Overview

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.

Full Product Details

Author:   Giulio Baù ,  Sara Di Ruzza ,  Rocío Isabel Páez ,  Tiziano Penati
Publisher:   Springer International Publishing AG
Imprint:   Springer International Publishing AG
Edition:   1st ed. 2022
Volume:   399
Weight:   0.692kg
ISBN:  

9783031131141

ISBN 10:   3031131142
Pages:   300
Publication Date:   10 February 2023
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.

Table of Contents

1) The contribution by Ugo Locatelli focuses on the explicit construction of invariant tori exploiting suitable Hamiltonian normal forms, with particular emphasis on applications to Celestial Mechanics. First, the algorithm constructing the Kolmogorov normal form is described in detail. Then the extension to lowerdimensional elliptic tori is provided. Both algorithms are then combined so as to accurately approximate the long-term dynamics of the HD 4732 extrasolar system. 2) The contribution by Gabriella Pinzari presents a review of some results of their research group, regarding the relation between some particular motions of the Three–Body problem (3BP) and the motions of the so–called Euler (or two–centre) problem, which is integrable. For the analysis of such relation, the authors make use of two novel results: on one hand, the two–centre problem (2CP) bears a remarkable property, here called renormalizable integrability, which states that the simple averaged potential of the 2CP and the Euler integral are one function of the other; on the other hand, the motions of the Euler integral are at least qualitatively explicit, and the averaged Newtonian potential is a prominent part of the 3BP Hamiltonian.3) The contribution by Alessandra Celletti deals with dissipative systems, a key topic in Celestial Mechanics. In particular the problem of the existence of invariant tori for conformally symplectic systems, which have the property to transform the symplectic form into a multiple of itself, is studied. Two different models are presented: a discrete system known as the standard map and a continuous system known as the spin–orbit problem. In both cases, both the conservative and dissipative versions are considered, in order to highlight the differences between the symplectic and conformally symplectic dynamics. 4) The contribution by Gwenael Boué provides basic tools to understand the rotational dynamics of extended bodies which could be either rigid or deformable by tides. The problem is described in a Lagrangian formalism as it was developed by H. Poincar´e in 1901. The case of rigid body is also presented in the corresponding Hamiltonian formalism. The mathematical description of the deformation of the extended body follows the approach used by C. Ragazzo and L. Ruiz in their two papers of 2015, 2017 due to the compactness and clarity of their formalism. In this Chapter, many applications to the rotation and the libration of celestial bodies are illustrated. 5) The contribution by Christos Efthymiopoulos concerns the phenomenon of Arnold diffusion. The authors begin with the famous example given by Arnold to describe the slow diffusion taking place in the action–space in Hamiltonian nonlinear dynamical systems with three or more degrees of freedom. The text introduces basic concepts related to our current understanding of the mechanisms leading to Arnold diffusion and at the same time performed a qualitative investigation of the phenomenon of Arnold diffusion with many examples. The problem of the speed of diffusion is investigated using methods of perturbation theory, with particular emphasis on Nekhoroshevs theorem. 6) The contribution by Giovanni F. Gronchi deals with the problem of initial orbit determination of a solar system body, i.e. the determination of a preliminary orbit from observations collected for example by a telescope. The two methods that are presented, named Link2 and Link3, try to link together two and three, respectively, short arcs of optical observations of the same object which can possibly be quite far apart in time. The conservation laws of Kepler’s problem are used to derive a polynomial equation of degree 9 (Link2) and 8 (Link3) for the distance of the body from the observer. 7) The contribution by Catalin Gales provides an overview of some recent developments in the study of dynamics of space debris with focus on specific resonant interactions, in particular those related to the tesseral resonances. After an historical introduction to the topic, the authors provide a long–term picture of the dynamics that can help in the modeling and mitigation of the space debris problem, both in term of Cartesian coordinates and in the Hamiltonian framework. Some key terms in the perturbing functions are classified, while the effect of the dissipative force of the atmospheric drag is also formulated. 8) The contribution by Massimiliano Guzzo presents the use of the Fast Lyapunov Indicators (FLI) in the Three–Body problem, with the eventual aim of computing transit orbits. The FLI belong to the family of the finite time indicators, which are able to extract the information of the solutions of the variational equations on short time intervals. First, the FLI are applied to two model problems: the standard map and the double gyre. Then, it is described a modification of the FLI which was originally introduced to improve the computation of stable and unstable manifolds in model systems and the Three–Body problem. 9) The contribution by Antonio Giorgilli provides an answer to a simple question, how did Kepler discover his celebrated laws?. The answer however is not that simple and the present paper guides the reader by a short walk along the main works of Kepler, notably the Astronomia Nova, trying to follow his search of the perfection of the World till the discovery of his celebrated laws. At the end of the road, the consciousness that the finish line had not yet been reached.

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Author Information

"Giulio Baù is a Associate Professor at the Department of Mathematics of the University of Pisa. His research activity deals with the development of new orbit propagation and determination methods for small celestial bodies (in particular asteroids and space debris), the study of their dynamics, and regularizations techniques in the $N$-body problem. Sara Di Ruzza is a researcher of Mathematical Physics at the Department of Mathematics at the University of Palermo. Her main field of interest is celestial mechanics. She worked on the spin-orbit problem, on some particular cases of the three-body problem both from an analytical and numerical point of view. She applied theory to real scenarios such as asteroid motion and space missions. The latest works are focused on the occurrence of chaos in the planar three-body problem. Rocío Isabel Páez is a senior postdoctoral researcher at the University College Cork in Ireland. She received her Ph.D. in Mathematics from the University of Rome ""Tor Vergata"" in 2016. Since then, she has held academic positions at the University of Rome ""Tor Vergata"", the Academy of Athens and the University of Padova. Her research area is in Applied Mathematics, with a focus on the application of advanced methods of perturbation theory and numerical simulations in dynamical astronomy. Tiziano Penati is Associate Professor of Mathematical Physics at the Department of Mathematics, University of Milan. His main field of interest is Hamiltonian perturbation theory, with special attention to normal form techniques. His work focuses on the investigation of time-periodic localized structures and metastability phenomena in Hamiltonian lattices, such as the Fermi-Pasta-Ulam-Tsingou, discrete Klein-Gordon or discrete Nonlinear Schroedinger models. Marco Sansottera is a Researcher of Mathematical Physics at the Department of Mathematics, University of Milan. His research activity is mainly focused on dynamical systems and celestial mechanics. In particular, he studied the stability properties of planetary systems, investigating the dynamics in the neighborhood of some invariant objects, such as maximal dimension KAM tori, equilibrium points and lower dimensional elliptic tori."

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