Perturbation approximation for orbits in axially symmetric funnels

The author would like to thank an anonymous referee for alerting me to the fact that this result is in Ref. 1.

The moment of inertia I of a ball rolling with frequency $ω$ adds a contribution $(1/2)Iω2$ to the energy. In Ref. 9, a no-slipping assumption is introduced by assuming the scalar relation $v=aω$⁠, where v is the velocity and a is the radius of the ball. Then the equations of motion for a point particle remain valid, with the gravitational constant g reduced by a factor $1/(1+I/ma2)=5/7$⁠, where m is the mass of the ball. Actually, the correct no-slipping assumption is a vector relation $v→=aω→×n→$⁠, where $n→$ is a unit vector normal to the surface of the funnel.¹ Then $v2=a2(ω2−ωn2)$⁠, where $ωn$ is the spin component normal to the surface, and the scalar condition is valid approximately when $ωn=0$⁠, for nearly circular orbits. On the other hand, when the spin in the transverse direction approximately vanishes for such orbits, the value of g is not reduced. I would like to thank G. White for a helpful discussion that clarified these points.

This result is in Newton's Principia, Prop. 41, Cor. 3.

In his correspondence between Hooke and Newton, the angle of aperture of Hooke's cone was not mentioned, but to account for his observation, Hooke's cone must have had an opening angle similar to the one shown in Fig. 3.

In accordance with Proposition 1 in the Principia, where he gave a proof of Kepler's area law for central forces, Newton set the time interval $dt∝r2 dϕ$⁠. The missing constant of proportionality is the conserved angular momentum l and correspondingly $F=l2u2(d2/dϕ2+1)u$⁠.

In Prop. 45, Newton applied his method to calculate the rate of precession of the elliptical orbit of the moon due to the perturbation of the gravitational force of the sun, but he failed to obtain agreement with the observed result, stating “The advance of the apsis of the moon is about twice as fast.”