The rotating saddle not only is an interesting system that is able to trap a ball near its saddle point, but can also intuitively illustrate the operating principles of quadrupole ion traps in modern physics. Unlike the conventional models based on the mass-point approximation, we study the stability of a ball in a rotating-saddle trap using rigid-body dynamics. The stabilization condition of the system is theoretically derived and subsequently verified by experiments. The results are compared with the previous mass-point model, giving large discrepancy as the curvature of the ball is comparable to that of the saddle. We also point out that the spin angular velocity of the ball is analogous to the cyclotron frequency of ions in an external magnetic field utilized in many prevailing ion-trapping schemes.
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The general saddle–shaped surface equation should be z = x2/a – y2/b where a and b are positive constants. In this article, however, for the sake of simplicity, we consider the special case a = b (symmetric saddle). [This is a loss of generality, so I have removed the statement that it is not.]
