Intro
Preface
Contents
About the Editors
Invariant KAM Tori: From Theory to Applications to Exoplanetary Systems
1 Introduction
2 Basics of KAM Theory
2.1 Near to the Identity Canonical Transformations by Lie Series
2.2 Statement(s) of KAM Theorem
2.3 Algorithmic Construction of the Kolmogorov Normal Form
2.4 On the Convergence of the Algorithm Constructing the Kolmogorov Normal Form
3 Construction of Invariant Elliptic Tori by a Normal Form Algorithm
3.1 Algorithmic Construction of the Normal Form for Elliptic Tori

3.2 On the Convergence of the Algorithm Constructing the Normal Form for Elliptic Tori
4 Construction of Invariant KAM Tori in Exoplanetary Systems with Rather Eccentric Orbits
4.1 Secular Model at Order Two in the Masses
4.2 Semi-analytic Computations of Invariant Tori
References
A New Analysis of the Three-Body Problem
1 Overview
2 Euler Problem Revisited
3 Perihelion Librations in the Three-Body Problem
4 Chaos in a Binary Asteroid System
References
KAM Theory for Some Dissipative Systems
1 Introduction

1.1 Consequences of the A-Posteriori Method for Conformally Symplectic Systems
1.2 Organization of the Paper
2 Conservative/Dissipative Standard Maps and Spin-Orbit Problems
2.1 The Conservative Standard Map
2.2 The Dissipative Standard Map
2.3 The Spin-Orbit Problems
3 Conformally Symplectic Systems and Diophantine Vectors
3.1 Discrete and Continuous Conformally Symplectic Systems
3.2 Diophantine Vectors for Maps and Flows
4 Invariant Tori and KAM Theory for Conformally Symplectic Systems
4.1 Invariant KAM Tori
4.2 Conformally Symplectic KAM Theorem

4.3 A Sketch of the Proof of the KAM Theorem
5 Breakdown of Quasi-periodic Tori and Quasi-periodic Attractors
5.1 Sobolev Breakdown Criterion
5.2 Greene's Method, Periodic Orbits and Arnold's Tongues
6 Collision of Invariant Bundles of Quasi-periodic Attractors
7 Applications
7.1 Applications to the Standard Maps
7.2 Applications to the Spin-Orbit Problems
References
Tidal Effects and Rotation of Extended Bodies
1 Introduction
2 Coordinate System
2.1 The 3-1-3 Euler Angles
2.2 Unitary Quaternion
2.3 Special Case: Axisymmetric Body

3 Generalised Velocity and Kinematic Equation
3.1 Kinematic Equation Satisfied by the Rotation Matrix
3.2 Kinematic Equation Satisfied by the 3-1-3 Euler Angles
3.3 Kinematic Equation Satisfied by Unitary Quaternions
3.4 Kinematic Equation Satisfied by a Unit Vector of the Figure Axis
4 Least Action Principle and Dynamical Equations
4.1 Parametrisation of the Tangent Space
4.2 Variation of the Action
4.3 Dynamical Equations
4.4 Rayleigh Dissipation Function
4.5 Spin Operator
4.6 Hamiltonian Formalism
4.7 Example: The Gyroscope