Ebook: Integrable Problems of Celestial Mechanics in Spaces of Constant Curvature

Introd uction The problem of integrability or nonintegrability of dynamical systems is one of the central problems of mathematics and mechanics. Integrable cases are of considerable interest, since, by examining them, one can study general laws of behavior for the solutions of these systems. The classical approach to studying dynamical systems assumes a search for explicit formulas for the solutions of motion equations and then their analysis. This approach stimulated the development of new areas in mathematics, such as the al­ gebraic integration and the theory of elliptic and theta functions. In spite of this, the qualitative methods of studying dynamical systems are much actual. It was Poincare who founded the qualitative theory of differential equa­ tions. Poincare, working out qualitative methods, studied the problems of celestial mechanics and cosmology in which it is especially important to understand the behavior of trajectories of motion, i.e., the solutions of differential equations at infinite time. Namely, beginning from Poincare systems of equations (in connection with the study of the problems of ce­ lestial mechanics), the right-hand parts of which don't depend explicitly on the independent variable of time, i.e., dynamical systems, are studied.

---

This book combines a most interesting area of study, celestial mechanics, with modern geometrical methods in physics. According to recently developed views and research, one of the basic qualitative characteristics of an integrable Hamiltonian system is a structure of the Liouville foliation. A number of interesting results have been obtained. In particular, some of the constructed topological invariants did not appear in integrable cases investigated by many researchers earlier on. The topology of the isoenergy surfaces is also strongly different from what authors presented before. Some new topological effects in the problems of dynamics on spaces of constant curvature have been discovered. At present there are no other books published in this particular area.
This book is intended for specialists and post-graduate students in celestial mechanics, differential geometry and applications, and Hamiltonian mechanics.

---

This book combines a most interesting area of study, celestial mechanics, with modern geometrical methods in physics. According to recently developed views and research, one of the basic qualitative characteristics of an integrable Hamiltonian system is a structure of the Liouville foliation. A number of interesting results have been obtained. In particular, some of the constructed topological invariants did not appear in integrable cases investigated by many researchers earlier on. The topology of the isoenergy surfaces is also strongly different from what authors presented before. Some new topological effects in the problems of dynamics on spaces of constant curvature have been discovered. At present there are no other books published in this particular area.
This book is intended for specialists and post-graduate students in celestial mechanics, differential geometry and applications, and Hamiltonian mechanics.
Content:
Front Matter....Pages i-xi
Basic Concepts and Theorems....Pages 1-36
Generalization of the Kepler Problem to Spaces of Constant Curvature....Pages 37-80
The Two-Center Problem on a Sphere....Pages 81-132
The Two-Center Problem in the Lobachevsky Space....Pages 133-157
Motion in Newtonian and Homogeneous Field in the Lobachevsky Space....Pages 159-172
Back Matter....Pages 173-184

---

This book combines a most interesting area of study, celestial mechanics, with modern geometrical methods in physics. According to recently developed views and research, one of the basic qualitative characteristics of an integrable Hamiltonian system is a structure of the Liouville foliation. A number of interesting results have been obtained. In particular, some of the constructed topological invariants did not appear in integrable cases investigated by many researchers earlier on. The topology of the isoenergy surfaces is also strongly different from what authors presented before. Some new topological effects in the problems of dynamics on spaces of constant curvature have been discovered. At present there are no other books published in this particular area.
This book is intended for specialists and post-graduate students in celestial mechanics, differential geometry and applications, and Hamiltonian mechanics.
Content:
Front Matter....Pages i-xi
Basic Concepts and Theorems....Pages 1-36
Generalization of the Kepler Problem to Spaces of Constant Curvature....Pages 37-80
The Two-Center Problem on a Sphere....Pages 81-132
The Two-Center Problem in the Lobachevsky Space....Pages 133-157
Motion in Newtonian and Homogeneous Field in the Lobachevsky Space....Pages 159-172
Back Matter....Pages 173-184
....