Dynamic Stability of The Time-averaged Orbiting Potential Trap: Exact Classical Analysis

We calculate exactly the modes of motion of the Time-averaged Orbiting Potential (TOP) trap with its four degrees of freedom, namely the three translations and the spin, taken into account. We find that, when gravity is neglected, there are two parameters in the problem namely, the angular velocity of the rotating field and its strength. We present the stability diagram in these parameters. We find the mode frequencies calculated from the time-averaged potential model used by the inventors of the TOP is an excellent approximation to our exact results. However, for other parameters, this may not be the case.