Newton's Theorem of Revolving Orbits - Mathematical Statement

Mathematical Statement

Consider a particle moving under an arbitrary central force F₁(r) whose magnitude depends only on the distance r between the particle and a fixed center. Since the motion of a particle under a central force always lies in a plane, the position of the particle can be described by polar coordinates (r, θ₁), the radius and angle of the particle relative to the center of force (Figure 1). Both of these coordinates, r(t) and θ₁(t), change with time t as the particle moves.

Imagine a second particle with the same mass m and with the same radial motion r(t), but one whose angular speed is k times faster than that of the first particle, where k is any constant. In other words, the azimuthal angles of the two particles are related by the equation θ₂(t) = k θ₁(t). Newton showed that the motion of the second particle can be produced by adding an inverse-cube central force to whatever force F₁(r) acts on the first particle

where L₁ is the magnitude of the first particle's angular momentum, which is a constant of motion (conserved) for central forces.

If k2 is greater than one, F₂ − F₁ is a negative number; thus, the added inverse-cube force is attractive, as observed in the green planet of Figures 1–4 and 9. By contrast, if k2 is less than one, F₂−F₁ is a positive number; the added inverse-cube force is repulsive, as observed in the green planet of Figures 5 and 10, and in the red planet of Figures 4 and 5.