What is the connection between Newton's Shell Theorem and Bertrand's Theorem?

Question

The most general force that can fulfil the first part of Newton's Shell theorem (any spherically symmetric body affects external bodies as if its mass were concentrated at its centre) is an inverse square force $F(r) \sim r^{-2}$, an harmonic oscillator $F(r) \sim r$ or the sum of both types of forces $F(r) \sim A r + B r^{-2}$ .

Interestingly these two types of forces (inverse square and harmonic oscillator) are also the only two types of forces which fulfil Bertrand's theorem:

Among central force potentials with bound orbits, there are only two types of central force potentials with the property that all bound orbits are also closed orbits, the inverse-square force potential and the harmonic oscillator potential.

Both theorems seem to deal with completely different problems (closed bound orbits vs. the affect of spherical bodies on external bodies). But because the solution to both problems are the same type of forces I wonder if there might be a deeper connection between these two theorems. What could be the connection between Newtons Shell theorem and Bertrand's theorem?

Answer 1

TL;DR: It is a coincidence.

1. Firstly, the power laws don't match for $n\neq 3$ spatial dimensions:

   • On one hand, Bertrand's theorem is confined to a 2D orbit plane (due to angular momentum conservation), and doesn't depend on the ambient spatial dimension $n$.

   • On the other hand, Newton's shell theorem is tied to Gauss' law. Gauss surfaces are hypersurfaces of dimension $n-1$.

2. Secondly, even if we restrict to $n=3$ spatial dimensions, the solutions are different:

   • On one hand, Bertrand's theorem only works for a $1/r^2$ force law and Hooke's law separately but not for non-trivial linear combinations thereof.

   • On the other hand, the converse Newton's shell theorem also works for linear combinations thereof.

3. Thirdly, the known proofs of Bertrand's theorem are longer and the requirement of closed orbits leads to a rationality condition, which has no counterpart in the converse Newton's shell theorem.