Darrigol O., Frisch U. «From Newton's mechanics to Euler's equations» (1123933), страница 4
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J. Bernoulli subsequently wrotehis second part, where he added the determination of the internal pressure toEuler’s treatment.(9)(10)The latter is the condition of equilibrium of the pendulumunder the action of the forces m A g − m A γ A and m B g − m B γ Bacting respectively on A and B. In d’Alembert’s terminology,the products m A g and m B g are the motions impressed (perunit time) on the bodies A and B under the sole effect ofgravitation (without any constraint). The products m A γ A andm B γ B are the actual changes of their (quantity of) motion (perunit time). The differences m A g − m A γ A and m B g − m B γ Bare the parts of the impressed motions that are destroyed by therigid connection of the two masses through the freely rotatingrod. Accordingly, d’Alembert saw in Eq. (10) a consequenceof a general dynamic principle following which the motionsdestroyed by the connections should be in equilibrium.22D’Alembert based his dynamics on three laws, whichhe regarded as necessary consequences of the principle ofsufficient reason.
The first law is that of inertia, according towhich a freely moving body moves with a constant velocityin a constant direction. The second law stipulates the vector21 For a different view, cf. Truesdell, 1954: XXXIII; Calero, 1996: 460–474.22 D’Alembert, 1743: 69–70. Cf. Vilain, 2000: 456–459. D’Alembertreproduced and criticized Johann Bernoulli’s derivation on p. 71. OnJacob Bernoulli’s anticipation of d’Alembert’s principle, cf. Lagrange, 1788:176–177, 179–180; Dugas, 1950: 233–234; Vilain, 2000: 444–448.O. Darrigol, U.
Frisch / Physica D 237 (2008) 1855–1869superposition of motions impressed on a given body. Accordingto the third law, two (ideally rigid) bodies come to rest aftera head-on collision if and only if their velocities are inverselyproportional to their masses. From these three laws and furtherrecourse to the principle of sufficient reason, d’Alembertbelieved he could derive a complete system of dynamicswithout recourse to the older, obscure concept of force as causeof motion.
He defined force as the motion impressed on abody, that is, the motion that a body would take if this forcewere acting alone without any impediment. Then the third lawimplies that two contiguous bodies subjected to opposite forcesare in equilibrium. More generally, d’Alembert regarded staticsas a particular case of dynamics in which the various motionsimpressed on the parts of the system mutually cancel eachother.23Based on this conception, d’Alembert derived the principleof virtual velocities, according to which a connected systemsubjected to various forces remains in equilibrium if the work ofthese forces vanishes for any infinitesimal motion of the systemthat is compatible with the connections.24 As for the principleof dynamics, he regarded it as a self-evident consequence ofhis dynamic concept of equilibrium.
In general, the effect ofthe connections in a connected system is to destroy part ofthe motion that is impressed on its components by meansof external agencies. The rules of this destruction should bethe same whether the destruction is total or partial. Hence,equilibrium should hold for that part of the impressed motionsthat is destroyed through the constraints. This is d’Alembert’sprinciple of dynamics. Stripped of d’Alembert’s philosophyof motion, this principle stipulates that a connected system inmotion should be at any time in equilibrium with respect to thefictitious forces f − mγ , where f denotes the force applied onthe mass point m of the system, and γ is the acceleration of thismass point.3.2.
Efflux revisitedAt the end of his treatise on dynamics, d’Alembertconsidered the hydraulic problem of efflux through the vesselof Fig. 2. His first task was to determine the condition ofequilibrium of a fluid when subjected to an altitude-dependentgravity g(z).
For this purpose, he considered an intermediateslice of the fluid, and required the pressure from the fluid abovethis slice to be equal and opposite to the pressure from the fluidbelow this slice. According to a slight generalization of Stevin’shydrostatic law, these two pressures are given by the integral ofthe variable gravity g(z) over the relevant range of elevation.Hence the equilibrium condition reads:25Z z1Z ζS(ζ )g(z)dz = −S(ζ )g(z)dz,(11)z0ζ23 D’Alembert, 1743: xiv–xv, 3. Cf.
Hankins, 1968; Fraser, 1985.24 The principle of virtual velocities was first stated generally by JohannBernoulli and thus named by Lagrange (1788: 8–11). Cf. Dugas, 1950:221–223, 320. The term ’work’ is, of course, anachronistic.25 D’Alembert, 1743: 183–186.orZ1861z1g(z)dz = 0.(12)z0According to d’Alembert’s principle, the motion of the fluidunder a constant gravity g must be such that the fluid is inequilibrium under the fictitious gravity g(z) = g−dv/dt, wheredv/dt is the acceleration of the fluid slice at the elevation z.Hence comes the equation of motionZ z1 dvg−dz = 0,(13)dtz0which is the same as Johann Bernoulli’s equation (6).
Inaddition, d’Alembert proved that this equation, together withthe constancy of the product Sv, implied the conservationof live forces in Daniel Bernoulli’s form (Eq. (3)). In hissubsequent treatise of 1744 on the equilibrium and motionof fluids, d’Alembert provided a similar treatment of efflux,including his earlier derivations of the equation of motion andthe conservation of live forces, with a slight variant: he nowderived the equilibrium condition (13) by setting the pressureacting on the bottom slice of the fluid to zero.26 Presumably, hedid not want to base his equations of equilibrium and motionon the concept of internal pressure, in conformity with hisgeneral avoidance of internal contact forces in his dynamics.His statement of the general conditions of equilibrium of afluid, as found at the beginning of his treatise, only required theconcept of wall-pressure.
Yet, in a later section of his treatised’Alembert introduced “the pressure at a given height”:P(ζ ) =Zζ(g − dv/dt)dz,(14)z0just as Johann Bernoulli had done, and for the same purpose ofderiving the velocity dependence of wall-pressure.27In the rest of his treatise, d’Alembert solved problemssimilar to those of Daniel Bernoulli’s Hydrodynamica,with nearly identical results. The only important differenceconcerned cases involving the sudden impact of two layers offluids.
Whereas Daniel Bernoulli still applied the conservationof live forces in such cases (save for possible dissipation intoturbulent motion), d’Alembert’s principle of dynamics thereimplied a destruction of live force. Daniel Bernoulli disagreedwith these and a few other changes. In a contemporary letterto Euler, he expressed his exasperation over d’Alembert’streatise:28I have seen with astonishment that apart from a few little things there is nothingto be seen in his hydrodynamics but an impertinent conceit.
His criticisms arepuerile indeed, and show not only that he is no remarkable man, but also thathe never will be.2926 D’Alembert, 1743: 19–20.27 D’Alembert, 1743: 139.28 D. Bernoulli to Euler, 7 Jul 1745, quoted in Truesdell, 1954: XXXVIIn.29 This is but an instance of the many cutting remarks exchanged betweeneighteenth-century geometers; further examples are not needed here.1862O. Darrigol, U. Frisch / Physica D 237 (2008) 1855–18693.3.
The cause of windsIn this judgment, Daniel Bernoulli overlooked thatd’Alembert’s hydrodynamics, being based on a generaldynamics of connected systems, lent itself to generalizationsbeyond parallel-slice flow. D’Alembert offered strikingillustrations of the power of his approach in a prize-winningmemoir published in 1747 on the cause of winds.30 As thermaleffects were beyond the grasp of contemporary mathematicalphysics, he focused on a cause that is now known to benegligible: the tidal force exerted by the luminaries (the Moonand the Sun).
For simplicity, he confined his analysis to thecase of a constant-density layer of air covering a spherical globewith uniform thickness. He further assumed that fluid particlesoriginally on the same vertical line remained so in the courseof time and that the vertical acceleration of these particleswas negligible (owing to the thinness of the air layer), andhe neglected second-order quantities with respect to the fluidvelocity and to the elevation of the free surface.
His strategywas to apply his principle of dynamics to the motion inducedby the tidal force f and the terrestrial gravity g, both of whichdepend on the location on the surface of the Earth.31Calling γ the absolute acceleration of the fluid particles, theprinciple requires that the fluid layer should be in equilibriumunder the force f + g + γ (the density of the air is onein the chosen units).
From earlier theories of the shape ofthe Earth (regarded as a rotating liquid spheroid), d’Alembertborrowed the equilibrium condition that the net force shouldbe perpendicular to the free surface of the fluid. He alsorequired that the volume of vertical cylinders of fluid shouldnot be altered by their motion, in conformity with his constantdensity model. As the modern reader would expect, from thesetwo conditions d’Alembert derived some sort of momentumequation, and some sort of incompressibility equation. He didso in a rather opaque manner. Some features, such as the lackof specific notation for partial differentials or the abundantrecourse to geometrical reasoning, disconcert modern readersonly.32 Others were problematic to his contemporaries: heoften omitted steps and introduced special assumptions withoutwarning.
Also, he directly treated the utterly difficult problemof fluid motion on a spherical surface without preparing thereader with simpler problems.30 As a member of the committees judging the Berlin Academy’s prizes onwinds and on fluid resistance (he could not compete as a resident member),Euler studied d’Alembert’s submitted memoirs of 1747 and 1749. The subjectset for the first prize, probably written by Euler, was “to determine the order& the law wind should follow, if the Earth were surrounded on all sides by theOcean; so that one could at all times predict the speed & direction of the windin all places.” The question is here formulated in terms of what we now callEulerian coordinates (“all places”), cf.
Grimberg, 1998: 195.31 D’Alembert, 1747. D’Alembert treated the rotation of the Earth and theattraction by the Sun and the Moon as small perturbing causes whose effectson the shape of the fluid surface simply added (D’Alembert, 1747: xvii, 47).Consequently, he overlooked the Coriolis force in his analysis of the tidaleffects (in D’Alembert, 1747: 65, he writes he will be doing as if it were theluminary that rotates around the Earth).32 D’Alembert used a purely geometrical method to study the free oscillationsof an ellipsoidal disturbance of the air layer.Fig. 6.
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