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Über dieses Buch

The present book represents to a large extent the translation of the German "Vorlesungen über Himmelsmechanik" by C. L. Siegel. The demand for a new edition and for an English translation gave rise to the present volume which, however, goes beyond a mere translation. To take account of recent work in this field a number of sections have been added, especially in the third chapter which deals with the stability theory. Still, it has not been attempted to give a complete presentation of the subject, and the basic prganization of Siegel's original book has not been altered. The emphasis lies in the development of results and analytic methods which are based on the ideas of H. Poincare, G. D. Birkhoff, A. Liapunov and, as far as Chapter I is concerned, on the work of K. F. Sundman and C. L. Siegel. In recent years the measure-theoretical aspects of mechanics have been revitalized and have led to new results which will not be discussed here. In this connection we refer, in particular, to the interesting book by V. I. Arnold and A. Avez on "Problemes Ergodiques de la Mecanique Classique", which stresses the interaction of ergodic theory and mechanics. We list the points in which the present book differs from the German text. In the first chapter two sections on the tri pie collision in the three­ body problem have been added by C. L. Siegel.



Chapter One. The Three-Body Problem

Ours, according to Leibniz, is the best of all possible worlds, and the laws of nature can therefore be described in terms of extremal principles. Thus, arising from corresponding variational problems, the differential equations of mechanics have invariance properties relative to certain groups of coordinate transformations. Because this is particularly important for celestial mechanics, in the preliminary sections we will develop as much of the transformation theory for the Euler-Lagrange and the Hamiltonian equations as is desirable for our purposes.
Carl Ludwig Siegel, Jürgen K. Moser

Chapter Two. Periodic Solutions

The theorem of Sundman discussed in chapter one represents the most far reaching result yet known about general solutions to the three- body problem. Unfortunately his ingenious method cannot be extended to the case n > 3. Although the results in § 6 can be used to exclude at the outset the simultaneous collision of all n bodies, the investigations of § 12, § 13 show that already a triple collision may give rise to essential singularities.
Carl Ludwig Siegel, Jürgen K. Moser

Chapter Three. Stability

We begin with the definition of stability and instability. Let ℜ be a topological space whose points we denote by þ, and let a be a certain point in ℜ. By a neighborhood here we will always mean a neighborhood of a in ℜ. Let þ1 = Sþ be a topological mapping of a neighborhood U1 onto a neighborhood B1 whereby a = Sa is mapped onto itself. The inverse mapping p-1 = S-1þ then carries B1 onto U1, and in general þn = Snþ (n = 0, ± 1, ± 2,…) is a topological mapping of a neighborhood Un onto a neighborhood Bn, having a as a fixed-point. For each point þ = þ0 in the intersection U1∩B1=M we construct the successive images þk+1 =Sþk (k = 0,1,…), as long as þk lies in U1 and similarly þ-k-1 =S-1þ-k as long as þ-k lies in B1. If the process terminates with a largest k+ 1 = n, then þ0,…, þn-1 all still lie in U1, but þn no longer does; similarly for the negative indices. In this way, to each þ in M there is associated a sequence of image points þk (k =…, — 1,0,1,…), which is finite, infinite on one side, or infinite on both sides.
Carl Ludwig Siegel, Jürgen K. Moser


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