Topological Analysis of Integrable Problems of Celestial Mechanics on a Sphere and Pseudosphere

Abstract

In this paper, we consider the problem of motion of a point on a two-dimensional sphere and pseudosphere (with metric of constant curvature) in the field generated by two attracting centers. This problem is a natural analog of the classical plane two-center problem integrated even by Euler. However, the integrability of similar problems on the sphere and the Lobachevskii plane was established only recently. Therefore, the topological properties of these problems are poorly studied as yet. The present paper fills this gap. We also study the Lagrange problem on the pseudosphere under the action of the Newtonian attraction of the stationary center and one more force constant in magnitude and direction. This problem is a limit case of the two-center problem. In the process of passing to the limit, the second center moves at infinity in the direction of the propulsion force (moreover, its mass grows so that the constancy of the propulsion is ensured, i.e., proportionally to the squared distance). This problem was considered by Lagrange for the first time and reduces to quadratures. The question on gravitational interactions in spaces of constant curvature was posed for the first time by N. N. Lobachevskii, who studied generalizations of the attraction law to the space of constant negative curvature. This theme was further developed in the work of N. E. Zhukovskii [18] on the motion of a pseudospheric (“plane”) plate on the Lobachevskii plane. On a three-dimensional sphere (of constant curvature 1), one also considers an analog of the Newtonian potential generated by a gravitational center, which in this case is cotan θ, where the angle θ is equal to the distance to the center. The integrability of the Kepler problem on the three-dimensional sphere S3 was proved by E. Schrodinger [12]. Schrodinger considered the motion of an electron in the potential field of the nucleus of a hydrogen-like atom from the standpoint of quantum mechanics. Three classical Kepler laws have natural generalizations for spaces of constant curvature. For the Lobachevskii space and for the sphere, these generalizations were formulated by A. N. Chernikov [5] and V. V. Kozlov [8]. In particular, any finite orbit here is an ellipse in one of whose foci the gravitational center (the Sun) is located. Along with this, V. V. Kozlov and O. A. Kharin proved the integrability of the problem on the motion of a material particle on the sphere S2 in the field of two stationary Newtonian centers [9]. In [14–17], the problem of two centers on the three-dimensional sphere S3 and in the three-dimensional Lobachevskii space H3 and also the Lagrange problem (the limit case of the two-center problem when one of the centers is moved off at infinity) in the Lobachevskii space H3 were studied. In these works, the integrabiity of the problems is demonstrated, bifurcation diagrams are constructed, and the classification of regions of feasible motion is carried out. The first step in studying these problems consists of reducing the problem from the three-dimensional to the two-dimensional case, i.e., reducing the number of degrees of freedom. The next step consists of passing to convenient coordinates (in the problems under consideration, for instance, to spherical-conical and pseudo-spherical-conical coordinates which are defined as roots of certain rational functions) in order to reduce the system to the Liouville form.