Analytical Dynamics of Particles and Rigid Bodies

Add links
Source: Wikipedia, the free encyclopedia.

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies
Analytical dynamics
Genre
PublisherCambridge University Press
Publication date
  • 1904 (1st ed.)
  • 1917 (2nd ed.)
  • 1927 (3rd ed.)
  • 1937 (4th ed.)
Pages456
LC Class
QA845
Identifiers refer to the 1989 reprint of the fourth edition unless otherwise noted

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies is a

William Hunter McCrea
.

The book was very successful and received many positive reviews.

Classical Mechanics, where Herbert Goldstein argued in 1980 that, although the book is outdated, it remains "a practically unique source for the discussion of many specialized topics."[2]

Background

Whittaker was 31 years old and working as a lecturer at

Trinity College, Dublin.[3]

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).[4] 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".[4][5] 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.[6] 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.[4]

Content

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 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",[7] while part two, consisting of the final four chapters, is based on Whittaker's report on the three-body problem.[8]
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.[9] Among other changes to the book, Whittaker expanded chapters fifteen and sixteen considerably and renamed chapters nine and sixteen.[9] 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.[7] The first edition had 188 total consecutively numbered sections, which increased in the second and third editions of the book.[8] 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.[9]

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.

William Hunter McCrea in a 1989 reprint, the volume represented the book in its ultimate form.[12][13][8]

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".

principle of least action and the principle of least curvature.[7] Chapters ten through twelve, the final three chapters of part one, discuss Hamiltonian dynamics at length.[14]

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.[9] 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".[9] 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.[9] The last chapter includes discussions of solutions of the problems of previous chapters by integration of series, particularly trigonometric series.[9]

Reception

Portrait of Whittaker by Arthur Trevor Haddon.

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".

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.[1]

First edition

The first edition of the book received several reviews, including George H. Bryan in 1905[16][17] and Edwin Bidwell Wilson in 1906,[18][19] as well as German reviews by Gustav Herglotz, also in 1906[20] and Emil Lampe in 1918.[21][22] 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."[22][21]

Bryan's initial book review, published in 1905, was a review of three books published by the Cambridge University Press at around the same time.[16][17] Bryan opens the review by writing that, though he is not does not care for the "University Presses competing with private firms", he believes "there can only be one opinion as to the series of standard treatises on higher mathematics emanating at the present time from Cambridge".[16][17] He then noted that England's "lack of national interest in higher scientific research, particularly mathematical research, stands far behind most other important civilised countries"[16] and thus it was necessary for the "University Press to publish advanced mathematical works."[16][17] He went on to write: "We may take it as certain that the present volumes will be keenly read in Germany and America, and will be taken as proofs that England contains good mathematicians."[16][17] Bryan criticised the chapter four, The Soluble Problems of Analytical Dynamics, for "mostly [representing] things which have no existence".[16][17] Sparking a controversy published under the title "Fictitious Problems in Mathematics", Bryan goes on to write: "It is impossible for a particle to move on a smooth curve or surface because, in the first place, there is no such thing as a particle, and in the second place there is no such thing as a smooth curve or surface."[16][17] Bryan went on to write that the book is "essentially mathematical and advanced" and "written mainly for the advanced mathematician".[16][17]

Wilson's review was published in 1906 and began with an expression of distaste for the "imminent encroachment by pure mathematics of territory that traditionally belonged to applied mathematics", but then quickly states that at that time "there seems no immediate danger" as three recent books published by the Cambridge University Press were "highly important volumes" that "exhibit great mathematical power and attainments directed firmly and unerringly along the direction of physical research".[18][19] Noting the novelty of many of the sections in the book, Wilson wrote that the book "breaks the barricade and opens the way to fruitful advance".[18][19] He then noted that the book is advanced and, though it is self-contained, it is not for a beginning student. He elaborated by writing that "the book is mathematical in nature, written with a precision and developed with a logic sure to appeal to mathematicians"[18] and the "diversity of method taken with the compact style makes the book hard reading for any but the somewhat advanced student".[18][19] Wilson also expressed a desire to have topics such as statistical mechanics added to the textbook.[18][19]

Fictitious Problems in Mathematics

George H. Bryan in the 1900s. Bryan's April 1905 review of the book spurred a flurry of responses published in Nature under the title "Fictitious Problems in Mathematics".

The review

Cambridge Tripos examinations.[23] Of particular contention was Bryan's statement that a "perfectly rough body placed on a perfectly smooth surface forms as interesting a subject for speculation as the well-known irresistible body meeting the impenetrable obstacle"[16][17] and that "[w]hat the average college don forgets is that roughness or smoothness are matters which concern two surfaces, not one body".[16][17] The controversy stretched from 18 May to 22 June with letters on the dispute published in five issues of Nature. A reviewer later wrote that "100 years after they were written, it is difficult not to view the whole polemic as prompted by a bout of hair-splitting on the part of Bryan", though it was acknowledged that Bryan's original claim was "undoubtedly correct" and the "polemic" was likely a misunderstanding.[23]

The 18 May issue of Nature contained two letters starting the controversy, the first was an anonymous response under the title "Fictitious Problems in Mathematics" from an author referring to themself only as An Old Average College Don,[24] while the second was a response from Brayan under the same title.[25][23] The old college don charged Bryan to point to a page number where such problems are used, while Bryan responded by saying that the problems are ubiquitous and finding the places where the correct definition is used is easier than pointing out all the places where it is wrong.[23] In the 25 May issue of Nature, Alfred Barnard Basset[26] and Edward Routh[27] joined the debate. Routh explained that when "bodies are said to be perfectly rough, it is usually meant that they are so rough that the amount of friction necessary to prevent sliding in the given circumstances can certainly be called into play"[23] and states that the statements are abbreviations meant to "make the question concise".[23] In a similar tone, Basset wrote that the wording is used to designate "an ideal state of matter".[23] The 1 June issue of Nature contained a response from Charles Baron Clarke[28] and another rebuttal Bryan.[29] Charles Baron Clarke insinuates that he is the "Old Average College Don" that wrote the first anonymous letter, and again emphasises his original complaint.[23] The final two letters of the controversy were published by Routh[30] and Bryan[31] on the eighth and twenty-second of June, respectively.

Second and third editions

The second and third editions received several reviews, including another one from

principle of least action.[32][21] After listing several other problems, Jourdain ends the review by stating that "all these criticisms do not touch the very great value of the book which has been and will be the chief path by which students in English speaking countries have been and will be introduced to modern work on the general and special problems of dynamics."[32][21] Bryan also reviewed the second edition of the book in 1918 in which he criticises the book for not including the dynamics of aeroplanes, a lapse Bryan believes was acceptable for the first but not for the second edition of the book.[34][23] After discussing more about aeroplanes and the development of their dynamics, Bryan closes the review by stating that the book "will be found of much use by such students of a future generation as are able to find time to extend their study of particle and rigid dynamics outside the requirements of aerial navigation"[34][23] and that it would serve as "a valuable source of information for those who are in search of new material of a theoretical character which they can take over and apply to any particular class of investigation."[34][23] George David Birkhoff wrote a review in 1920 stating that the book is "invaluable as a condensed and suggestive presentation of the formal side of analytical dynamics".[35][21] Birkhoff also includes several criticisms of the book, including stating it was incomplete in some respects, pointing to the methods used in chapter sixteen on trigonometric series.[35][21]

The third edition, published in 1927, was reviewed by

Hamiltonian systems using trigonometric series, the third edition replaced the method used in previous editions with a new one published by Whittaker in 1916 which Cherry states "must be regarded as suggestive rather than conclusive", noting that not all applicable proofs are included.[10] He finishes by saying that the "optimistic view" the book takes toward the convergence of trigonometric series can be criticised, closing his review by saying "though the question is a difficult one, all the evidence suggests that the series are generally divergent and only exceptionally convergent."[10] Another reviewer expressed regret that the work of George David Birkhoff was not included in the third edition.[11]

Fourth edition

The final edition of the book, published in 1937, has received several reviews, including a 1990 review in German by Rüdiger Thiele.[37] Another reviewer of the final edition noted that the discussion of the three-body problem is brief and advanced such that it "will be difficult reading for one not already acquainted with the subject"[12] and that the references to then-recent American articles were incomplete, pointing to specific examples relating to the stability of the equilateral triangle positions for three finite masses.[12] The same reviewer then argued that "this does not detract from the merit of the text, which this reviewer regards as the best in its field in the English language."[12] Another reviewer in 1938 claims that the attainment of a fourth edition "shows that it has become the standard work on the topics with which it deals."[13] According to Victor Lenzen in 1952, the book was "still the best exposition of the subject on the highest possible level".[38]

In the second edition of his

Cambridge Tripos examination.[2][39] Despite the book's problems and its need to be updated, he went on to write: "It remains, however, a practically unique source for the discussion of many specialized topics."[2][39]

Influence

Paul Dirac in 1933. Dirac is said to be "indebted" to the book for its critical discussion of Poisson brackets, which were required for his work on quantum mechanics.

The book quickly became a classic textbook in its subject and is said to have "remarkable longevity", having remained in print almost continuously since its initial release over a hundred years ago.

John L. Synge and Byron A. Griffith, and Classical Mechanics by Herbert Goldstein.[41]

During the 1910s, Albert Einstein was working on his general theory of relativity when he contacted Constantin Carathéodory asking for clarifications on the Hamilton–Jacobi equation and canonical transformations. He wanted to see a satisfactory derivation of the former and the origins of the latter. Carathéodory explained some fundamental details of the canonical transformations and referred Einstein to E. T. Whittaker's Analytical Dynamics. Einstein was trying to solve the problem of "closed time-lines" or the geodesics corresponding to the closed trajectory of light and free particles in a static universe, which he introduced in 1917.[42]

Paul Dirac, a pioneer of quantum mechanics, 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.[1] In September 1925, Dirac received proofs of a seminal paper by Werner Heisenberg on the new physics. Soon he realised that the key idea in Heisenberg's paper was the anti-commutativity of dynamical variables and remembered that the analogous mathematical construction in classical mechanics was Poisson brackets.[43]

In a 1980 review of other works, Ian Sneddon stated 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".[44][39] 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)".[45][39]

Publication history

The treatise has remained in print for more than a hundred years, with four editions, a 1989 reprint with a new foreword by

William Hunter McCrea
, and translations in German and Russian.

Original editions

The original four editions of textbook were published in Great Britain by the Cambridge University Press in 1904, 1917, 1927, and 1937.[8]

Reprints and international editions

This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by adding missing items with reliable sources.

In addition to the four editions and 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 that was based on the book's second edition as well as a Russian edition that was printed in 1999.

William Hunter McCrea
was published in 1989.

See also

References

  1. ^ a b c d e f g Coutinho 2014, pp. 356–358 Section 1 Introduction
  2. ^
    ISBN 0-201-02918-9
    .
  3. ^ a b c Coutinho 2014, pp. 357–358 Section 2.1 The author
  4. ^ a b c Coutinho 2014, pp. 359–360 Section 2.2 The report
  5. ^ Report of the Sixty-Eighth Meeting of the British Association for the Advancement of Science Held at Bristol in September 1898. John Murray. 1899.
  6. ^ Whittaker, E. T. (1899). "Report on the Progress of the Solution of the Problem of Three Bodies". Report of the Sixty-Ninth Meeting of the British Association for the Advancement of Science Held at Dover in September 1899. London: John Murray. pp. 121–159.
  7. ^ a b c d e f Coutinho 2014, pp. 361–366 Section 3.1 The principles of dynamics
  8. ^ a b c d e Coutinho 2014, pp. 361–362 Section 2.3 The book
  9. ^ a b c d e f g Coutinho 2014, pp. 377–380 Section 3.3 Celestial mechanics
  10. ^
    JSTOR 3603797
    .
  11. ^
    ISSN 0002-9904
    .
  12. ^
    doi:10.1090/s0002-9904-1938-06728-6
    .
  13. ^
    ISSN 0025-5572
    .
  14. ^ Coutinho 2014, pp. 366–377 Section 3.2 Hamiltonian systems and contact transformations
  15. ^ Coutinho 2014, pp. 391–396 Section 5.1 Style
  16. ^
    S2CID 3978067
    .
  17. ^ a b c d e f g h i j Coutinho 2014, pp. 383–385 Section 4.2 A British point of view: G. H. Bryan
  18. ^
    doi:10.1090/s0002-9904-1906-01372-6
    .
  19. ^ a b c d e Coutinho 2014, pp. 380–382 Section 4.1 An American point of view: E. B. Wilson
  20. S2CID 118545646
    .
  21. ^ a b c d e f g h Coutinho 2014, pp. 388–391 Section 4.4 Other Reviews
  22. ^ a b Lampe, Emil (1918). "Review of the first edition of 'A Treatise on the Analytical Dynamics of Particles and Rigid Bodies'". Jahrbuch über die Fortschritte der Mathematik.
  23. ^ a b c d e f g h i j k l m Coutinho 2014, pp. 385–388 Section 4.3 The "Fictitious Problem" polemic
  24. S2CID 3975272
    .
  25. S2CID 4011940
    .
  26. S2CID 4047422
    .
  27. S2CID 4013954
    .
  28. S2CID 4018113
    .
  29. S2CID 4038064
    .
  30. S2CID 5767307
    .
  31. S2CID 4016099
    .
  32. ^
    JSTOR 3603175
    .
  33. JSTOR 43426359
    .
  34. ^
    S2CID 4163255
    .
  35. ^
    doi:10.1090/s0002-9904-1920-03290-8
    .
  36. ^ 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.
  37. doi:10.1002/zamm.19900700141
    .
  38. ISSN 0021-1753
    .
  39. ^ a b c d Coutinho 2014, p. 391
  40. ISBN 978-0-387-31255-2
    , retrieved 3 October 2020
  41. S2CID 124642355
    .
  42. ISBN 3-540-20352-4
    .
  43. ISBN 978-0-465-02210-6
    .
  44. ISSN 0273-0979
    .
  45. ISSN 0273-0979
    .

Further reading

External links
Retrieved from "https://en.wikipedia.org/w/index.php?title=Analytical_Dynamics_of_Particles_and_Rigid_Bodies&oldid=1192927239"