johann Bernoullito Newton’s second law together with the idiosyncratic conceptof translatio; d’Alembert to his own dynamical principle ofthe equilibrium of destroyed motions. With this more generalprinciple and his feeling for partial differentials, d’Alembertleapt from parallel-slice flows to higher problems that involvedtwo-dimensional anticipations of Euler’s equations. Althoughhis method implicitly contained a general derivation of theseequations in the incompressible case, his geometrical style andhis abhorrence of internal forces prevented him from taking thisstep.Despite d’Alembert’s reluctance, another important elementof this history turns out to be the rise of the concept of internalpressure.
The door on the way to general fluid mechanicsopened with two different keys, so to speak: d’Alembert’sprinciple, or the concept of internal pressure. D’Alembert(and Lagrange) used the first key, and introduced internalpressure only as a derivative concept. Euler used the secondkey, and ignored d’Alembert’s principle. As Euler guessed (andas d’Alembert suggested en passant), Newton’s old secondlaw applies to the volume elements of the fluid, if onlythe pressure of fluid on fluid is taken into account.
Euler’sequations derive from this deceptively simple consideration,granted that the relevant calculus of partial differentials isknown. Altogether, we see that hydrodynamics rose throughthe symbiotic evolution of analysis, dynamical principles, andphysical concepts. Euler pruned the unnecessary and unclearelements from the abundant writings of his predecessors,and combined the elements he judged most fundamental inthe clearest and most general manner. He thus obtained anamazingly stable foundation for the science of fluid motion.The discovery of sound foundations only marks thebeginning of the life of a theory. Euler himself suspectedthat the integration of his equations would in general be aformidable task.
It soon became clear that their applicationto problems of resistance or retardation led to paradoxes.In the following century, physicists struggled to solve theseparadoxes by various means: viscous terms, discontinuitysurfaces, instabilities. A quarter of a millennium later, somevery basic issues remain open, as many contributions to thisconference amply demonstrate.Ackeret, Jakob 1957 ‘Vorrede’, in L.
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Pauls and twoanonymous reviewers for many useful remarks. We alsoreceived considerable help from J. Bec and H. Frisch.O. Darrigol, U. Frisch / Physica D 237 (2008) 1855–1869Euler, Leonhard 1760 ‘Recherches sur le mouvement des rivières’[writtenaround 1750–1751]. MASB, 16 [printed in 1767], 101–118. Also in Operaomnia, ser. 2, 12, 272–288, E332.Euler, Leonhard 1756–1757 ‘Principia motus fluidorum’ [written in 1752].
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