11R10. Generating Families in the Restricted Three-Body Problem, II: Quantitative Study of Bifurcations. Lecture Notes in Physics, Vol m65. - M Henon (CNRS, Observatoire de la Cote d’Azur, BP 4229, Nice, 06304 Cedex 4, France). Springer-Verlag, Berlin. 2001. 301 pp. ISBN 3-540-41733-8. $44.00.
Reviewed by FH Lutze (Dept of Aerospace and Ocean Eng, VPI, Blacksburg VA 24061-0203).
This monograph belongs to a series called the Lecture Notes in Physics whose editorial policy states, “…reports new developments in physical research and teaching - quickly, informally, and at a high level. The type of material considered for publication in the monograph series includes monographs presenting original research or new angles in a classical field. The timeliness of a manuscript is more important than its form, which may be preliminary or tentative. Manuscripts should be reasonably self contained.” This monograph satisfies all these requirements, except for the last one. The current work starts with Chapter 11; the first ten appear in the first monograph volume with the same name. Furthermore, the “introductory” chapter in the current volume refers to definitions, figures, and results from the early chapters that are not reproduced in this volume. In addition, similar references appear throughout this work. Consequently, one would not be able to initiate a study of this subject using this second volume alone.
This work is an extension of the earlier monograph, by the author, in this same series. In that volume, the study of generating families in the restricted three-body problem was initiated and treated qualitatively. There, generating families are defined as the limits of families of periodic orbits for the case where the mass ratio approached 0. The main problem is in determining the junctions between the branches at a bifurcation orbit, where two or more families of generating orbits intersect. This monograph focuses on that problem.
The first chapter (Ch 11) sets up the definitions (often times referring to Chs 1–10) and the governing equations. In addition, a general method of approach is presented by the numbers, that is referred to by the same numbers in later chapters. Chapters 12–14 present the application of this approach to the total and partial bifurcations of Type 1. Chapter 15 presents a geometric approach to solving the problem that is interesting, but runs into problems with increasing numbers of arcs. Chapter 16 generalizes the results of Chapter 15, overcoming the numbers problem, and verifies results obtained from Chapters 12–14. Chapters 17–23 apply the method of Chapters 12–14 to total and partial bifurcations of Type 2. These are significantly more complicated and have additional phenomena called total and partial Transitions, requiring more details to sort out. Essentially, this monograph carries us through the details of the search for all possible combinations and permutations of possible junctions for Type 1 and Type 2 bifurcations.
Generating Families in the Restricted Three-Body Problem, II: Quantitative Study of Bifurcations is generally well written and in a logical order. Occasionally, there is lack of justification for some results. It would seem that there could be more figures in the book to support some of the statements and equations that are used and to help interpret some of the results. It is clear that the objective here is to get results, not interpret them. Finally, an analysis of Type 3 bifurcations is suggested as being much more complicated, introducing a dozen or so new types of arcs, and is not pursued in this volume. This reviewer would suggest that this volume should not be purchased without the first one. Further, the subject is very limited, but interesting, and requires a great deal of patience to follow all the possible paths to determine the solutions. While some of the techniques may be applied to general dynamical systems, most are reserved for this particular problem.