Is Newton’s second law really Newton’s?

Isaac Newton, Philosophiae Naturalis Principia Mathematica or Mathematical Principles of Natural Philosophy [Natural Philosophy meaning Physics], first published in 1687, with a second edition in 1713, and a third, just a year before Newton died, in 1726. Devoted to “rational mechanics”—“the science,” as Newton puts it, “expressed in exact propositions and demonstrations, of the motions that result from any forces whatever and of the forces that are required for any motions whatever”—the Principia consists of two preliminary sections (Definitions and Axioms, or the Laws of Motion) followed by three books: Books I and II (both titled The Motion of Bodies, but with Book II concentrating on resisted motion) together present mathematical principles, while Book III (The System of the World) applies these principles to planets and moons of our solar system. All quotations from the Principia come from the recent English translation: I. Newton, The Principia, Mathematical Principles of Natural Philosophy, translated into English from the Latin of the third (1726) edition by I. Bernard Cohen and Anne Whitman, assisted by Julia Budenz, and preceded by “A guide to Newton’s Principia” by I. Bernard Cohen (University of California Press, Berkeley, ).

Reference 1, p. .

,” in , edited by

I. B.

Cohen

and

G. E.

Smith

(, , ), pp. –;

M.

Blay

, “,” in , edited by

J. Z.

Buchwald

and

I. B.

Cohen

(, , ), pp. –; and

H.

 

Erlichson

, “,”  , – (), p. 844.

See, for example, these excellent mechanics texts:

J. M.

Knudsen

and

P. G.

Hjorth

, , revised and enlarged 3rd ed. (, , ), p. ;

J. B.

Marion

and

S. T.

Thornton

, , 4th ed. (, , ), p. ;

K. R.

 

Symon

, , 3rd ed. (, , ), p. .

For more detail on the interpretation of the Principia’s second law presented here, see B. Pourciau, “Newton’s interpretation of Newton’s second law,”  60, 157–207 ().

“At rest” in Newton’s sense of being at rest relative to “that immovable space in which the bodies truly move.” Newton was well aware of the weaknesses in his own theory—that it depends on “immovable space … [which] makes no impression on the senses,” (Ref. 1, p. 414) for example, or that it provides no explanation for the proportionality of inertial and gravitational mass.

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Today we would detect any deviation from uniform straight line motion or rest with a nonzero (vector) acceleration. Working instead with the moving and resting deflections, Newton could accommodate deviations generated by instantaneous impulses, when the acceleration, being infinite, does not exist. (For an instantaneous impulse, the arc segment PQ would be a line segment.) Such impulses play a central role early in the Principia, where they are used in series to approximate the motion of a body under the influence of a (continuous) centripetal force in the argument for Proposition I.

To clarify a possible confusion: A charged particle at rest at a point P in a magnetic field will remain at rest, while the same particle projected through P with a nonzero velocity (not parallel to the magnetic field) will be deflected from its uniform straight line motion. This difference does not violate the principle that the resting and moving deflections must be the same (for a given force on a given body). Although the origin or mechanism (the magnetic field) remains the same for the particle in both cases, the force (or “thrust”) experienced by the particle is not the same. If we were to imagine applying the same “thrust” experienced by the moving particle to the resting particle, then the resting deflection would equal the moving deflection, according to this natural sequel to the first law. That this natural sequel is relativistically false does not worry us, because we are working within an earlier paradigm: the mechanics of the Principia.

, edited by

H.

 

Turnbull

(, , ), Vol. , pp. –.

Christiaan Huygens’ The Pendulum Clock or Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks, translated and with notes by R. Blackwell, based on the original 1673 edition (Iowa State U.P., Ames, ), p. 33.

Reference 1, p. .

, edited and with extensive commentaries by

D. T.

 

Whiteside

(, , ), Vol. , pp. –.

See Ref.13, pp. 539–543, or

I. B.

 

Cohen

, “,” in , edited by

R.

 

Palter

(, , ), pp. –.

Nowhere else in his work on mechanics, published or unpublished, does Newton use the phrase “in the meaning of this law” in a passage that describes the second law of motion. Indeed, nowhere else does he describe how he interprets the second law. The passage we have quoted is the only known record, in his own words, of Newton’s understanding of his second law.

Newton often writes that two line segments are “parallel and equal,” or that a quantity “takes place along” a specified line. Such directed line segments are all over the Principia. The anachronistic overhead arrow we have used captures this notion perfectly, with no distortion of Newton’s intended meaning.

In Newton’s mechanics, a given force and its observable sign are conceptually distinct, the observable sign being the deflection (or the acceleration): intuitively, a thrust is seen as distinct from the observed effect of that thrust. In any development of classical mechanics where this distinction disappears, the Compound Second Law becomes a vacuous tautology. Within such developments, a meaningful analogue may be phrased in terms of reference frames and Galilean relativity: Two observers, moving with constant relative (vector) velocity, record the same deflection….

Reference 1, p. .

Because the mathematics of the Principia, for the most part, is a geometrical version of limits and calculus, Newton preferred to work with proportions rather than equations. But we lose nothing and we gain a more modern presentation treating these proportions as equalities. Hence, we write the “motive force is…,” rather than the “motive force is proportional to….”

A.

Koyré

, “,”  , – (),

reprinted in

A.

Koyré

, (, , ), pp. –.

Also see

J. G.

 

Yoder

, (, , ), Chaps. II–IV.

For example, we can gauge how fast 1 – cos h approaches 0 as h → 0⁺ by comparing it to the quantities h, h², and h³, which approach 0 increasingly fast: (1 – cos h)/h → 0, $(1-cosh)/h2→12$⁠, and (1 – cos h)/h³ → +∞. These limits tell us that 1 – cos h approaches 0 faster than h, at the same rate as h², and slower than h³. Newton would write that 1 – cos h “will be as” h² (as h → 0). We have the good approximation $1-cosh≈12h2$ for h sufficiently small; so we can also think of the Newtonian code “will be as” as signaling an “ultimate proportion”: 1 – cos h is nearly proportional to h² for sufficiently small h. It does not matter that the limit of the ratio is 1/2, only that the limit is nonzero and finite.

Reference 1, p. .

Reference 1, pp. –.

A (finite) force, by definition, generates a finite acceleration: thus $2(LQ→/h2)=2(LQ→/h)/h$ approaches the finite limit a(t₀). When a ratio has a finite limit and its denominator [h] approaches zero, then the numerator [$LQ→/h$] must also approach zero. The same is true for $PG→/h$⁠. Thus, as we claimed earlier, the motive force $M(PG→/h)$ and the change in motion $M(LQ→/h)$ are relative, not absolute, measures of the force.

J.

 

Lagrange

, , translated and edited by and

Victor N.

 

Vagliente

(, , ), p. .

The major treatises on mechanics in the 18th century—by Varignon, Hermann, Euler, MacLaurin, d’Alembert, Euler (again), Lagrange, and Laplace—all acknowledge the enormous influence of Newton’s Principia, yet none of these works states or cites the second law as it appears in the Principia. Given that Newton evidently regarded the Principia’s second law as a fundamental axiom of mechanics, this omission is odd. But the explanation is easy. Confused by the ambiguous account of the second law in the Principia, these scientists misinterpreted the second law in ways that made it appear irrelevant to their developments of mechanics, as a law applying to impulses only, for example. Nevertheless, all of them did apply the property asserted by the Principia’s second law (as Newton understood it, in the form of the Compound Second Law), using it as a fundamental assumption of their own mechanics, without realizing that they should have cited the second law in the Principia. See, for example, J. Hermann, Phoronomia, sive de viribus et motibus corporum solidorum et fluidorum, libri duo (R. & G. Wetstenios, Amsterdam, ), pp. 68–69; C. MacLaurin, A Treatise of Fluxions in Two Books (T. W. and T. Ruddimans, Edinburgh, ), Vol. 1, p. 347; and L. Euler, Theoria motus corporum solidorum seu rigidorum: ex primus nostrae cognitionis principiis stabilita et ad omnes motus, qui in huis modi corpora cadere possunt, accommodata, Leonhardi Euleri Opera Omnia, Series II: Opera Mechanica et Astronomica Vol. 3, edited by C. Blanc, originally published by Orell Füssli in 1765 (Birkhäuser, Berlin, ), pp. 68–69. For more along these lines, read B. Pourciau, “Newton’s second law (as Newton understood it) from Galileo to Laplace,” unpublished.

Reference 1, p. .

For details on this and on Proposition VI more generally, read B. Pourciau, “,” Hist. Math. 34, 140–172 ().

See Ref. 28, p. 160. Our $PG→$ and $LQ→$ notation in Proposition VI hides a technical point: if these two deflections were generated by exactly the same force (in direction and magnitude) during the brief time h, then we would have $LQ→=PG→$ by the Compound Second Law and the ratio would be M even for h > 0 and not just in the limit. But in Proposition VI, the force which generates $LQ→$ is only approximately the same as the force that generates $PG→$⁠. Nevertheless, we still obtain M in the limit as h → 0.

Reference 1, p. .

We have concluded that Proposition VI is the Principia’s version of f = ma. Although alike underneath, Proposition VI and f = ma are not alike on the surface; hence the need for the modifier “version of.” A natural question arises: Who was the first to write the equation f = ma in something like its present form? The answer, like so many answers in the history of mathematics and physics, is Leonhard Euler. His “Discovery of a new principle in mechanics,” published in 1750, recorded the equation in Cartesian coordinates: 2Mddx = Pdt², 2Mddy = Qdt², 2Mddz = Rdt (L. Euler, “,” Mémoires de l’Académie des Sciences de Berlin 6, 185–217 (). Also see Leonhardi Euleri Opera Omnia, Serie II, Vol. 5, pp. –.) Euler could legitimately regard this principle as new, even though its ancestor had appeared 63 years earlier as Newton’s Proposition VI, because, for the first time, it was written in a fixed orthogonal coordinate system and applied, not just to point-masses, but to extended bodies. For further details, read G. Maltese, “The Ancients’ Inferno: The Slow and Tortuous Development of ‘Newtonian’ Principles of Motion in the Eighteenth Century,” in Essays on the History of Mechanics: In Memory of Clifford Ambrose Truesdell and Edoardo Benvenuto, edited by Antonio Becchi (Birkhäuser Verlag, Basel, ), pp. 199–221. Also see C. Truesdell, The Rational Mechanics of Flexible or Elastic Bodies 1638–1788, Introduction to Leonhardi Euleri Opera Omnia, Series II, Vol. 10 and 11 (Orell Füssli, Zürich, ), p. 251.