A. P. Ivanov, “Proof of stability in the Brower–Paul problem”, Dokl. RAN. Math. Inf. Proc. Upr., 2021, Volume 501,Pages <nobr>42

MATHEMATICS

Proof of stability in the Brower–Paul problem

A. P. Ivanov^ab

^a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
^b National Engineering Physics Institute "MEPhI", Moscow, Russia

Abstract: 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.

Keywords: ball on a rotating saddle, stability, KAM theory.

UDC: 517.93

Presented: V. V. Kozlov
Received: 16.09.2021
Revised: 16.11.2021
Accepted: 17.11.2021

DOI: 10.31857/S2686954321060084