Curvature in orbital dynamics

J. B. Brackenridge and M. Nauenberg, “Curvature in Newton’s dynamics,” in The Cambridge Companion to Newton, edited by I. B. Cohen and G. E. Smith (Cambridge U.P., Cambridge, 2003), pp. 85–137.

M.

Nauenberg

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J. B. Brackenridge, “The critical role of curvature in Newton’s developing dynamics,” in The Investigation of Difficult Things: Essays on Newton and the History of the Exact Sciences in Honour of D. T. Whiteside, edited by P. M. Harman and Alan E. Shapiro (Cambridge U.P., Cambridge, 1992), pp. 231–261.

M.

Nauenberg

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An English translation from the Latin by Mary Ann Rossi of Secs. 1, 2, and 3 of Book I from the first (1687) edition of Newton’s Principia is given in J. B. Brackenridge, The Key to Newton’s Dynamics: The Kepler Problem and the Principia (University of California Press, Berkeley, 1995), pp. 229–267.

M. Nauenberg, “Newton’s perturbation methods for the three-body problem and their application to lunar motion,” in Isaac Newton’s Natural Philosophy, edited by J. Z. Buchwald and I. B. Cohen (MIT, Cambridge, MA, 2001), pp. 189–224.

M. Nauenberg, “Newton’s Portsmouth perturbation method and its application to lunar motion,” in The Foundations of Newtonian Scholarship, edited by R. H. Dalitz and M. Nauenberg (World Scientific, Singapore, 2000), pp. 167–194.

N. Guicciardini, Reading the Principia: The Debate on Newton’s Mathematical Methods for Natural Philosophy from 1687 to 1736 (Cambridge U.P., Cambridge, 1999), pp. 205–216.

P. Frost, Newton’s Principia, First Book, Sections I, II, III, with Notes and Illustrations and a Collection of Problems Principally Intended as Examples of Newton’s Method (Macmillan, Cambridge, 1854).

E. T. Whittaker, A Treatise on the Analytic Dynamics of Particles and Rigid Bodies (Dover, New York, 1937), 4th ed., p. 75. Here the radial acceleration $a=f/m$ for a central force $f$ is given in the form $a=h2r/p3ρ,$ where ρ is the radius of curvature, $p=r sin α,$ and $h=l/m.$ This expression, which corresponds to Eq. (13), is attributed by Whittaker to Siacci, Atti della R. Acc di Torino, XIV, p. 715.

E. J. Routh, A Treatise on the Dynamics of a Particle (Cambridge U.P., Cambridge, 1898), p. 199. Reprinted by Dover Publications, 1960.

Isaac Newton, The Principia, Mathematical Principles of Natural Philosophy, translated by I. B. Cohen and Anne Whitman (University of California Press, Berkeley, 1999), Prop. 6, Cor. 3.

A derivation along these lines was first given by Alexis Clairaut in 1752 and two years later by Jean le Rond d’Alembert in connection with his first successful theory of lunar motion. See F. Tisserand, Traité de Méchanique Céleste (Gauthier-Villars et fils, Paris, 1894), Vol. 3, pp. 40 and 60. See also C. Wilson, “The problem of perturbation analytically treated: Euler, Clairaut, d’Alembert,” in R. Taton and C. Wilson, The General History of Astronomy: Planetary Astronomy From the Renaissance to the Rise of Astrophysics (Cambridge U.P., Cambridge, 1995), pp. 89–107.

B.

Pourciau

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M.

Nauenberg

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In a cryptic remark in his 1664 notebook on circular motion, “Waste Book,” Newton remarked that “If the body $b$ moves in an Ellipsis, then its force in each point (if its motion [velocity] in that point be given) may be found by a tangent circle of equal crookedness with that point of the Ellipsis.” The word “crookedness” was Newton’s early term for “curvature,” and by “tangent circle” he meant the “osculating circle” first introduced by Huygens, and re-discovered and named by Leibniz some 30 years later (Ref. 2).

This relation for the acceleration was first considered by Newton as a generalization of the case of circular motion and had been obtained somewhat earlier by Christian Huygens; see Ref. 2.

This expression for the radius of curvature ρ in polar coordinates was first obtained by Newton in 1671. See The Mathematical Papers of Isaac Newton, edited by D. T. Whiteside (Cambridge U.P., Cambridge, 1969), Vol. III, pp. 169–173, 1670–1673.

The general lack of understanding of the geometrical significance of the differential form, Eq. (14), is revealed by the reference in a recent textbook on classical mechanics to the relation $u=1/r$ as a “magic transformation.” See L. Hand and J. D. Finch, Analytic Mechanics (Cambridge U.P., Cambridge, 1998), p. 142.

In Newton’s own words, Corollary 3 of Prop. 6, “If the orbit $APQ$ either is a circle or touches a circle concentrically or cuts it concentrically–that is, if it makes with the circle an angle of contact or of section which is the least possible–and has the same curvature and the same radius of curvature at point $P,$ and if the circle has a chord drawn from the body through the center of forces, then the centripetal force will be inversely as the solid $SY2×PV.$ For $PV$ is equal to $QP2/QR.$” See Ref. 12, p. 455.

M.

Nauenberg

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M. Nauenberg, “Robert Hooke’s seminal contribution to orbital dynamics,” Physics in Perspective (to be published).

This general form for an orbit was first given in a geometrical form by Newton in his Principia, Book 1, Proposition 43. In this proposition, Newton proved that for this orbit the central force had the form $f(r)∝ν2/r2+(1−ν2)r0/r3.$ See Ref. 12, p. 538.

Some of the difficulties that Kepler experienced in establishing from Tycho Brahe’s observations of the orbit of Mars that this orbit deviated from a circle and instead fitted an ellipse was due to the fact that these deviations were second-order effects in the eccentricity of Mars.

The Correspondence of Isaac Newton, edited by H. W. Turnbull (Cambridge U.P., Cambridge, 1960), Vol. 2, pp. 307–308.

Averaging over the period of the moon, this perturbation gives an additional repulsive force $fp=cr$ between the earth and the moon that depends linearly on the earth–moon distance $r,$ where $c=GM0ML/2R3,$ $G$ is Newton’s constant, $M0$ is the mass of the sun, $ML$ is the mass of the moon, and $R$ is the earth–sun distance (Refs. 6 and 7). If we substitute for $f$ in Eq. (21), the net attractive force on the moon $f=GM/r2−fp,$ we obtain $ν≈1−(3/4)m2,$ where $m=M0r3/MR3$ is the ratio of the period of the moon around the earth divided by the period of the earth around the sun. “Thus,” Newton found, “… in each revolution the upper apsis will move forward through $1°31′45″$” and he admitted reluctantly that the advance of “the apsis of the moon is twice as swift.” This discrepancy remained as one of the major unsolved problems in Newton’s theory of gravitation, until it was finally resolved in 1752 by a more detailed calculation of Alexis Clairaut (see Appendix B)—Ref. 13. Early on, Newton realized that averaging the solar perturbation was inadequate, and he made considerable progress in a more detailed calculation along the same direction taken later by Clairaut. But his work remained unpublished until it was found among his papers in the Portsmouth collection (Ref. 4).

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1980), p. 92, Figs. 3–13. On the cover of this edition there also is a “schematic illustration of the nature of the orbits for bounded motion,” which appears in the text as Figs. 3–7, in which there is a change of sign in the curvature of the orbit. But the text associated with this figure implies that the central force is purely attractive, which is wrong. Although it is not mentioned in this textbook, this figure corresponds to the perturbative solution given by Eq. (19) for $ε≈0.49$ and $ν≈3.1.$ Clearly this solution is not a perturbation solution, but it is a correct solution for the central force given by Eq. (24), which changes sign at $r=r0(ν2−1)/ν2≈0.896r0.$ These errors are also repeated in the third edition, H. Goldstein, J. L. Safko, and C. P. Poole (Addison-Wesley, Reading, MA, 2002), and in the first edition (Addison-Wesley, Reading, MA, 1950), which also shows the sketch of an untenable orbit for attractive central forces in Figs. 3–37.

M.

Tiersen

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At the time, Newton had not yet discovered the conservation law of angular momentum. To obtain the radius of curvature ρ, he had to apply Eq. (10). Then he could have calculated the change of velocity at each step only approximately from the tangential component of the force $f cos α,$ which would have led to an error in his graphical calculation. Newton completed his diagram by drawing secondary braches from the mirror reflection of the initial branch, but he made a further error by not properly aligning the center (Refs. 4 and 22).

According to R. S. Westfall and H. E. Erlichson, Newton’s diagram in his Dec. 13, 1679 letter to Hooke was obtained by an application of the geometrical construction in Newton’s proof of Kepler’s area law, as shown in Prop. 1, Book 1 of the Principia.

See

R. S.

Westfall

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H. E.

Erlichson

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H. W. Turnbull, The Correspondence of Isaac Newton (Cambridge U.P., Cambridge, 1960), Vol. II, p. 69.

The negative sign appears below because the angle θ is assumed to increase in the counterclockwise direction, and therefore $dr/dθ$ is negative in Fig. 1.