Proof of stability in the Brower–Paul problem

We study the stability of equilibrium in the problem known as “a ball on a rotating saddle”, which was first considered by the famous Dutch mathematician Brauer in 1918. He showed that, in the case of a smooth surface, the saddle point, unstable in the absence of rotation, can be stabilized in a certain range of angular velocities. Later, this system was considered by Bottema from a standpoint of bifurcation theory. The physical analogue of this problem is the Nobel Laureate Paul’s ion trap: here, the rotating solid support is replaced by a quadrupole with a periodically changing voltage and gravity is replaced by an electrostatic field. The stability conditions were obtained in a linear approximation, and their sufficiency has not yet been proven. In this paper, such a proof is carried out by methods of Hamiltonian mechanics.