Geodesic
Proof of stability in the Brower--Paul problem
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 501 (2021), pp. 42-45

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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. 
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     author = {A. P. Ivanov},
     title = {Proof of stability in the {Brower--Paul} problem},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {42--45},
     publisher = {mathdoc},
     volume = {501},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2021_501_a7/}
}
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A. P. Ivanov. Proof of stability in the Brower--Paul problem. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 501 (2021), pp. 42-45. http://geodesic.mathdoc.fr/item/DANMA_2021_501_a7/