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A treatise on the analytical dynamics of particles and rigid bodies. With an introduction to the problem of three bodies. With a foreword by Sir William McCrea. Reprint of the fourth edition. (English) Zbl 0665.70002
Cambridge Mathematical Library. Cambridge (UK) etc.: Cambridge University Press. xvii, 456 p. £15.00; $ 24.95 (1988).

The unmodified reprint of the more than fifty year old version of Whittaker’s magnum opus arises questions: why and for whom ? Answers get more urgent as the book is neither an original historic document nor a first course for a freshman in analytical dynamics. First published in 1904 it reflects the style of late 19th century and the state of the mathematical apparatus of those days. The excessive use of elliptic functions in the elaboration of explicite formulas, for example, is a fact maybe surprising to the modern reader. The bulk of calculations contrasting with the lack of figures (not so bad as with Lagrange 200 years ago: there none, here three!), the newcomer in the discipline will be bewildered, but soon will appreciate the clearness of physical thought and the development of formulas seldom found in other texts. Nonetheless, as a first reading one would better give him another text.

The publishers added to the reprint a lucid foreword by William McCrea. This gives some biographical remark to the author, historical comments on the development of the text and the main reasons why this book should not be missed by anybody working in the area of analytical dynamics or only wanting to understand the vivacity of the discipline, which has seen some breakthroughs even Whittaker couldn’t foresee.

The first eight chapters develop Lagrangian dynamics (I. Kinematical preliminaries, II. The equations of motion, III. Principles available for the integral, IV. The soluble problems of particle dynamics, V. The dynamical specification of bodies, VI. The soluble problems of rigid dynamics, VII. Theory of vibrations, VIII. Nonholonomic systems, dissipative systems). Chapter IX presents the variational principles, chapters X to XII are concerned with Hamiltonian dynamics (Hamiltonian systems and their integral invariants, the transformation theory of dynamics, properties of the integrals of dynamical systems). Chapter XIII and XIV introduce into the problem of three bodies (The reduction of the problem of three bodies, the theorems of Bruns and Poincaré). Chapter XV contains the general features of orbits mainly of a particle moving in a plane under a conservative force, Chapter XVI is devoted to the integration by series.

Besides the examples integrated and elaborated in the text, there are 250 “miscellaneous examples” not discussed in the book, in their own respect a wealthy area.

From McCrea’s foreword: “The book serves too as a summary of all the classical work in the field within its scope. It is written as a working monograph, not a history. But the critical footnotes and the references therein summarize the historical development of the subject... Specialists in the field seem in fact to be convinced that Analytical Dynamics does contain mention of all significant work in the field up to the time of the last edition... Because of its completeness the book must continue to serve as a work of reference in its field up to about 1936... It will continue to serve as an amplification for any other reading on the subject.”

Reviewer: K.Horneffer

70-01Textbooks (mechanics of particles and systems)
70-02Research monographs (mechanics of particles and systems)