##
**Lectures on celestial mechanics. (Translation by C. I. Kalme).**
*(English)*
Zbl 0312.70017

Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Band 187. Berlin-Heidelberg-New York: Springer-Verlag. xii, 290 p. DM 78.00; $ 22.60 (1971).

The famous yellow series by Springer enriched the mathematical literature once again by publishing the enlarged and translated version of Dr. Carl Ludwig Siegel’s “Vorlesungen über Himmelsmechanik”. The original, in German, published in 1956 as Volume 85 of the series “Die Grundlehren der mathematischen Wissenschaften” has always been regarded as one of the clearest exposition of certain mathematical aspects of celestial mechanics. Volume 85 was based on Professor Siegel’s 1951/52 lectures in Göttingen and on Dr. Jürgen Moser’s notes taken. An extension of this cooperation resulted in the publication of the book under joint authorship that is presently under review.

The clarity and straightforward simplicity of Siegel’s style is retained, the translation is excellent and the additions to the original version are most welcome. Neither Siegel nor Moser are astronomers; the few subjects treated in the book are fundamental in celestial mechanics without either astronomical applications or completeness. The book is essentially a study of the solutions of the differential equations of celestial mechanics and of Hamiltonian systems. The background is formed by Poincaré, Sundman, Birkhoff, and Lyapunov. The three chapters of the book represent, what may be considered the three major subjects in celestial mechanics: the problem of three bodies, periodic solutions and stability.

The first subject is treated as a masterful combination of Sundman’s work (1913) and Siegel’s famous paper on the problem of triple collision (1941). This latter subject is missing in the original version and its inclusion in the new edition renders the book one of the most important “modern” contributions to celestial mechanics. Indeed, the mathematical problems associated with the essential singularities of triple collisions become the computational difficulties of triple close approaches, which latter occupy a principal part of modern research on the problem of three bodies. Einstein’s dictum of working on scientific problems because of one’s “inner need” bore delightful fruits in Siegel’s case. When clarifying the analytical aspects of triple collisions he (unknowingly) established the foundations of one of the modern dynamical explanations of binary star formation.

The second chapter on periodic orbits is changed but little from the original version. It may be considered one of the best treatments of periodic orbits in the literature of celestial mechanics: Lagrange, Hill, Poincaré and Birkhoff offer the foundations together with Siegel’s original contributions.

The third chapter on stability benefits considerably from recent results by Kolmogorov, Arnold and Moser regarding quasi-periodic solutions.

By expanding the original version, the book, once again, may be considered as one of the important contributions to modern celestial mechanics. In style and clarity the book surpasses almost all other corresponding volumes in the literature of celestial mechanics. Poincaré’s “Méthodes nouvelles”, Birkhoff’s “Dynamical systems”, and Wintner’s “Analytical foundations” for instance, may be considered of fundamental importance. Distinctions between conjectures and theorems, proofs and indications are less than clear and definitions are often subject to the reader’s guesses in the first two volumes, while the third’s pedantry, precision, and often unnecessary complications are mind-boggling.

In comparison, the book under review offers an enjoyable style and ease of understanding. This, however might not be as important as the selection of the subject matter. The field of celestial mechanics is rather large: the spectrum extends from numerical analysis to group theory, and applications include orbits of comets, rotation of planets, stability and control of spacecraft, etc. The incomparable attempt for an all encompassing encyclopedic treatment in Y. Hagihara’s recent work (four volumes published, two to follow) is unique and in a class by itself. Most other treatments make selections.

The following, admittedly incomplete list of authors of major books on celestial mechanics, (excluding those already mentioned) is offered to put the Siegel/Moser book in proper prospective: Newton, Euler, Lagrange, Gauss, Laplace, Tisserand, Poincaré, Charlier, Whittaker, Hill, Darwin, Moulton, Plummer, Brown and Shook, Smart, Stumpff, Khilmi, Brouwer and Clemence, Finlay-Freundlich, Pars, Chebotarev, Szebehely, Duboshin, Sternberg, Thiry, McCuskey, Baker, Pollard, Danby, Kovalevski, Sterne, etc.

Unquestionably, the only guiding principle for selection (if any) is to follow the Einsteinian “inner need” without expecting possible rewards and disregarding the often empty trends of fashionable none-science. That the subjects of the book under review are of fundamental importance could hardly be argued. That these days there are more fashionable and financially rewarding problems than the problem of three bodies (one of the book’s major topics) is also unquestionable. The authors’ choices, nevertheless, are sound since their selected topics penetrate and interconnect a large number of subfields, applications, details and fundamentals. Celestial mechanics is made more enjoyable, easier to comprehend, indeed, more popular by the authors.

This referee’s copy of the original German version is in shambles from the twenty years of use by him and by his students. His copy of the new translated and enlarged edition is still holding up but not for long. The yellow series is maintaining its tradition: Knapp, Hurwitz, Klein, Polya/Szegö, Courant/Hilbert, Runge/König, Lichtenstein, Kellog, Hamel, Blaschke, Collatz, Bieberbach, (just to mention a few that were used by all of us) were joined recently by Stiefel/Scheifele and now by Siegel/Moser. The profession is indeed in debt to Springer for the excellent printing and for publishing the new translated edition at a reasonable price.

The clarity and straightforward simplicity of Siegel’s style is retained, the translation is excellent and the additions to the original version are most welcome. Neither Siegel nor Moser are astronomers; the few subjects treated in the book are fundamental in celestial mechanics without either astronomical applications or completeness. The book is essentially a study of the solutions of the differential equations of celestial mechanics and of Hamiltonian systems. The background is formed by Poincaré, Sundman, Birkhoff, and Lyapunov. The three chapters of the book represent, what may be considered the three major subjects in celestial mechanics: the problem of three bodies, periodic solutions and stability.

The first subject is treated as a masterful combination of Sundman’s work (1913) and Siegel’s famous paper on the problem of triple collision (1941). This latter subject is missing in the original version and its inclusion in the new edition renders the book one of the most important “modern” contributions to celestial mechanics. Indeed, the mathematical problems associated with the essential singularities of triple collisions become the computational difficulties of triple close approaches, which latter occupy a principal part of modern research on the problem of three bodies. Einstein’s dictum of working on scientific problems because of one’s “inner need” bore delightful fruits in Siegel’s case. When clarifying the analytical aspects of triple collisions he (unknowingly) established the foundations of one of the modern dynamical explanations of binary star formation.

The second chapter on periodic orbits is changed but little from the original version. It may be considered one of the best treatments of periodic orbits in the literature of celestial mechanics: Lagrange, Hill, Poincaré and Birkhoff offer the foundations together with Siegel’s original contributions.

The third chapter on stability benefits considerably from recent results by Kolmogorov, Arnold and Moser regarding quasi-periodic solutions.

By expanding the original version, the book, once again, may be considered as one of the important contributions to modern celestial mechanics. In style and clarity the book surpasses almost all other corresponding volumes in the literature of celestial mechanics. Poincaré’s “Méthodes nouvelles”, Birkhoff’s “Dynamical systems”, and Wintner’s “Analytical foundations” for instance, may be considered of fundamental importance. Distinctions between conjectures and theorems, proofs and indications are less than clear and definitions are often subject to the reader’s guesses in the first two volumes, while the third’s pedantry, precision, and often unnecessary complications are mind-boggling.

In comparison, the book under review offers an enjoyable style and ease of understanding. This, however might not be as important as the selection of the subject matter. The field of celestial mechanics is rather large: the spectrum extends from numerical analysis to group theory, and applications include orbits of comets, rotation of planets, stability and control of spacecraft, etc. The incomparable attempt for an all encompassing encyclopedic treatment in Y. Hagihara’s recent work (four volumes published, two to follow) is unique and in a class by itself. Most other treatments make selections.

The following, admittedly incomplete list of authors of major books on celestial mechanics, (excluding those already mentioned) is offered to put the Siegel/Moser book in proper prospective: Newton, Euler, Lagrange, Gauss, Laplace, Tisserand, Poincaré, Charlier, Whittaker, Hill, Darwin, Moulton, Plummer, Brown and Shook, Smart, Stumpff, Khilmi, Brouwer and Clemence, Finlay-Freundlich, Pars, Chebotarev, Szebehely, Duboshin, Sternberg, Thiry, McCuskey, Baker, Pollard, Danby, Kovalevski, Sterne, etc.

Unquestionably, the only guiding principle for selection (if any) is to follow the Einsteinian “inner need” without expecting possible rewards and disregarding the often empty trends of fashionable none-science. That the subjects of the book under review are of fundamental importance could hardly be argued. That these days there are more fashionable and financially rewarding problems than the problem of three bodies (one of the book’s major topics) is also unquestionable. The authors’ choices, nevertheless, are sound since their selected topics penetrate and interconnect a large number of subfields, applications, details and fundamentals. Celestial mechanics is made more enjoyable, easier to comprehend, indeed, more popular by the authors.

This referee’s copy of the original German version is in shambles from the twenty years of use by him and by his students. His copy of the new translated and enlarged edition is still holding up but not for long. The yellow series is maintaining its tradition: Knapp, Hurwitz, Klein, Polya/Szegö, Courant/Hilbert, Runge/König, Lichtenstein, Kellog, Hamel, Blaschke, Collatz, Bieberbach, (just to mention a few that were used by all of us) were joined recently by Stiefel/Scheifele and now by Siegel/Moser. The profession is indeed in debt to Springer for the excellent printing and for publishing the new translated edition at a reasonable price.

Reviewer: V. Szebehely

### MSC:

70F15 | Celestial mechanics |

70-02 | Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems |

70F07 | Three-body problems |

34Cxx | Qualitative theory for ordinary differential equations |

37Cxx | Smooth dynamical systems: general theory |

70H08 | Nearly integrable Hamiltonian systems, KAM theory |