Two Integrable Systems on a Two-Dimensional Sphere

摘要

As was shown in [1], an integrable analogue of the plane Euler problem of particle motion in the field of two fixed Newtonian centers can be formulated on a two-dimensional sphere S~2. Integrability was proved by the method of separation of variables. The integrability of this problem in the three-dimensional case, i.e., for a particle moving on a three-dimensional sphere S~3, was proven in [2]. The proof consists of the elimination of a cyclic variable so that the three-dimensional problem on S~3 reduces to a two-dimensional problem on S~2. But in this case an additional Hookean center originates at the pole on the perpendicular to the equatorial plane of the two centers. In this paper, we explicitly write out algebraic integrals in the more general case of a material point moving in the field of two Newtonian centers and three Hookean centers placed on mutually orthogonal axes so that two of the Hookean centers lie on the plane of the Newtonian centers and the third one on the perpendicular to this plane (see Fig. 1).