The Euler Equations

Abstract

One of the most important applications of Newtonian mechanics occurs in the study of the motion of a rigid body. To apply Newtonian principles to a system of particles rigidly attached to each other is by no means trivial; and when the mathematization of the problem has been carried out, the differential equations which result are among the most elusive in classical mechanics, almost as interesting as those of the three-body problem. Because of its importance this problem attracted the attention of many great mathematicians. Indeed, anyone who has researched the history of the problem will probably agree that it is harder to find mathematicians who have not worked on the problem than to find mathematicians who have. This chapter examines the developments most closely related to the problem as studied by Kovalevskaya, then analyzes her work in detail. Finally, since her work represents the final chapter in the story of closed-form solutions, a comparison will be made with alternative methods, of which the work of Klein is taken as representative.