The story of this book began in the late 1960s, when Prof. Buarque Borges invited me to give a graduate course at the Aeronautics Institute of Technology. The course was to deal with the perturbation theories used in Celestial Mechanics, but they should be presented in a universal way, so as to be understandable by investigators and students from related fields of science. This hint marked the rest of the story. The course evolved and for the past 30 years was taught almost yearly at the University of São Paulo and, occasionally, in visited institutions abroad. A long visit of Prof. Gen-Ichiro Hori to the University of São Paulo was the occasion for many illuminating discussions on the subject.


Soon, in this story, came the project of a book. But two major obstacles did not allow it to progress at that time. One of them was the concurrence of many other time-consuming duties. The drafts of many chapters could only be written during visits abroad: to Austin, Grasse, La Plata, Oporto, Nice, Paris, Vienna, and the book could only be completed now, after my formal retirement. The other obstacle, more determinant, was the fact that theories able to treat Bohlin's problem, a resonant Hamiltonian system with two degrees of freedom, where the second degree of freedom is degenerate, were not available. So, the book project had to wait for new investigations!


In accordance with the initial proposal, the aim of the book is to present the main canonical perturbation theories used in Celestial Mechanics without any involvement with the particularities of the astronomical problems to which they are applied; one does not need to know Astronomy to read it. The other objective is to provide, in one book, all the information necessary for the application of the theories. For instance, not only is it told how to actually obtain action-angle variables, but they are explicitly given for important dynamical systems such as the simple pendulum, the Ideal Resonance Problem and the first Andoyer Hamiltonian. In addition, every theory presented in the book is followed by case studies and examples able to illustrate the directions for their use in applications. For the sake of making the book useful as a handbook in investigations using perturbation theories, special care was taken to avoid errors in the given equations. All my students have, in the past, communicated to me the errors found in the drafts. I have myself checked every equation, but I am not foolish to say that no flaws remain. Transcriptions, transpositions, and the work on LaTeX source files are non-robust operations that may have added new errors.  A Web page will be created to inform readers of any flaws finally remaining in the text.


The book is composed of 10 chapters and four appendices. The two first chapters are devoted to some results of Hamilton-Jacobi theory. This short presentation, where only points of practical interest are given a longer development, is not a substitute for a full text on Analytical Dynamics. Many sections were directly inspired by the seminal classes of the late Prof. Abrahão de Moraes, which I had the privilege of attending in my undergraduate years, and by books with which I became acquainted in frequent visits to his personal library. One of them was Charlier's Die Mechanik des Himmels, the book referenced in many papers on fundamental Physics in the first decades of past century.


Chapters 3 and 4 are devoted to perturbation theories where canonical transformations are obtained by means of Jacobi's generating function. These chapters include the Poincaré theory for perturbed non-degenerate Hamiltonians, the von Zeipel--Brouwer theory for perturbed degenerate Hamiltonians, the procedures of frequency relocation and quadratic convergence used by Kolmogorov in the proof of his theorem, the theory used in Delaunay's lunar theory and the solution of Garfinkel's Ideal Resonance Problem. It is worth emphasizing that the definition of degeneracy used throughout this book, due to Schwarzschild, is less strict than the definition of degeneracy used in Kolmogorov's theorem.


Chapter 5 introduces Lie mappings and Chap. 6 reconsiders the study of perturbed non-degenerate Hamiltonian systems with canonical transformations written as Lie series. Lie series theories in action--angle variables are completely equivalent to those founded on Jacobian transformations and the choice of one or another is a matter of work economy only. Their comparison is done in two typical examples.


Chapter 6 introduces Hori's theory with unspecified canonical variables and this is the point where the equivalence to the old theories disappears. Hori's theory shows that every perturbation theory has a dynamical kernel, the Hori kernel. From the algorithmic point of view, the Hori kernel is a Hamiltonian system that repeats itself at every order of approximation, and whose Hamilton-Jacobi equation needs to be completely solved. From the dynamical point of view, it forces the solutions given by perturbation theories to have the same topology as the Hori kernel. However, generally, the Hori kernel and the given Hamiltonian have different topologies and this difference gives rise to the well-known small divisors.


In Chap. 7, it is shown how Hori's theory with unspecified canonical variables allows the construction of formal solutions using non-singular Poincaré variables, thus allowing the study of perturbed systems near the singularities of the actions. In Chaps. 8 and 9, the understanding of the role played by the Hori kernel is the key to dealing with resonant systems with two or more degrees of freedom presenting simultaneously resonant and degenerate angles. The Hori kernels in these chapters are systems whose restrictions to one degree of freedom are the simple pendulum and the first Andoyer Hamiltonian, respectively.

The techniques discussed in Chap. 2 are used to extend the action-angle of these models to the two-degrees-of-freedom Hori kernel. Finally, in Chap. 10, the theories presented in the previous chapters are applied to some quasi-harmonic Hamiltonian systems.


Appendix A is devoted to presenting Bohlin's theory and an extension of Delaunay's theory and to discuss the difficulties presented by these theories when applied to systems with more than one degree of freedom involving simultaneously resonant and degenerate arguments.


Appendices B and C present the complete solutions of two integrable Hamiltonians fundamental in resonance studies: the simple pendulum and the first Andoyer Hamiltonian. The action-angle variables of these two Hamiltonians are constructed with the help of elliptic functions. Expansions in terms of trigonometric functions valid in a neighborhood of the libration center are also given. Appendix C also includes the construction of solutions in the neighborhood of the pendulum separatrix and the associated whisker and standard mappings. Appendix D presents the main features of some higher-order Andoyer Hamiltonians.


One last comment on the contents of this book is that it is not aimed at being an encyclopedia on the subject and does not cover every approach of the problem. On the contrary, several sections and even one chapter not belonging to the backbone of the subject were dropped during the revision. Canonical perturbation theories are an old subject, and many approaches exist that were not even mentioned in the book.


The list of references, at the end of the book, also deserves some comments. One characteristic feature of this list concerns the old references where important concepts in present-day theories were introduced. It is human nature to highlight the more recent contributions showing the importance of some old concepts and to forget the founding fathers that introduced them much earlier. Special attention was paid to give to them the acknowledgement that they deserve and to inform new generations of their achievements. In what concerns the recent references, we included only some items that have a very direct relationship to what is written in this book. We considered it important not to let these few items disappear amid an exhaustive bibliography. This choice was made having in mind that search engines on the internet may give, nowadays, more and better bibliographical information than a long list at the end of a book.



São Paulo, June 2006

Sylvio Ferraz-Mello