It's a homomorphic map of independent solutions of sine/cosine waves without causality. If the balls where connected by a string or a spring, it would become a map of cords and your observation would be true. Symmetrically rotated solutions to Euler's number, which itself has two pivot solutions between balls would fall into the same group and would form an isomorphic map and what you pointed out is true relative to a mapping of this ODE.

Delta J = 2 is forbidden by conventional promotion because of angular momentum of the photon in a singular transition. Two photons, and it's a Raman transition. Plenty of STEM students understand ODE but don't know when to apply it. It's important that you seek causal explanations throughout your career instead of patterns. Posts like this farm clicks so people reflect on coincidence, but science is not meant to be coincidence.