CANONICAL
PERTURBATION
THEORIES,

DEGENERATE SYSTEMS AND RESONANCE

PREFACE

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

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

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