Progress in Mathematical Physics: Integrable Systems in Celestial Mechanics (Volume 51)

Direct involvement with the subject area of the present work dates from my years with NASA at its Electronics Research Center (ERC) in Cambridge, M- sachusetts, in the 1960s. However, my approach to the problems of mathem- ical physics had been shaped earlier in my time as a graduate student in the Mathematics Department at MIT. The passage of time tends merely to further enhance my appreciation of that graduate study program, where I had the b- e?t of the intensive courses from Norman Levinson, C.-C. Lin, Jurgen ¨ Moser, and Eric Reissner. In the case of Reissner, my years as research assistant were a formative apprenticeship ― one could say on the “shop-?oor”. The stimulus to organize my convictions in book form came from my friends at Birkh¨ auser Boston, and I wish to thank Ann Kostant for providing me with the opportunity and support in producing it; a special thanks goes to Edwin Beschler, formerly of Birkh¨ auser, for his consistent encouragement over the years. In the course of writing, I had the good-humored support and invaluable help from my one-time teacher and long-time friend, Vincent Hart. He read each chapter as it was written and his sharp eye picked up my many slips and errors. More signi?cantly, his persistent questioning on the original form of Chapter 3 forced me to address all parameter ranges and provide solution forms to cover all possibilities for the case of negative energy.

This work presents a unified treatment of three important integrable problems relevant to both Celestial and Quantum Mechanics. Under discussion are the Kepler (two-body) problem and the Euler (two-fixed center) problem, the latter being the more complex and more instructive, as it exhibits a richer and more varied solution structure. Further, because of the interesting investigations by the 20th century mathematical physicist J.P. Vinti, the Euler problem is now recognized as being intimately linked to the Vinti (Earth-satellite) problem.

Here the analysis of these problems is shown to follow a definite shared pattern yielding exact forms for the solutions. A central feature is the detailed treatment of the planar Euler problem where the solutions are expressed in terms of Jacobian elliptic functions, yielding analytic representations for the orbits over the entire parameter range. This exhibits the rich and varied solution patterns that emerge in the Euler problem, which are illustrated in the appendix. A comparably detailed analysis is performed for the Earth-satellite (Vinti) problem.

Key features:

* Highlights shared features in the unified treatment of the Kepler, Euler, and Vinti problems

* Raises challenges in analysis and astronomy, placing this trio of problems in the modern context

* Includes a full analysis of the planar Euler problem

* Highlights the complex and surprising orbit patterns that arise from the Euler problem

* Provides a detailed analysis and solution for the Earth-satellite problem

The analysis and results in this work will be of interest to graduate students in mathematics and physics (including physical chemistry) and researchers concerned with the general areas of dynamical systems, statistical mechanics, and mathematical physics and has direct application to celestial mechanics, astronomy, orbital mechanics, and aerospace engineering.