One of the most famous results in The Principia is the theorem that states that if a body is acted on solely by a force that attracts to a fixed point, and obeys the inverse square law, its orbit is a conic. And that, conversely, if a body that is acted on solely by a force that acts towards a fixed point, and moves in an orbit that is a conic with one focus at that point, then the force obeys the inverse square law.
Recall that Kepler’s first law asserts that the orbits of the planets about the sun are ellipses, with the sun at a focus.
The Principia contains three proofs of this theorem. Let me follow the usage of Newton’s time, and call the direct problem: Given the orbit, find the force, and the inverse problem: Given the force, find the orbit.
He solves the direct problem in three propositions, namely Propositions 11, 12, and 13 of Section 3, Book 1, which deal in turn with the case of an orbit that is an ellipse, a hyperbola, and a parabola. Between them, these propositions give a proof of the inverse problem. That is because these cases (together with the case of a circular orbit, centred on the centre of attraction, and a linear orbit going directly towards or away from the centre) provide an orbit to fit any initial condition. That is to say, given the position and velocity (speed together with direction of motion) of a planet at a given time, its orbit about the sun is then determined.
Technically, there are no singularities. It is abundantly clear that Newton understood this point, though he does not state this explicitly in The Principia. He gives two proofs of Propositions 11 and 12. He deals with the inverse problem in Proposition 41 of Section 8, Book 1. In this Proposition 41 he deals with the case of an arbitrary centripetal force, reducing the solution to the calculation of integrals, and hence, of necessity, not giving the orbit explicitly. But in the case of the inverse square law these integrals are easily calculated, as Newton demonstrates elsewhere.
In this blog I concentrate on the second proof (idem aliter) of Proposition 11. The proof consists of a single sentence. It is such a beautiful and simple proof that, if you do not have time to study any more of The Principia, I suggest that this is the proof you should study
To state that this proof consists of a single sentence is absolutely true, though the essentially equivalent proof of Proposition 12, in line with Newton’s elegant variation, consists of two sentences. It is also true that this proof depends on some simple results that he has proved in Section 2.
Let me sketch these preliminary results. Suppose first that the body moves in an ellipse whose centre is also the centre of attraction. In this case the force is directly proportional to the distance of the body from the centre. This is easy to prove.
As a distraction from the matter in hand, I set the following problems:
Problem 1: Why is it much easier to deal with the case of an attractive force that is directly proportional to the distance, rather than one that obeys the inverse square law; or with the motion of a body subject to a constant force moving through a resisting medium when the resistance is proportional to the speed, rather than when the resistance is proportional to the square of the speed?
Newton also uses elementary results that he has proved that relate the force applied at a given point of the orbit towards a given centre to the curvature of the orbit at that point.
Now he has the ingredients for his second proof. The awkward problem of the curvature of the ellipse is completely avoided by comparing the forces required to cause the body to move in the same ellipse, but with the force being directed in one case to the centre of the ellipse, and in the other case to a focus.
Problem 2: Supply the missing details for this second proof.
The slickest modern proof of Proposition 11 also starts with the case when the centre of attraction is the centre of the ellipse.
Take the plane in which the body moves to be the set of complex numbers, so the problem is now in one complex dimension, and use a conformal transformation to take the ellipse with centre of attraction at the centre of the ellipse to an ellipse where the centre of attraction is a focus.
It should be possible to deconstruct this argument, so that complex numbers are not mentioned, and to see if this gives rise to Newton’s proof. A more heroic attitude to these propositions was shown by the great physicist Richard Feynman, who, having difficulty in following Newton’s proof, constructed his own proof in Newton’s calculus-free style (it turned out that Feynman’s proof had already been found by Hamilton).
No doubt Feynman was trying to follow Newton’s first proof of Proposition 11. He should have looked at the second proof. I think that the basic problem for Feynman was that he was unaware that Newton was taking his notation from Proposition 10 of Section 2, which deals with the above mentioned proof when the force is proportional to the distance. He states that he had problems with Newton’s use of the geometry of conics, but I don’t believe that was the real issue.
Do not despair. I give all the geometry that Newton assumes in a brief appendix to my translation.
Problem 3: Newton’s alternative proofs of Propositions 11 and 12 have no analogue for Proposition 13, the case of a parabolic orbit, where no second proof appears. Why is this? Can you find a one-sentence proof of this case?
Problem 4: An elementary result states that the oscillations of a simple pendulum are approximately isochronous. That means, loosely speaking, that, provided that the oscillations are small, the period is independent of the magnitude of the oscillations.
This is the fundamental basis of the pendulum clock, as conceived by Galileo and constructed by Huygens. Now consider what would happen if the bob were to swing, with small oscillations, in an orbit that did not lie in a vertical plane, so the orbit approximates to an ellipse. Would the oscillations still be approximately isochronous?
Problem 5: If a string pendulum swings in a vertical plane the oscillations can be made perfectly isochronous by constraining the string by suspending it from the cusp of an inverted cycloid, as in the illustration. How would you make a string pendulum swing in a perfectly isochronous oscillation that was not in a vertical plane? Or how would you design a saucer in which a marble would roll in an isochronous orbit?
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