Analytical Dynamics of Particles and Rigid Bodies

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies

Title page of the first edition of the book, published in 1910.
Author	E. T. Whittaker
Language	English
Subject	Mathematical physics Analytical dynamics
Genre	Non-fiction Academic history Scientific writing
Publisher	Cambridge University Press
Publication date	1904, 1917, 1927, 1937
Publication place	England

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies is a textbook on analytical dynamics originally published in 1904 by British mathematician Sir Edmund Taylor Whittaker FRS FRSE covering topics in mathematical physics and analytical dynamics, focusing on the three-body problem. The book quickly became a classic textbook in its subject and has remained in print for most of its lifetime, over more than a hundred years, where it has been said to have "remarkable longevity". Among other notabilities, Paul Dirac is said to be "indebted" to the book, as it contained the only material he could find on Poisson brackets, which he needed to finish his work on quantum mechanics in the 1920s. While it is a historic textbook on the subject, presenting what was the state of the art at the time of publication, it was noted in a 2014 biography of the book's development, published in the Archive for History of Exact Sciences, that it is not "used merely as a historical document", but that of the 114 books and papers that cited the book between 2000 and 2012, "only three are of a historical nature".

Background

The book was written in 1904, just two years after the publication of Whittaker's celebrated mathematical analysis textbook A Course of Modern Analysis. The second half of the textbook was based on a report Whittaker completed for the Council of the British Association.

Contents

Table of contents (3rd and 4th eds.)
Chapter	Title
1	Kinematical Preliminaries
2	The Equations of Motion
3	Principles Available for the Integration
4	The Soluble Problems of Analytical Dynamics
5	The Dynamical Specification of Bodies
6	The Soluble Problems of Rigid Dynamics
7	Theory of Vibrations
8	Non-Holonomic Systems, Dissipative Systems
9	The Principles of Least Action and Least Curvature
10	Hamiltonian Systems and their Integral-Invariants
11	The Transformation-Theory of Dynamics
12	The Properties of the Integrals of Dynamic Systems
13	The Reduction of the Problem of Three Bodies
14	The Theorems of Bruns and Poincaré
15	The General Theory of Orbits
16	Integration by Series

The first edition had sixteen chapters with 188 total total consecutively numbered sections. The chapter structure has remained constant throughout the life of the book, although chapter IX was renamed in the second edition and chapter XVI was renamed in the third. Chapter IX, The Principles of Least Action and Least Curvature, was originally titled The principles of Hamilton and Gauss before it was renamed in the second edition, while chapter XVI, Integration by series, was originally titled Integration by trigonometric series. In addition to altering chapter titles, many new sections were added to the second and third editions of the book. It has been noted that the book can be naturally divided into two parts: Part one, consisting of the twelve chapters, covers the basic principles of dynamics while part two, consisting of the final four chapters, the second part is based on Whittaker's report on the three-body problem. While the first part, the first twelve chapters, remained mostly constant throughout the book's multiple editions, the second part was expanded considerably in the second and third editions. The fourth, final edition published in 1937 represented the book in its final form and differed from the third edition only in correcting some errors and supplying additional references.

Part I of the book is said to give a "state-of-the-art introduction to the principles of dynamics as they were understood in the first years of the twentieth century". The first chapter, on kinematic preliminaries, discusses the mathematical formalism required for describing the motion of rigid bodies. The second chapter begins the advanced study of mechanics, with topics beginning with relatively simple concepts such as as motion and rest, frame of reference, mass, force, and work, but it soon goes on to discuss kinetic energy, introduce Lagrangian mechanics, and discuss impulsive motions. Chapter three discusses the integration of equations of motion at length, the conservation of energy and its role in reducing degrees of freedom, and separation of variables. Chapters one through three focus exclusively on systems of point masses. In chapter four the first concrete examples of dynamic systems are introduced and the methods of the previous chapters are employed in solving them, including the pendulum, central forces, and motion on a surface. Chapter five introduces the moment of inertia and angular momentum in preparation for the study of the dynamics of rigid bodies. Chapter six focuses on the solutions of problems in rigid body dynamics, with exercises including "motion of a rod on which an insect is crawling" and the motion of a spinning top. Chapter seven covers the theory of vibrations, a standard component of mechanics textbooks. Chapter eight introduces dissipative and [Nonholonomic systems]], up to which point all the systems discussed were holonomic and conservative. Chapter nine discusses action principles, such as the principle of least action and the principle of least curvature. Chapters ten through twelve, the final three chapters of part one, discuss Hamiltonian dynamics at length.

Chapter thirteen begins part two and focuses on the applications of the material in part one to the three body problem, where he introduces both the general problem and several restricted examples. Chapter fourteen includes a proof of Brun's theorem.

Long term impact

According to Victor Lenzen in 1952, the book is "still the best exposition of the subject on the highest possible level".

In a 1980 review of other works, Ian Sneddon states that the "theoretical work of the century and more after the death of Lagrange was crystallized by E. T. Whittaker in a treatise Whittaker ( 1904) which has not been superseded as the definitive account of classical mechanics".

In another 1980 review of other works, Shlomo Sternberg states that the books reviewed "should be on the shelf of every serious student of mechanics. One would like to be able to report that such a collection would be complete. Unfortunately, this is not so. There exist topics in the classical repertoire, such as Kowalewskaya's top which are not covered by any of these books. So hold on to your copy of Whittaker (1904)".

In the 2006 engineering textbook Principles of Engineering Mechanics, the book is "highly recommended to advanced readers" and is said to remain "one of the best mathematical treatments of analytical dynamics".

In a 2015 article on modern dynamics, Miguel Ángel Fernández Sanjuán noting the impact of the book says "Traditionally, the idea of dynamics in textbooks has evolved with time. When we think about textbooks used for the teaching of mechanics in the last century, we may think on the book A Treatise on the Analytical Dynamics of Particles and Rigid Bodies" as well as Pinciples of Mechanics by John L. Synge and Byron A. Griffith, and Classical Mechanics (book) by Herbert Goldstein

Publication history

The original four editions of book are listed below:

• Whittaker, E. T. (1904) A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. (1st ed.), Cambridge.
• Whittaker, E. T. (1917). A treatise on the analytical dynamics of particles and rigid bodies; with an introduction to the problem of three bodies (2nd ed.). Cambridge University Press. OCLC 352133.
• Whittaker, E. T (1927). A treatise on the analytical dynamics of particles and rigid bodies: with an introduction to the problem of three bodies (3rd ed.). Cambridge: The University Press. OCLC 1020880124.
• Whittaker, E. T (1937). A treatise on the analytical dynamics of particles and rigid bodies: with an introduction to the problem of three bodies (4th ed.). Cambridge University Press. OCLC 959757497.

Reprints and international editions

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In addition to four editions and to the reprints which have kept the book in circulation in the English language for the past hundred years, the book has a German edition that was printed in 1924 which was based on the book's second edition and a Russian edition that was printed in 1999.

• Whittaker, E. T.; Mittelsten, F.; Mittelsten, K. (1924). Analytische Dynamik der Punkte und Starren Körper: Mit Einer Einführung in das Dreikörperproblem und mit Zahlreichen Übungsaufgaben. Grundlehren der mathematischen Wissenschaften (in German). Berlin Heidelberg: Springer-Verlag. ISBN 978-3-662-24567-5.
• Whittaker, E. T (1937). A treatise on the analytical dynamics of particles and rigid bodies: with an introduction to the problem of three bodies (in Spanish) (4th ed.). Cambridge: Cambridge University Press. OCLC 1123785221.
• Whittaker, E. T. (1988). A treatise on the analytical dynamics of particles and rigid bodies : with an introduction to the problem of three bodies (4th ed.). Cambridge University Press. ISBN 0-521-35883-3. OCLC 264423700.
• Whittaker, E. T. (1988). A treatise on the analytical dynamics of particles and rigid bodies : with an introduction to the problem of three bodies (4th ed.). ISBN 978-0-511-60879-7. OCLC 967696618.
• Whittaker, E. T. (1999). A treatise on the analytical dynamics of particles and rigid bodies : with an introduction to the problem of three bodies (4th ed.). Cambridge. ISBN 978-1-316-04314-1. OCLC 1100677089.{{cite book}}: CS1 maint: location missing publisher (link)
• Whittaker, E. T. (1999). A treatise on the analytical dynamics of particles and rigid bodies : with an introduction to the problem of three bodies (4th ed.). Cambridge University Press. ISBN 0-521-35883-3. OCLC 629676472.

References

Further reading

• Coutinho, S. C. (1 May 2014). "Whittaker's analytical dynamics: a biography". Archive for History of Exact Sciences. 68 (3): 355–407. doi:10.1007/s00407-013-0133-1. ISSN 1432-0657. S2CID 122266762.

Notable reviews

• Wilson, E. B. (1906). "Book Review: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies; with an Introduction to the Problem of Three Bodies". Bulletin of the American Mathematical Society. 12 (9): 451–459. doi:10.1090/s0002-9904-1906-01372-6.
• Herglotz, G. (December 1906). "A treatise ou the analytical Dynamics of particles and rigid bodies; with au introduction to the problem of three bodies: Von E. T. Whittaker. 40, XIV+414 S. Cambridge. Univ. press., 1904". Monatshefte für Mathematik und Physik (in German). 17 (1): A23–A24. doi:10.1007/BF01697683. ISSN 0026-9255.
• Bryan, G. H. (April 1905). "The Algebra of Invariants The Dynamical Theory of Gases A Treatise on the Analytical Dynamics of Particles and Rigid Bodies". Nature. 71 (1852): 601–603. doi:10.1038/071601a0. ISSN 0028-0836.
• Birkhoff, G. D. (1920). "Book Review: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies; with an Introduction to the Problem of Three Bodies". Bulletin of the American Mathematical Society. 26 (4): 183–184. doi:10.1090/s0002-9904-1920-03290-8.
• Jourdain, Philip E. B. (October 1917). "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies; With an Introduction to the Problem of Three Bodies". The Mathematical Gazette. 9 (131): 145. doi:10.2307/3603175.
• Cherry, T. M. (1928). "Review of Dynamical Systems, ; A Treatise on the Analytical Dynamics of Particles and Rigid Bodies". The Mathematical Gazette. 14 (195): 198–199. doi:10.2307/3603797. ISSN 0025-5572.
• Wilson, E. B. (1906). "Treatise on the Analytical Dynamics of Particles and Rigid Bodies, by E. T. Whittaker". Bull. Amer. Math. Soc. 12 (9): 451–458. doi:10.1090/s0002-9904-1906-01372-6.
• Thiele, R. (1990). "Whittaker, E.T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. With an Introduction to the Problem of Three Bodies. Cambridge etc., Cambridge University Press 1988. XVII, 456 pp., £ 15.00 P/b. ISBN 0-521-35883-3 (Cambridge Mathematical Library)". ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik (in German). 70 (1): 78–78. doi:10.1002/zamm.19900700141.

Other reviews

• B., G. H. (January 1918). "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: with an introduction to the Problem of Three Bodies". Nature. 100 (2515): 363–364. doi:10.1038/100363a0. ISSN 0028-0836.
• Longley, W. R. (September 1928). "Review: E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies; with an Introduction to the Problem of Three Bodies". Bulletin of the American Mathematical Society. 34 (5): 671–671. ISSN 0002-9904.
• Marcolongo, R. (1930). "Whittaker, E. T. - A Treatise On The Analytical Dynamics Of Particles And Rigid Bodies With An Introduction To The Problem Of Three Bodies". Scientia, Rivista di Scienza. 24 (47): 273.
• A. H. W. (October 1938). "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies with an Introduction to the Problem of Three Bodies. By E. T. Whittaker. Pp. xiv, 456. 25s. 1937. (Cambridge)". The Mathematical Gazette. 22 (251): 415–415. doi:10.1017/S0025557200058587. ISSN 0025-5572.
• Buchanan, Herbert Earle (1938). "Review: Treatise on the Analytical Dynamics of Particles and Rigid Bodies, by E. T. Whittaker". Bull. Amer. Math. Soc. 44 (5): 316. doi:10.1090/s0002-9904-1938-06728-6.