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The theory of the top. Volume I. Introduction to the kinematics and kinetics of the top. Preface by Michael Eckert. Translated from the German original by Raymond J. Nagem and Guido Sandri. (English) Zbl 1167.70001
Basel: Birkhäuser (ISBN 978-0-8176-4720-9/hbk). xviii, 279 p. (2008).
Without doubt, this book is a classic in physics (mechanics), written at the end of the nineteenth century by Felix Klein, a mathematician of world fame and by Arnold Sommerfeld, a rising star in physics at that time. Felix Klein was, by the way, also responsible for the ”golden age” of mathematics in Göttingen in the first third of the twentieth cenntury, because he convinced the Prussian minister of culture at that time to establish a center of excellence in mathematics in Göttingen, for which the next two new hired professors were D. Hilbert and H. Minkowski.
Klein taught besides the usual classes in pure mathematics also classes on very applied subjects, like “’elementary geometry”, “the theory of top” or “’technical mechanics” for students in mathematics who wished to become high school teachers. In these classes usually one of his assistents was charged in elaborating the manuscript into a booklet. In the case of the theory of top it was Sommerfeld’s duty, who at this time (1895) was assistent in mathematics. Indeed the outcome was not a booklet but a book of about thousand pages edited in four volumes, where the last volume appeared about 15 years after the first one, when Sommerfeld was already professor, now of physics, and not in Göttingen anymore.
It is interesting to note that Klein had the impression that the whole project derailed, because in the third and fourth volume, which dealt with applications, and in particular in the fourth volume, where technical applications were treated, almost no use was made of the theoretical framework developed in the first two volumes, that is, the advanced function-theoretic methods and the exact representation of the motion by elliptic functions. For the technical applications, the concepts of momentum and rigid body dynamics were sufficient. Also the representation of the singularity-free rotation, for which four parameters instead of three were necessary and their quaternion geometric meaning was elaborated in the first volume, was not relevant to the applications, even if it was a masterpiece of the presentation.
In the opinion of the reviewer, to give in the formulation of the kinetic equations preference to the method of impact forces over the now usual method of continuous forces is certainly a weak point in this book. Today we know that this formulation used by Klein in this book completely disappeared from the literature on classical mechanics. Although the difference between these two formulations – impact forces versus continuous forces – is not really substantial, it requires some effort to get acquainted with these equations.
The key equations in applications of the theory of top are Euler’s equations describing the rotations of a rigid body. Here V. I. Arnol’d in his book [Mathematical methods of classical mechanics: Textbook. 3rd ed., rev. and compl. (Russian). Moskva: Nauka (1989; Zbl 0692.70003)] gives an extremely elegant and easily comprehensible derivation. It is based on a clever notation giving the same letter in lower case or upper case to a quantity depending whether it is represented in the inertial or in the body fixed frame, respectively. It is interesting to note that Klein also uses this notation and may have been in some sense influencial for Arnol’d’s approach.
Another item should be mentioned here in admiration of Klein’s work, and this is his definition of the concept of stability, which also meets all requirements which nowadays are made for this important subject. Some examples, like the force-free motion and the stabilty of the rotation axis are worked out in detail.
It is very positive that, by producing the English edition of this book, a large group of contemporary scientists and especially mathematicians can admire and appreciate what a broad understanding mathematicians like Felix Klein not only of their field but also of physics had.
Very interesting ample notes of the translators (about 50 pages), where also various motions are worked out, physically interpreted and excellently illustrated by numerous figures, and a nice preface by Michael Eckert make this volume even more interesting to read than the German original.
Reviewer: Hans Troger (Wien)

70-02 Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems
70E05 Motion of the gyroscope
70E17 Motion of a rigid body with a fixed point
70E50 Stability problems in rigid body dynamics