Synopses & Reviews
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.
Synopsis
This work focuses on the two integrable systems of relevance to celestial mechanics, both of which date back to the 18th century. 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; in short, the Euler problem and the earth-satellite problem are in a dual complementary relation.
The present work shows that the solutions to all of these integrable problems can be put in a form that admits the general representation of the orbits and follows a definite shared pattern. A key feature of this treatment involves a full analysis of the planar Euler problem (yielding an exact solution form in terms of Jacobian elliptic functions) via a clear generalization of the form of the solution in the Kepler case (expressed in terms of trigonometric functions). In contrasting the issues involved in the Kepler-Euler problems, the author raises current issues in analysis and astronomy, placing these classical problems in a modern context, with original insights that have hithertofore not appeared in book form.
Synopsis
Shows that exact solutions to the Kepler (two-body), the Euler (two-fixed center), and the Vinti (earth-satellite) problems can all be put in a form that admits the general representation of the orbits and follows a definite shared pattern Includes a full analysis of the planar Euler problem via a clear generalization of the form of the solution in the Kepler case Original insights that have hithertofore not appeared in book form
Table of Contents
General Introduction.- The Kepler Problem (Two-Body Problem): the central Newtonian potential.- Bernoulli solution.- Features of the central Newtonian potential.- The Non-Central Newtonian Potential.- The Euler problem: two fixed centers of attraction.- The Vinti problem: earth-satellite theory.- Implications for perturbation theories.- Relativistic context.- Index.