A KAM theorem for the spatial lunar problem [Elektronische Ressource] / vorgelegt von Britta Sylvia Sommer

Britta Sommer

Sujets

• Physik

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2003
• Nombre de lectures: 18
• Langue: English

Extrait

A KAM Theorem for the Spatial Lunar Problem
Von der Fakultat fur Mathematik, Informatik und Naturwissenschaften der˜ ˜
Rheinisch-Westfalischen Technischen Hochschule Aachen zur Erlangung des˜
akademischen Grades eines Doktors der Naturwissenschaften genehmigte
Dissertation
vorgelegt von
Diplom-Mathematikerin
Britta Sylvia Sommer
aus Aachen
Berichter: Universitatsprofessor Dr. Volker En…˜
Universit Dr. Dr.h.c. Hubertus Th. Jongen˜
Universitatsprofessor Dr. Richard H. Cushman, Univ. Calgary˜
Tag der mundlic˜ hen Prufung:˜ 25. Juni 2003
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online
verfugbar.˜Acknowledgement
It is my pleasure and privilege to thank those who helped me in one way or
another to accomplish this thesis. First of all, I am very grateful to my supervisor
Professor Dr. Volker En… for his continuous interest and engaged support of my
work. Thank you for many instructive discussions and the constructive criticism
which considerably helped to improve this work. I thank Dr. Heinz Han…mann
who never complained when I rushed into his o–ce to discuss the latest flndings
and who listened patiently to whatever idea I came up with. Working and learning
mathematics with you in the past years has been fun and inspiration. You introdu-
ced me to this subject and helped me in many discussions to elaborate the main
facts. I wish to thank Professor Dr. Richard Cushman for his interest in my work,
his invaluable comments and suggestions, the improvement of my writing and for
his encouragement. I also thank Professor Dr. Volker En…, Professor Dr. Richard
Cushman and Professor Dr. Dr.h.c. Hubertus Th. Jongen for undertaking the labor
of refereeing this thesis.
The flgures were drawn using the software package ESFERAS and MAPLE 7. The
formulae have been checked with MAPLE 7.
3Inhaltsverzeichnis
1. Introduction 7
1.1. The n-body problem and King Oscar’s prize . . . . . . . . . . . . . . 7
1.2. The lunar problem and perturbed Hamiltonian systems . . . . . . . . 9
1.3. KAM theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4. A KAM theorem for the lunar problem . . . . . . . . . . . . . . . . . 14
1.5. Summary of the chapters . . . . . . . . . . . . . . . . . . . . . . . . . 16
2. Integrable Hamiltonian systems and their perturbations 21
2.1. Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.1. Scaling Hamiltonian systems . . . . . . . . . . . . . . . . . . . 24
2.2. The lunar problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.1. From the full three body problem to the restricted one. . . . 27
2.2.2. From the restricted to the lunar problem . . . . . . . . . . . 32
2.2.3. Regularization of the Kepler Hamiltonian . . . . . . . . . . . 34
2.2.4. Excursion: KS transformation and the lunar problem . . . . . 36
2.3. Constrained normal forms . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.1. Cosymplectic Submanifolds . . . . . . . . . . . . . . . . . . . 39
2.3.2. Constrained normalization using Lie-series . . . . . . . . . . . 42
2.4. Reducing the degrees of freedom . . . . . . . . . . . . . . . . . . . . . 46
2.4.1. Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4.2. of the Kepler symmetry . . . . . . . . . . . . . . . 50
2.4.3. Reduction of the angular momentum symmetry . . . . . . . . 52
3. The lunar problem - Analysis of the perturbed Keplerian problem 55
3.1. Constrained normal forms - Application to the lunar problem . . . . 55
3.1.1. Calculating the constrained normal form of the lunar problem 55
3.1.2. Normalization of the reduced Hamiltonian . . . . . . . . . . . 60
3.1.3. The twice normalized, twice reduced system . . . . . . . . . . 61
3.2. Action-angle coordinates for the lunar problem . . . . . . . . . . . . . 64
5Inhaltsverzeichnis
3.2.1. An intermediate canonical coordinate transformation . . . . 65
3.2.2. The action-angle-variables . . . . . . . . . . . . . . . . . . . . 68
2 23.2.3. The bifurcation at c =l = 3=5 . . . . . . . . . . . . . . . . . . 74
4. A KAM theorem for certain perturbed Keplerian systems 75
4.1. A (very) short introduction to KAM theory . . . . . . . . . . . . . . 75
4.2. From the geometry of the system to the perturbation analysis . . . . 80
4.3. Preliminaries and known results . . . . . . . . . . . . . . . . . . . . . 83
4.3.1. Preliminary deflnitions . . . . . . . . . . . . . . . . . . . . . . 86
4.4. The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4.1. The Propositions . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.4.2. Proof of the Main Theorem 4.4.1 . . . . . . . . . . . . . . . . 95
4.4.3. Proof of Proposition 4.4.1.1 . . . . . . . . . . . . . . . . . . . 96
4.4.4. Whitney-smoothness{Proof of Proposition 4.4.1.2 . . . . . . . 111
4.4.5. Proof of Proposition 4.4.1.3 . . . . . . . . . . . . . . . . . . . 112
5. Application of Theorem 4.4.1 to the lunar problem 115
5.1. Dealing with the order of the perturbation . . . . . . . . . . . . . . . 115
5.2. The non-degeneracy conditions . . . . . . . . . . . . . . . . . . . . . 116
5.2.1. Improving Kummer’s results . . . . . . . . . . . . . . . . . . . 116
5.2.2. Qualitative results for angular momentum c = 0 . . . . . . . . 118
5.2.3. Quantitative results for angular momentum c… 0 . . . . . . . 120
6. Concluding remarks 129
7. Summary 131
8. Zusammenfassung 135
A. Technical lemmas 137
6
61. Introduction
This thesis deals with a problem which is situated at the very point were three
flelds of the theory of dynamical systems meet. One is the so-called three body
problem. It describes the behaviour of three bodies under the in uence of their
mutual gravitational attraction. The remaining two flelds are parts of perturbation
theory. The flrst one deals with the analysis of perturbed Hamiltonian systems by
means of averaging. Keywords of this part of the theory are \normalization" and
\reduction". The so-called Kolmogorov-Arnol’d-Moser (KAM) Theory is the second
fleld within perturbation theory which will interest us. It allows us to give certain
stability results for perturbed Hamiltonian systems. It seems appropriate to give a
short introduction into these matters before actually discussing the main issue of
this thesis.
1.1. The n-body problem and King Oscar’s prize
The dynamics of two bodies that move under the in uence of their mutual
gravitational attraction is well understood. Already Newton solved the problem
of the behaviour of one planet rotating about the sun (cf. [Newton]). It seems
natural to ask what happens when we add a third body to the system. Actually,
this one body su–ces to alter the equations of motion in such a way that no exact
solution can be given any more. This is one of the reasons why n-body problems
have been such an interesting fleld of research until today. As we can no longer give
any exact solutions, we have to flnd a difierent strategy to analyze the dynamics
of the system. An outstanding point is the question concerning the stability
of orbits that occur. We are living in such a system and this turns the questi-
onwhethertherecanbeforever-stableorbitsofamoonoraplanetintoacrucialone.
"Given a system of arbitrarily many mass points that attract each other
according to Newton’s law, try to flnd, under the assumption that no two
points ever collide, a representation of the coordinates of each point as a
series in a variable that is some known function of time and for all of whose
values in the series converges uniformly. This problem, whose solution would
considerably extend our understanding of the solar system, would seem
capable of solution using analytic methods presently at our disposal; we
7Kapitel 1. Introduction
can at least suppose as much, since Lejeune Dirichlet communicated shortly
before his death to a geometer of his acquaintance [Leopold Kronecker],
that he had discovered a method for integrating the difierential equations
of Mechanics, and that by applying this method, he had succeeded in
demonstrating the stability of our planetary system in an absolutely rigorous
manner. Unfortunately, we know nothing about his method, except that
the theory of small oscillations would appear to have served as his point of
departure for this discovery. We can nevertheless suppose, almost with cer-
tainty, that this method was based not on long and complicated calculations,
but on the development of a fundamental and simple idea that one could
reasonably hope to recover through preserving and penetrating research. In
the event that this problem nevertheless remains unsolved at the close of
the contest, the prize may also be awarded for a work in which some other
problemofMechanicsistreatedinamannerindicatedandsolvedcompletely."
This announcement was to be read in Acta Mathematica, vol. 7, of 1885-1886. King
Oscar II of Sweden and Norway had been convinced by the famous mathematician
G˜osta Mittag-Le†er to establish a substantial prize and medal to be awarded to
the flrst person who obtained the global general solution of then{body problem. Its
prestigewasequivalenttoaNobelPrizetodayandmanymathematiciansattempted
tosolvetheproblem.ThejurywasformedbyKarlWeierstrass,CharlesHermiteand
Mittag-Le†er himself. Only flve of twelve entries to the competition even attempted