Acknowledgments
1 Introduction
1.1 Integrability of differential systems
1.2 Historical note
1.3 Original results
1.3.1 Homogeneous potentials and N-Body Problems
1.3.2 Non-integrability of Hill's Problem
1.4 General structure, notation and conventions
2 Theoretical background
2.1 Useful results from Algebraic Geometry
2.1.1 Preliminaries
2.1.2 Linear algebraic groups and Lie algebras
2.1.3 Rational invariants
2.2 Notions of integrability
2.2.1 Integrability of Hamiltonian systems
2.2.2 Integrability of linear differential systems
2.3 Morales-Ramis theory
2.3.1 The general theory
2.3.2 Special Morales-Ramis theory: homogeneous potentials
2.4 Basics in Celestial Mechanics
2.4.1 The N-Body Problem
2.4.2 Hill's Lunar Problem
3 The meromorphic non-integrability of some N-Body Problems
3.1 Introduction
3.2 Preliminaries
3.2.1 Statement of the main results
3.2.2 Setup for the proof
3.3 Proofs of Theorems 3.2.2 and 3.2.3
3.3.1 Proof of Theorem 3.2.2
3.3.2 Proof of Theorem 3.2.3.
3.3.3 Proof isolate: N = 2m equal masses
4 The meromorphical non-integrability of Hill's Lunar problem
4.1 Introduction
4.1.1 Statement of the main results
4.2.1 Change of variables
4.2.2 Solution of the new equation
4.2.3 Singularities of ∮2(t)
4.3.1 Layout of the system
4.3.2 Fundamental matrix of the variational equations
4.3.3 Relevant facts concerning ψ(t)
4.5 Concluding statements
5 Conclusions and work in progress
5.1 Overview
5.2 Perspectives on Conjectures 5.1.1 and 5.1.2
5.2.1 The N-body problem with arbitrary masses
5.2.2 Candidates for a partial result
5.3 Hamiltonians with a homogeneous potential
5.3.1 Higher variational equations
Appendices
B Resum
B.1 Introducció
B.1.1 Dues nocions d'integrabilitat en sistemes dinàmics
B.1.2 Alguns problemes de la Mecànica Celeste
B.2 Resultats originals
B.2.1 Existència d'una integral primera addicional
B.2.2 Problemes de N Cossos.
B.2.3 La no-integrabilitat del Problema de Hill
Bibliography
Index
编辑手记
展开
