Celestial Mechanics, Volume 1: Dynamical Principles and Transformation Theory

The launching of space vehicles has given rise to a broadened interest in the problems of celestial mechanics, and the availability of computers has made practical the solution of some of the more numerically unwieldy of the problems. These circumstances only further enhance the importance of the appearance of "Celestial Mechanics, " to be published in five volumes. This treatise is by far the most extensive of its kind and it rigorously develops the full mathematical theory. The emphasis is on the results obtained during the past hundred years, although the classical mechanics from the time of Laplace is also reviewed.The author writes that "The trend of celestial mechanics at this stage of machine computers should be directed toward studying the general behavior of trajectories on rigorous mathematical foundations, so as to check the results of numerical computation, and at the same time toward estimating the errors involved in using nonconvergent series expansions within the accuracy to be expected from the limited observation time. Thus the science of celestial mechanics is now confronting a new revolutionary era by abruptly moving from reliance on the old weapon of series expansions toward rigorous mathematical discussion of the physical behavior of the trajectories for an unlimited time. The artificial satellites and planets are offering us the more difficult and more general problems of studying motions widely different from those of natural objects hitherto discussed in celestial mechanics. This new aspect completely surpasses the apparent elegance of the old-fashioned celestial mechanics, and its importance cannot be over-emphasized."This first volume is divided into six chapters. The first develops the principles of analytical dynamics. From the variational principle of Hamilton, and using the calculus of variations, Hamilton's canonical form of the differential equations is deduced. As an introduction to the topological study of dynamical trajectories, Riemannian geometry is described as it pertains to the quadratic form of kinetic energy. The second chapter deals with quasi-periodic motions. The third is devoted to particular solutions of the three-body and many-body problems. Euler's and Lagrange's types of particular solutions are obtained for the "n"-body problem in a general manner, and the nature of the motion is fully analyzed. The isosceles-triangular solution is discussed on the basis of the theory of analytic functions. The fourth chapter takes up Lie's theory of continuous groups of transformations, with application to the "n"-body problem. A summary of the modern theory of Lie groups is given with a view to suggesting a new trend of development. In Chapter 5, the differential equations of the "n"-body problem are reduced by using the known integrals. The last chapter concerns Burns's and Poincare's theorems.This work was prepared editorially at NASA's Goddard Space Flight Center.