Euler, the theory of gravitation and some geophysical aspects

Acta Gead. Geoph. Hung., Vol. 36(4), pp. 463-470 (2001)

EULER, THE THEORY OF GRAVITATION AND SOME GEOPHYSICAL ASPECTS

W SCHRODER1 and H-J TREDER2

[Manuscript received October 15, 1999, revised January 24, 2000]

Euler's results concerning the theory of gravitation are discussed in this paper in order to give an insight into the scientific problems of Euler's age and his relation to Newton's work. Geophysical aspects are emphasized in his work.

Keywords: Euler; Newton; theory of gravitation

I.

Euler was, together with A C Clairaut (1713-1765) one of the founders of celestial mechanics, the grand masters of which became later J L Lagrange (1736-1813) and PS Laplace (1749-1827) (Ertel 1953). Euler's and Clairaut's pioneering merit was to formulate and discuss the Newtonian principles of mechanics and the Newtonian gravity law in the analytical form of the infinitesimal calculus. Isaac Newton (1643-1727) himself formulated his "Principia" basically without referring to his fluxion calculus in a purely geometrical way and correspondingly he presented celestial mechanics, including the n-body-problem and the computation of disturbances in a synthetic form (Treder 1983, 1997). In fact, Newton naturally deduced at first his results concerning the movement of celestial bodies using the by him just then discovered infinitesimal calculus. Newton supposed, however, that it would not be proper to present simultaneously the new physical content and the new mathematical formalism. That is why Clairaut and Euler, later also Lagrange and Laplace had to transform the technically clumsy computations in "Principia" and to interpret them algorithmically in the language of the infinitesimal calculus.

As President of the Mathematical Section of the Prussian Academy of Sciences, Euler designed a conception in 1742 for the astronomical research - namely as President, he supervised the Berlin Astronomical Observatory, too. This concept is presented in a letter to the perpetual secretary of the Academy, to Ph de Jarige. The corresponding part of this letter tells us:

"The real theory of astronomy consists mainly of a basic intellection of the socalled Newtonian philosophy which explains very magnificently not only all Motos Coelestes being known, but it also gives a possibility to the astronomy to make new discoveries and to get more exactly acquainted with the real movements of all celestial bodies. This science will enable the Astronomus to direct all his observations toward a final aim, but also to extract all possible benefit from them. As in

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464 W SCHRODER and H-J TREDER

contrast a single Observator spends the time with such observations which may be either superfluous or of a form that they do not give any results."

Euler considered as the main aim of astronomy to exhaust the consequences of the Newtonian mechanics and gravity theory for astronomy, but he had a second idea, too. Euler rejected both the Cartesian philosophy of nature and LeibnizWolff's monadology. But he was according to his concept of the world no devoted Newtonian; Euler did not doubt in the axioms of Newtonian mechanics, but he had doubts about the principles of the Newtonian gravity theory.

Euler had similarly doubts about the physical principles of the Newtonian optics. As a matter of fact, Euler was wrong with his objections against Newton, never

theless his criticism led beyond Newton's position. That is why Euler accepted the Newtonian gravity law in a pseudo-Cartesian interpretation following his teacher Johann I Bernoulli (1667-1748), and Euler disputed the truth content of Newton's physical optics referring to the wave theory of light by R Descartes (1596-1650) and C Huygens (1627-1697) (see also Walther and Walther 1999).

All the arguments against Newton's optics in Euler's famous popular scientific book, "Letters to a German Princess" are de facto unwarrantable. Euler's thesis that the Sun would very quickly loss its mass M according to Newton's emanation theory of the light, and therefore this loss would be detectable by celestial mechanical methods, is incorrect already due to the fact that Euler forgot that at velocities of the particles v = c on the basis of classical mechanics a connection exists between energy E and mass M:

(1)

The mass loss b.M of the Sun is according to the Newtonian emanation theory given by the momentary radiation -itE:

d 2 dE -M=-·-dt v2 dt

(2a)

i.e. the mass M of the Sun would disappear according to the Newtonian theory in a time interval b.t:

(2b)

This value, b.t ~ 1012 years is the same - with the exception of the factor 2 -as given by modem physics on the basis of Einstein's special relativity theory as extreme limit of the lifetime of the Sun. The actual lifetime of the Sun is much shorter, ~101O years, as only one percent of the solar mass can be converted into radiation energy.)

Nevertheless, Euler's remarks concerning the problems of a purely particle theory of the light against the undulation theory by Huygens were signals for a further development. Euler showed that - on the basis of Huygens' principle - the wave theory of the light can explain the existence of "light beams", moreover it describes deflection and interference phenomena of the light, experimentally known since P Grimaldi and Newton more correctly than the Newtonian emission theory.

Acta Geod. Geoph. Hung. 36, 2001

EULER, THE THEORY OF GRAVITATION 465

When Euler rejected Newton's emanation theory of the light due to neglecting its wave characteristics, he had in mind the conception of a mechanical theory of light which represented the light as wave-like oscillations of a cosmic medium, he doubted in the absolute validity of the Newtonian laws of gravity, too, just from the point of view of particle physics, and he referred to Huygens's ideas suggested in the supplement of his treatise on the wave theory of light.

II.

In a struggle with himself through several decades Newton reached the conclusion that gravity is an inherent property of matter. From Newton's correspondence, but also from the "Questiones" in the supplement of his Optics Newton developed the idea of a dynamic atomism according to which particles are singular points in the three-dimensional space. The interaction of these particles is not defined by their "space-filling", but through their "space-taking", i.e. by their different "spheres of influence". These spheres of influence are defined according to the force law in an idealistic case of a purely gravitational particle of the mass m and of the radius ro as:

-fm -3-T + amr5(T - TO)

r (3)

where the first term is the Newtonian attraction and the second term corresponds to the hardness of the corpuscles. Newton reached finally the conclusion that his gravity law

K _ -fm 1 m II 1,11- 3 .

r1 ,11 (4)

is necessarily joined to the structure of space and to the dynamic constitution of matter. The force flux ,....., -!2m of the Newtonian gravity is divergence-free in empty space,

div f~T = div (grad f~ ) (5a)

and Laplace deduced from this in the language of the analytic theory of potential the equation

~ (~) = O. (5b)

Newton also remarked that the divergence-free character of the "force flux" expressed the "integral constant nature of the effect" of the particles (content of the Gaussian integral law!) and thus it enabled simultaneously to substitute all a spherically symmetric mass distribution (be it the Sun, be it a planet) by the mass concentrated in its mass centre,

l2=m·r5(T)

and using this, celestial mechanics gets point mechanics. Moreover Newton has shown that his gravity force ,....., /3 T results in closed paths of motion, namely in Keplerian ellipses, only in cases of one- and two-body-problems.



(N ewton was the first to determine the motion of the perihelia of the planets in the one-body-problem due to deviations from the Newtonian laws of motion or from the Newtonian law of gravity.)

Together with Cartesians and also with atomists like P Gassendi (1592-1655) Euler supposed that - neglecting the "ghosts" discussed by Euler, but irrelevant in physics - changes in the motion of masses in direction and in magnitude, i.e. the acceleration of the mass motion according to G Galilei and Newton are possible only by the "immediate contact" of two (and of several) bodies. Euler identified accordingly against Newton the ("space-filling" with "space-taking" and supposed together with Descartes, Gassendi and Huygens that only spatial contact can result in accelerations of inert masses. The postulate is that according to .the "theorem of the impossible third" two bodies must not occupy simultaneously the same place and accordingly they have to displace each other. That is why Euler supposed as only physical forces those due to pressure and impact. Euler has, however found out of David Bernoulli's (1700-1782) work that "pressure" is to be considered as the surface sum of many atomic impacts. Therefore Euler considered impact as the only force effect being no "scholastic qualitas occulta". Impact was explained by Euler as by Descartes and Huygens as consequence of the logical "theorem of the impossible third". RJ Boscovitsch (1711-1787) and I Kant (1724-1804) were the first ones to remark as followers of the Newtonian dynamism (as later PS Laplace) that a "logical axiom" alone cannot accelerate "even the lightest feather" (Treder 1997).

Euler developed at the same time and in continuous exchange of ideas with the experienced scientist MV Lomonossov (1711-1765) in Petersburg his supposition that the Newtonian gravity force is based on the impact of "intermundary" particles (see Lomonossov 1961). Euler and Lomonossov supposed that the "cosmic vacuum", Newton's absolute space is filled by a homogeneous and isotropic gas of such intermundary particles. According to D Bernoulli's deduction of the pressure forces an estimation results for the impacts of "ponderable masses" so that when two (or more) masses are opposed in the cosmos, then an anisotropy of the distribution of impacts results and the pressure between the two masses gets less than the outer pressure.

It follows then from elementary geometric laws of the three-dimensional Euclidean space that the two bodies move with acceleration toward each other, and just with an acceleration which corresponds to the Newtonian T2-law. It is, however, important here that the impacts of the intermundary particles with the particles of ponderable masses must not be elastic. This point was discussed by an indirect pupil of Euler, the Geneva mathematician GL Le Sage in his book "Lucretee Newtonien" (1782). In the case of purely elastic impacts exactly the same amount of intermundary particles will be reflected into the space between the bodies, as are shielded by these bodies.

There are, however, deviations from the Newtonian gravity theory which were searched in vain experimentally by Lomonossov and in celestial mechanics by Euler: According to the conception that gravity is based on (inelastic) impacts, the gravity effect of the bodies is proportional not with the inert masses of the bodies and



with their volume in the case of homogeneous mass distributions, respectively, but with their "effective" cross-section. Therefore neither the Newtonian law of gravity, nor the Galilean law of fall which would testify the proportionality of inertia and gravity, can be exactly valid. Lomonossov accordingly proposed with Euler's consent to postulate a far-reaching "hole-content" of matter against intermundary particles so that up to the then accuracy limit of measurements the proportionality of gravity with effective cross-section of matter cannot be distinguished from the proportionality with the inertial masses.

If, however, more accurate measurements would be possible - then deviations would be present from the Galilean law of fall and from the Newtonian law of gravity would be experimentally - or in celestial mechanics in the three- or several-bodyproblem. As a matter of fact, the Newtonian rv ~-law is substituted based on

T

the principles by Euler and Lomonossov by another law deduced by Laplace in his "Mecanique celeste" in the sense of the Eulerian principle:

m

- jmlmI II (J) 3 ' TI,II exp -). gdr TI,II

41T gT3

3 g = mass density.

(6)

The Newtonian gravity constant j is substituted by the effective cross-sections -). of the atomic particles, by the average density JL of the intermundary gas and by the average velocity V of the intermundary particles: j = ).2::2. This corresponds to the idea found in Euler's, Lomonossov's and D Bernoulli's papers. The basic crux of the atomistic gravity theories as presented by Euler is naturally the energy law. Already Euler surely knew what was later explicitly shown by Le Sage. An acceleration of ponderable masses appears only in the case of inelastic impacts. Laplace's approximate formula for the Euler-Le Sage atomistic gravity theory is only valid if the impacts are completely inelastic. Partially inelastic impacts would change both the gravity constant j and the absorption constant )..

In spite of the fact that Euler as L Bernoulli's and D Bernoulli's pupil was a follower of Leibniz in the sense that he substituted as "measure of the living force" the kinetic energy ~mv2 and accordingly supposed all "perpetua mobilia" as academically indisputable, nevertheless Euler was not aware of a basic problem of the energy law. He did not consider what happens with the energy spent for inelastic impacts. Based on the law of impacts of the classical mechanics and the law of the conservation of the living force after Leibniz, a gravity theory in Euler's and Le Sage's sense is possible only if living force gets lost during impacts.

Independently of the physical problems of the intermundary reduction of the Newtonian gravity to the cosmic anisotropy of inelastic impacts of intermundary particles on macroscopic masses, the questions clearly seen by Euler emerge concerning corrections ofthe Newtonian celestial mechanics on the one hand for quickly moving particles, on the other hand mainly for three- and several-body- problems.



III.

Euler supposed and hoped that in the framework of the increasing accuracy of the measurements in celestial mechanics of the solar system, i.e. in the framework of the validity of a Newtonian disturbance computation, deviations will emerge from the T-2 -gravity law involved in the Newtonian principles in the sense of an approximate validity of the Laplacian force law (Eq. 6). In the forties of the 18th century Euler believed to be able to show on the basis of his analytic representation of the Newtonian celestial mechanics that in the satellite system of Jupiter the Newtonian law of gravity is only approximately valid (i.e. by neglecting the absorption factor ~ exp( -.>..Qr).

A C Clairaut was the first together with Euler who analytically dealt with the Newtonian synthetic representation of the system Sun-Earth-Moon as with a rest ringed body problem and Clairaut - following Huygens and Newton - had also seen the problems and the deviations of the figures of Earth and Moon from a Newtonian rotation body (see Ertel 1955).

Clairaut generalized the Huygens-Newtonian theory of rotating fluids so that he could discuss all such existing arbitrary mass distributions. Nevertheless, when Clairaut deduced the theory of the figure of the Earth applying Newtonian mechanics and gravity theory on the problem Sun-Earth-Moon, he forgot his own corrections to the Newtonian computations concerning the symmetry properties, of the Earth (see Ertel 1953).

The theory of the Moon is both the most difficult and empirically most easily controllable consequence of the motion equation of a body in a general gravity field. Clairaut stated at first that the Newtonian gravity law is not able to explain completely correctly the theory of the Moon in the framework of the results presented in Newton's "Principia", it seemed to be necessary to use small, but basically important corrections to it. Euler supposed that these corrections by Clairault hint at "non-Newtonian" gravity effects. Clairaut, however, substituted Newton's approximation of the Earth figure by his own theory of the terrestrial figure based on Newtonian principles and found that in the framework of this more exact representation the theory of the lunar motion fully supports the Newtonian gravity law within the accuracy of the then measurements. At first Euler refused to accept this; later, however he convinced himself about the correctness of Clairaut's Moon theory.

Nevertheless, Euler's corpuscular gravity theory had remained till the 19th century as a possible disputable alternative to Newton's dynamism. But the increasing accuracy in the proportionality between inertia and heavy masses, and with it, the proportionality of the gravity force with volume led to the conclusion - as shown already by Laplace - that the absorption coefficient.>.. has to be very small:

(7)

and as

(8)



the average energy density ~ p, V 2 of the "intermundary particles" must be very high.

The theorem of the conservation of energy was before all the crux of this hypothesis. Namely the impact of the intermundary particles on ponderable bodies must be, as already mentioned, inelastic, thus kinetic energy continuously gets lost by these impacts. Euler himself referred to the validity of the theorem of the mechanical energy when he rejected a further treatment of "perpetua mobilia". The problem of the energy theorem did not emerge at that time in the case of not purely mechanical processes, as of inelastic impacts: Later physicists (von Helmholtz and Mayer) demanded, however, the validity of the energy theorem in the case of inelastic impacts, too, when kinetic energy is transformed into heat That is how it is easily shown that the mechanical gravity theory according to Euler is absolutely impossible.

Today (naturally very small) effects are looked for in celestial mechanics and gravimetry in the framework of the further development of Einstein's General Theory of Relativity which are connected with the absorption or suppression of the Newtonian gravity flux as supposed by L Euler and M Lomonossov. These yield "post-Newtonian" and "post-Einsteinian" corrections to Einstein's theory of gravity which suppose a non-linear (and non-minimum) coupling between matter and gravity. Already Euler's idea about the essence of gravity led to a non-linear dependence of the gravity force from the gravitating masses m corresponding to Laplace' force theorem (see Treder 1983).

Euler was the first who discovered the problem of the genuine non-linear character of physical laws. His criticism of Newton's corpuscular theory of light contained a hint that according to Newton's theory a non-linear scattering of light would exist, namely due to elastic impacts between light corpuscles (This is described today by quantum electrodynamics). For Euler, his correction of the Newtonian acoustics was important. Euler remarked that the linear wave equation as given by Newton:

82

u-2 8t2 ¢ = 6.¢

can be valid only for small amplitudes: in the case of greater amplitudes, nonlinear equations result. They follow from the first non-linear system of equations in physics obtained by Euler by transferring the Newtonian point mechanics to continua. In the Eulerian equations of motion for ideal fluids, the inertial term Q' ~~ is substituted according to the difference found by Euler between partial and total time dependence (i.e. generally between total and partial derivates):

dv (8V ) Q dt = Q 8t + v grad v . (9)

Therefore if one proceeds from points to continua in the framework of Newtonian principles, then a non-linear equation of motion results. Therefore the theory of light as desired by Euler and later developed by A J Fresnel as a wave motion in an elastic medium is non-linear in contrast to the linear electromagnetic light theory by J C L Maxwell (Treder 1983).



References

Ertel H 1953: Entwicklungsphasen der Geophysik, Akademie-Verlag Berlin Ertel H 1955: Hydrostatische Homotropie und Legendres Dichtegesetz, Akademie-Verlag,

Berlin Euler L 1961: Gesammelte Werke, Birkhauser, Basel Lomonossov M V 1961: Schriften, Vol. 1/1 I, Akademie-Verlag, Berlin Treder H-J 1961: Grosse Physiker und ihre Probleme, Akademie-Verlag, Berlin Treder H-J, Schroder W 1997: Physics and geophysics with special historical case studies,

Science Edition, Bremen Walther Th, Walther H 1999: Was ist Licht? Beck, Miinchen


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Euler, the theory of gravitation and some geophysical aspects