Newton's Theorem of Revolving Orbits

Generalization

Isaac Newton first published his theorem in 1687, as Propositions 43–45 of Book I of his Philosophiæ Naturalis Principia Mathematica. However, as astrophysicist Subrahmanyan Chandrasekhar noted in his 1995 commentary on Newton's Principia, the theorem remained largely unknown and undeveloped for over three centuries.

The first generalization of Newton's theorem was discovered by Mahomed and Vawda in 2000. As Newton did, they assumed that the angular motion of the second particle was k times faster than that of the first particle, θ₂ = k θ₁. In contrast to Newton, however, Mahomed and Vawda did not require that the radial motion of the two particles be the same, r₁ = r₂. Rather, they required that the inverse radii be related by a linear equation

$\frac{1}{r_{2}(t)} = \frac{a}{r_{1}(t)} + b$

This transformation of the variables changes the path of the particle. If the path of the first particle is written r₁ = g(θ₁), the second particle's path can be written as

$\frac{a r_2}{1 - b r_2} = g\left( \frac{\theta_2}{k} \right)$

If the motion of the first particle is produced by a central force F₁(r), Mahomed and Vawda showed that the motion of the second particle can be produced by the following force

$F_2(r_2) = \frac{a^3}{\left( 1 - b r_2 \right)^2} F_{1}\left( \frac{a r_2}{1 - b r_2} \right) + \frac{L^2}{mr^3} \left( 1 - k^2 \right) - \frac{bL^2}{mr^2}$

According to this equation, the second force F₂(r) is obtained by scaling the first force and changing its argument, as well as by adding inverse-square and inverse-cube central forces.

For comparison, Newton's theorem of revolving orbits corresponds to the case a = 1 and b = 0, so that r₁ = r₂. In this case, the original force is not scaled, and its argument is unchanged; the inverse-cube force is added, but the inverse-square term is not. Also, the path of the second particle is r₂ = g(θ₂/k), consistent with the formula given above.

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Newton's Theorem of Revolving Orbits - Generalization