Distinct Feature of Inverse Square Force and Harmonic Force: A Simplistic Proof of Bertrand’s Theorem

Abstract

Bertrand’s theorem is essential to the discussion of the motion of particles under central force. The theorem states that among all central forces, the only two in which bound orbits are also closed orbits are the harmonic force and inverse-square force. As existing proofs of the theorem require advanced mathematics, this paper presents a much more simplistic proof of Bertrand’s theorem without having to solve any complicated integrals or using mathematical tools beyond the scope of high school curriculum. The first part of the proof shows that the radial force must be a single power law, and the second part of the proof — which is the main crux of this paper — uses the fact that the radial distance of the motion must be a power of a sinusoidal function of the polar angle in nature. With a simple mathematical manipulation, the proof is then completed. This derivation is expected to be of great interest to high school teachers and students as it enhances the understanding of this distinct feature of inverse square force and harmonic force.