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Canonical Perturbation Theories Degenerate Systems and Resonance ASTROPHYSICS AND SPACE SCIENCE LIBRARY VOLUME 345 EDITORIAL BOARD Chairman W.B. BURTON, National Radio Astronomy Observatory, Charlottesville, Virginia, U.S.A. ([email protected]); University of Leiden, The Netherlands ([email protected]) MEMBERS J.M.E. KUIJPERS, Faculty of Science, Nijmegen, The Netherlands F. BERTOLA, Universitá di Padova, Italy J.P. CASSINELLI, University of Wisconsin, Madison, U.S.A. C.J. CESARSKY, European Southern University, Garching bei München, Germany P. EHRENFREUND, Leiden University, The Netherlands O. ENGVOLD, Institute of Theoretical Astrophysics, University of Oslo, Norway A. HECK, Strassbourg Astronomical Observatory, France V.M. KASPI, McGill University, Montreal, Canada P.G. MURDIN, Institute of Astronomy, Cambridge, U.K. F. PACINI, Istituto Astronomia Arcetri, Firenze, Italy V. RADHAKRISHNAN, Raman Research Institute, Bangalore, India F.H. SHU, University of California, Berkeley, U.S.A. B.V. SOMOV, Astronomical Institute, Moscow State University, Russia R.A. SUNYAEV, Space Research Institute, Moscow, Russia E.P.J. VAN DEN HEUVEL, Astronomical Institute, University of Amsterdam, The Netherlands H. VAN DER LAAN, Astronomical Institute, University of Utrecht, The Netherlands Canonical Perturbation Theories Degenerate Systems and Resonance Sylvio Ferraz–Mello Sylvio Ferraz-Mello Instituto de Astronomia, Geofísicae e Ciências Atmosféricas Universidade de São Paulo Rua do Matão, 1226 CEP 05508-900 São Paulo, Brasil [email protected] http://www.astro.iag.usp.br/~sylvio/ Library of Congress Control Number: 2006931783 ISBN-10: 0-387-38900-8 e-ISBN-13: 978-0-387-38905-9 ISBN-13: 978-0-387-38900-4 e-ISBN-10: 0-387-38905-9 Printed on acid-free paper. © 2007 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identifi ed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 springeronline.com Preface A uńica maneira de cumprir o trabalho era tê-lo comocoisalerdaecont´ınua,mansa,semcomeço nem fim, as maõs sempre sujas da massa. Joaõ Guimara˜es Rosa, Buriti 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 Technol- ogy. 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 un- derstandablebyinvestigatorsandstudents fromrelatedfields ofscience.This hintmarkedtherestofthestory.Thecourseevolvedandforthepast30years was taught almost yearly at the University of Saõ Paulo and, occasionally,in visited institutions abroad. A long visit of Prof. Gen-Ichiro Hori to the Uni- versity of Saõ 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 whichtheyareapplied;onedoesnotneedtoknowAstronomytoreadit.The Page:V job:b macro:svmono.cls date/time:20-Oct-2006/9:21 VI Preface other objective is to provide, in one book, all the information necessary for theapplicationofthetheories.Forinstance,notonlyisittoldhowtoactually obtain action–angle variables,but they are explicitly given for important dynamical systems such as the simple pendulum, the Ideal Resonance Problem andthe firstAndoyerHamiltonian.Inaddition,everytheorypresentedinthe book is followedby casestudies andexamplesable to illustrate the directions fortheiruseinapplications.Forthesakeofmakingthebookusefulasahand- book in investigations using perturbation theories, special care was taken to avoid errors in the given equations. All my students have, in the past, com- municated 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 thatmayhaveaddednewerrors.AWebpagewillbecreatedtoinformreaders 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 sectionsweredirectlyinspiredbytheseminalclassesofthelateProf.Abrahaõ 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 per- sonal 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 transformationsareobtainedbymeansofJacobi’sgeneratingfunction.These chaptersincludethePoincarétheoryforperturbednon-degenerateHamiltoni- ans, the von Zeipel–Brouwer theory for perturbed degenerate Hamiltonians, the procedures of frequency relocation and quadratic convergence used by Kolmogorovin 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 emphasizingthatthe definitionofdegeneracyusedthroughoutthisbook,due to Schwarzschild, is less strict than the definition of degeneracy used in Kol- mogorov’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 Page:VI job:b macro:svmono.cls date/time:20-Oct-2006/9:21 Preface VII whose Hamilton–Jacobi equation needs to be completely solved. From the dynamicalpointofview,itforcesthesolutionsgivenbyperturbationtheories to have the same topology as the Hori kernel. However, generally, the Hori kernelandthe givenHamiltonianhavedifferenttopologiesandthis 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 singulari- ties 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 Hamil- tonian, 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 quasiharmonic 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 Hamiltoniansfundamentalinresonancestudies:thesimplependulumandthe first Andoyer Hamiltonian. The action–angle variables of these two Hamilto- nians are constructed with the help of elliptic functions. Expansions in terms of trigonometric functions valid in a neighborhood of the libration center are alsogiven.AppendixCalsoincludestheconstructionofsolutionsintheneigh- borhoodofthependulumseparatrixandtheassociatedwhiskerandstandard mappings. Appendix D presents the main features of some higher-order An- doyer Hamiltonians. One last comment on the contents of this book is that it is not aimed at beinganencyclopediaonthesubjectanddoesnotcovereveryapproachofthe problem.Onthecontrary,severalsectionsandevenonechapternotbelonging to the backbone of the subject were dropped during the revision. Canonical perturbationtheoriesareanoldsubject,andmanyapproachesexistthatwere not even mentioned in the book. Thelistofreferences,attheendofthebook,alsodeservessomecomments. 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 acknowledgementthat 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 directrelationshiptowhatiswritteninthisbook.Weconsidereditimportant Page:VII job:b macro:svmono.cls date/time:20-Oct-2006/9:21 VIII Preface not to let these few items disappear amid an exhaustive bibliography. This choicewasmadehavinginmindthatsearchenginesontheinternetmaygive, nowadays,moreandbetter bibliographicalinformationthanalonglistatthe end of a book. Acknowledgements. I thank my family for continuous support. Many friends and colleagues have given me suggestions that helped to improve the book. I thank all of them and, particularly, Prof. Jean Kovalevsky, who, long ago, introduced me to canonical perturbation theories and Profs. André Brahic, Rudolf Dvorak, Claude Froeschlé, Juan Carlos Muzzio and Bruno Sicardy, who have often invited me to their institutions, allowing me to have time to write. I thank all my students. They have read almost all the drafts of this book and collaborated with valuable comments that resulted in many improvements in the written text. I thank the copy editor Mike Nugent for his invaluable contribution for the editorial quality of this book. During the work on this project, I had the support of USP – University of Saõ Paulo, Observato´rio Nacional, Bureau des Longitudes (now IMCCE), Observatoire de Paris–Meudon, Wien Universität, FAPESP – Research Foundation of the State of Saõ Paulo and CNPq – National Council for Scientific and Techno- logical Development. Saõ Paulo, June 2006 Sylvio Ferraz-Mello Page:VIII job:b macro:svmono.cls date/time:20-Oct-2006/9:21 Contents Preface......................................................... v 1 The Hamilton–Jacobi Theory.............................. 1 1.1 Canonical Pertubation Equations.......................... 1 1.2 Hamilton’s Principle ..................................... 2 1.2.1 Maupertuis’ Least Action Principle .................. 4 1.2.2 Helmholtz Invariant ............................... 5 1.3 Canonical Transformations ............................... 6 1.4 Lagrange Brackets....................................... 9 1.5 Poisson Brackets ........................................ 11 1.5.1 Reciprocity Relations .............................. 12 1.6 The Extended Phase Space ............................... 13 1.7 Gyroscopic Systems...................................... 15 1.7.1 Gyroscopic Forces ................................. 15 1.7.2 Example ......................................... 16 1.7.3 Rotating Frames .................................. 17 1.7.4 Apparent Forces................................... 17 1.8 The PartialDifferential Equation of Hamilton and Jacobi..... 18 1.9 One-Dimensional Motion with a Generic Potential........... 20 1.9.1 The Case m<0................................... 23 1.9.2 The Harmonic Oscillator ........................... 23 1.10 Involution. Mayer’s Lemma. Liouville’s Theorem ............ 24 2 Angle–Action Variables. Separable Systems................ 29 2.1 Periodic Motions ........................................ 29 2.1.1 Angle–Action Variables ............................ 30 2.1.2 The Sign of the Action............................. 32 2.2 Direct Construction of Angle–Action Variables .............. 33 2.3 Actions in Multiperiodic Systems. Einstein’s Theory ......... 35 2.4 Separable Multiperiodic Systems .......................... 37 2.4.1 Uniformized Angles. Charlier’s Theory ............... 37 2.4.2 The Actions ...................................... 38 2.4.3 Algorithms for Construction of the Angles............ 39 Page:IX job:b macro:svmono.cls date/time:20-Oct-2006/9:21