Concern with the class of problems investigated in this monograph began for me as a graduate student at MIT (1958-62) when serving as research assistant to Professor Eric Reissner who initiated me into the subject and whose influence - whether directly or dialectically - is probably discernable in the contours of the work. My fIrst attempt at a systematic derivation of the equations of shell theory was made while on a summer assistantship with Professor Norman Levinson in 1960. Beyond gaining a sobering reali- zation of the complexities involved, I made little progress at that time. In 1962-64 while a Temporary Member at the Courant Institute of Mathematical Sciences (NYU) I made a fresh start, while benefIting from my association and discus- sions with Professor Fritz John. With the conviction that the full integration of the equations with respect to the thickness coordinate, by means of the Legendre repre- sentations, must lead to a clarifIcation of the position of the two-dimensional theory in its three-dimensional context, the necessary computations were completed during that period. Several years passed while I became reconciled with the thought that the material needed to be organized as a monograph. This was done during 1969-70 while at the NASA Electronics Research Center in Cambridge, MA.

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 complexa "and surprising orbita "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) andresearchers 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.

Concern with the class of problems investigated in this monograph began for me as a graduate student at MIT (1958-62) when serving as research assistant to Professor Eric Reissner who initiated me into the subject and whose influence - whether directly or dialectically - is probably discernable in the contours of the work. My fIrst attempt at a systematic derivation of the equations of shell theory was made while on a summer assistantship with Professor Norman Levinson in 1960. Beyond gaining a sobering reali- zation of the complexities involved, I made little progress at that time. In 1962-64 while a Temporary Member at the Courant Institute of Mathematical Sciences (NYU) I made a fresh start, while benefIting from my association and discus- sions with Professor Fritz John. With the conviction that the full integration of the equations with respect to the thickness coordinate, by means of the Legendre repre- sentations, must lead to a clarifIcation of the position of the two-dimensional theory in its three-dimensional context, the necessary computations were completed during that period. Several years passed while I became reconciled with the thought that the material needed to be organized as a monograph. This was done during 1969-70 while at the NASA Electronics Research Center in Cambridge, MA.