Some Elementary Exercises in Celestial Mechanics

Abstract

For centuries celestial mechanics has been an exceptional rich source of problems and results in mathematics. To some extent this is still the case. Today one can discern, rather artificially, three problem fields. The first one is the study of classical problems like perturbed Kepler motion, orbits in the threebody problem, the theory of asteroids and comets, etc. The second one is a small but relatively important field in which the astrophysicists are interested; we are referring to systems with evolution like for instance changes caused by tidal effects or by exchange of mass. The third field is what one could call ‘mathematical celestial mechanics’, a subject which is part of the theory of dynamical systems. The distinction between the fields is artificial. There is some interplay between the fields and hopefully, this will increase in the future. An interesting example of a study combining the first and the third field is the paper by Brjuno [41]. A typical example of an important mathematical paper which has found already some use in classical celestial mechanics is Moser’s study on the geometrical interpretation of the Kepler problem [193]. Surveys of mathematical aspects of celestial mechanics have been given in [194] and [3].