New frontiers of celestial mechanics : theory and applications : I-CELMECH Training School, Milan, Italy, February 3-7, 2020

This volume contains the detailed text of the major lectures delivered during the I-CELMECH Training School 2020 held in Milan (Italy). The school aimed to present a contemporary review of recent results in the field of celestial mechanics, with special emphasis on theoretical aspects. The stability of the Solar System, the rotations of celestial bodies and orbit determination, as well as the novel scientific needs raised by the discovery of exoplanetary systems, the management of the space debris problem and the modern space mission design are some of the fundamental problems in the modern developments of celestial mechanics. This book covers different topics, such as Hamiltonian normal forms, the three-body problem, the Euler (or two-centre) problem, conservative and dissipative standard maps and spin-orbit problems, rotational dynamics of extended bodies, Arnold diffusion, orbit determination, space debris, Fast Lyapunov Indicators (FLI), transit orbits and answer to a crucial question, how did Kepler discover his celebrated laws? Thus, the book is a valuable resource for graduate students and researchers in the field of celestial mechanics and aerospace engineering. 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 5 Lagrangian of a Rigid Body Interacting with a Point Mass 5.1 Kinematic Energy 5.2 Potential Energy 5.3 Lagrangian and Hamiltonian of the System 5.4 Equations of Motion 6 Libration in the Vicinity of the Synchronous State 6.1 Rotating Frame 6.2 Equilibrium State in the Case of a Circular Orbit with No Inclination 6.3 Eigenfrequencies 6.4 Driven Solution 7 Tidal Deformation 7.1 Inertia Matrix and Stokes Coefficients 7.2 Tisserand Frame 7.3 Lagrangian of the Problem 7.4 Love Number 7.5 Constant Deformation 7.6 Maximal Triaxiality 8 Tidal Evolution 8.1 Equations of Motion 8.2 Secular Tidal Torque Out of Spin-Orbit Resonances 8.3 Secular Evolution Out of Spin-Orbit Resonances 8.4 Libration in the Vicinity of the Synchronous Rotation 9 Conclusion References Arnold Diffusion and Nekhoroshev Theory 1 Introduction 2 Arnold's Example 2.1 Existence of KAM Tori 2.2 Semi-analytical (`Melnikov') Approach 2.3 Geometric Approach 3 A Priori Stable Systems—Nekhoroshev Theory 3.1 Nekhoroshev Theory and Exponential Stability 3.2 A Simple Example 3.3 Diffusion in the Web of Resonances 4 Construction of the Nekhoroshev Normal Form: Semi-analytical Estimates 4.1 Construction of the Nekhoroshev Normal Form 4.2 Removal of Deformation Effects 4.3 Modeling the Jumps in the Adiabatic Action Variables References Orbit Determination with the Keplerian Integrals 1 Introduction 2 Linkage with the Keplerian Integrals 2.1 Kepler's Problem and Its First Integrals 2.2 Polynomial Equations for the Linkage 2.3 Consistency of Eqs. (14) 3 An Optimal Property of the Polynomial mathfraku 3.1 A Gröbner Basis for the Ideal I 3.2 Selecting the Solutions 3.3 Numerical Test with Link2 4 Joining Three TSAs 4.1 Straight Line Solutions 4.2 Selecting the Solutions 4.3 Numerical Test with Link3 5 Conclusions and Future Work References Resonant Dynamics of Space Debris 1 Introduction 2 Equations of Motion and Resonances 2.1 Cartesian Equations of Motion 2.2 Hamiltonian Equations of Motion 2.3 Effects of J2 2.4 Classification of the Arguments in the Disturbing Functions 2.5 Dissipative Effects: The Atmospheric Drag 3 Tesseral Resonances in GEO and MEO Regions 3.1 The 1:1 Resonance 3.2 The 2:1 Resonance 4 Tesseral Resonances in the LEO Region 5 Secular Resonances 5.1 Types of Secular Resonances 5.2 Effects of Secular Resonances 6 Conclusions and Perspectives References The Unaccomplished Perfection of Kepler's World 1 Apology 2 Before Kepler 2.1 The Universe of Nicolaus Copernicus (1473–1543) Compared to Claudius Ptolemæus (c. 100–170) 2.2 The Universe of Tycho Brahe (1546–1601) 3 The Discovery of Kepler's Laws 3.1 Ad Imitationem Veterum 3.2 Towards the ``Law of Areas'' 3.3 Kepler's Principle Revisited 3.4 The Elliptic Orbit of Mars, the Inobservabile Sidus 3.5 Kepler's Equation 3.6 The Perfection of the World 4 The ``Tabulæ Rudolphinæ'' 4.1 The Need for ``Secular Equations'' 4.2 After Kepler 4.3 A Final, Strictly Personal Remark References