A Treatise on the Analytical Dynamics of Particles and Rigid Bodies is a treatise and textbook on analytical dynamics by British mathematician Sir Edmund Taylor Whittaker. Initially published in 1904 by the Cambridge University Press, the book focuses heavily on the three-body problem and has since gone through four editions and has been translated to German and Russian. Considered a landmark book in English mathematics and physics, the treatise presented what was the state-of-the-art at the time of publication and, remaining in print for more than a hundred years, it is considered a classic textbook in the subject. Section 1 Introduction In addition to the original editions published in 1904, 1917, 1927, and 1937, a reprint of the fourth edition was released in 1989 with a new foreword by William Hunter McCrea.
The book was very successful and received many positive reviews. A 2014 "biography" of the book's development wrote that it had "remarkable longevity" and noted that the book remains more than historically influential. Among many others, G. H. Bryan, E. B. Wilson, P. Jourdain, G. D. Birkhoff, T. M. Cherry, and R. Thiele have reviewed the book. The 1904 review of the first edition by G. H. Bryan, who wrote reviews for the first two editions, sparked controversy among Cambridge University professors related to the use of Cambridge Tripos problems in textbooks. The book is mentioned in other textbooks as well, including Classical Mechanics, where Herbert Goldstein argued in 1980 that, although the book is outdated, it remains "a practically unique source for the discussion of many specialized topics."
Background
Whittaker was 31 years old and working as a lecturer at Trinity College, Cambridge when the book was first published, less than ten years after he graduated from Cambridge University in 1895. Section 2.1 The author Whittaker was branded Second Wrangler in his Cambridge Tripos examination upon graduation in 1895 and elected as a Fellow of Trinity College, Cambridge the next year, where he remained as a lecturer until 1906. Whittaker published his first major work, the celebrated mathematics textbook A Course of Modern Analysis, in 1902, just two years before Analytical Dynamics. Following the success of these works, Whittaker was appointed Royal Astronomer of Ireland in 1906, which came with the role of Andrews Professor of Astronomy at Trinity College, Dublin.
The second half of the treatise is an expanded version of a report Whittaker completed on the three-body problem at the turn of the century at the request of the British Science Association (then called the British Association for the Advancement of Science). In 1898, the council of the British Association passed a resolution that "Mr E. T. Whittaker be requested to draw up a report on the planetary theory". Section 2.2 The report A year later, Whittaker delivered his report, titled “Report on the progress of the solution of the problem of three bodies”, in a lecture to the Association, who published it in 1900. He changed the name from the original "report on the planetary theory" to, in his own words, show "more definitely the aim of the Report", which covered the advances in theoretical astronomy that occurred between 1868 and 1898.
Content
Table of contents (3rd and 4th eds.) Chapter Title1Kinematical Preliminaries2The Equations of Motion3Principles Available for the Integration4The Soluble Problems of Analytical Dynamics5The Dynamical Specification of Bodies6The Soluble Problems of Rigid Dynamics7Theory of Vibrations8Non-Holonomic Systems, Dissipative Systems9The Principles of Least Action and Least Curvature10Hamiltonian Systems and their Integral-Invariants11The Transformation-Theory of Dynamics12The Properties of the Integrals of Dynamic Systems13The Reduction of the Problem of Three Bodies14The Theorems of Bruns and Poincaré15The General Theory of Orbits16Integration by Series
The book is a thorough treatment of analytical dynamics, covering topics in Hamiltonian mechanics and celestial mechanics and the three-body problem. It has been noted that the book can be divided naturally into two parts: Part one, consisting of the twelve chapters, covers the basic principles of dynamics, giving a "state-of-the-art introduction to the principles of dynamics as they stood in the first years of the twentieth century", while part two, consisting of the final four chapters, is based on Whittaker's report on the three-body problem. While the first part remained mostly constant throughout the book's multiple editions, the second part was expanded considerably in the second and third editions.
History
The book's structure remained constant throughout its development, with fifteen total chapters, though the second and third editions added new sections throughout. Among other changes to the book, Whittaker expanded chapters fifteen and sixteen considerably and renamed chapters nine and sixteen. The title of chapter nine, The Principles of Least Action and Least Curvature, was The principles of Hamilton and Gauss before being renamed in the second edition and the title of chapter sixteen, Integration by series, was Integration by trigonometric series before being renamed for the third edition. The first edition had 188 total consecutively numbered sections, which increased in the second and third editions of the book. Among the most heavily altered, chapter fifteen went from fourteen sections to twenty-two while chapter sixteen doubled its section count from nine to eighteen.
Most of the differences between the second and third editions were adding outlines of and references to works published after the book's second edition. The edition included a major rewrite of chapters fifteen and sixteen to update the book considering developments that had occurred in the eleven years since the publication of the second edition. The first fourteen chapters of the third edition were photolithographically reproduced from the second edition, with some corrections and added references. The new material contained a section on Synge’s geometry of dynamics and tensor analysis. The fourth edition, published in 1937, differed from the third edition only in correcting some errors and supplying references to works published after the previous edition; aside from a new foreword by William Hunter McCrea in a 1989 reprint, the volume represented the book in its ultimate form.
Synopsis
Part I of the book has been 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". Section 3.1 The principles of dynamics 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 motion and rest, frame of reference, mass, force, and work before discussing kinetic energy, introducing Lagrangian mechanics, and discussing 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 only on systems of point masses. The first concrete examples of dynamic systems, including the pendulum, central forces, and motion on a surface, are introduced in chapter four, where the methods of the previous chapters are employed in solving problems. Chapter five introduces the moment of inertia and angular momentum to prepare 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. Section 3.2 Hamiltonian systems and contact transformations
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. Section 3.3 Celestial mechanics Chapter fourteen includes a proof of Brun's theorem and a similar proof of a theorem by Henri Poincaré on "the non-existence of a certain type of integrals in the problem of three bodies". Chapter fifteen, The General Theory of Orbits, describes two-dimensional mechanics of a particle subject to conservative forces and discusses special-case solutions of the Three-body problem. The last chapter includes discussions of solutions of the problems of previous chapters by integration of series, particularly trigonometric series.
Reception
Receiving generally positive reviews throughout, the book has gone through four editions, each with multiple reviews. A reviewer of the first edition noted that the book contains "the outlines of a long series of researches for which hitherto it has been necessary to consult English, French, German, and Italian transactions". Section 5.1 Style One of those first edition reviews, by George H. Bryan in 1905, began a controversy among Cambridge University professors related to the use of Cambridge Tripos problems in textbooks. In 1980, Herbert Goldstein mentioned the book in his famous textbook Classical Mechanics where he noted that it was outdated, but remained a useful reference for some specialised topics. While it is a historic textbook on the subject, presenting what was the state-of-the-art at the time of publication, a 2014 "biography" of the book's development pointed out that the book remains influential for more than historical purposes.
First edition
The first edition of the book received several reviews, including George H. Bryan in 1905 and Edwin Bidwell Wilson in 1906, Section 4.1 An American point of view: E. B. Wilson as well as German reviews by Gustav Herglotz, also in 1906 and Emil Lampe in 1918. Section 4.4 Other Reviews Lampe called the treatise an "excellent work" and states that Cambridge's treatment of analytical dynamics "has had, as a consequence, that the English student is directed with great energy towards the study of mechanics in which he displays excellent performance, as can be gauged from the many, and not at all easy, problems appended at the end of each chapter of this book."