Comment on "A Nonholonomic Model of the Paul Trap"

TL;DR

This paper clarifies the equivalence of two equations of motion for a rigid ball in a rotating-saddle trap and corrects a stability condition error related to the ball's radius in a recent study.

Contribution

It demonstrates the equivalence of the Lagrangian and Newtonian derived equations and corrects a mistake in the stability analysis regarding the influence of ball radius.

Findings

The two equations of motion are identical.

The stability condition depends on the ball radius, contrary to previous claims.

A correction is provided for the stability analysis in the recent literature.

Abstract

A recent article by Borisov et al. [Regular and Chaotic Dynamics 23.3 (2018): 339-354.] studies the motion of a rigid ball in a rotating-saddle trap. The authors claim that they derive a new equation of motion from the Lagrangian formalism, which is different from the one we obtained from the Newtonian formalism in our recent work [American Journal of Physics 85.11 (2017): 821-829.]. We show here that these two equations of motion are the same. In addition, besides the reduced spin frequency and the moment of inertia coefficient, the stability condition given by the article is independent of the ball radius---this result is incorrect. The mistake is due to the fact that the center of mass and the contact point are not distinguished in the explicit expression of the local normal vector.