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Über dieses Buch

This original monograph aims to explore the dynamics in the particular but very important and significant case of quasi-integrable Hamiltonian systems, or integrable systems slightly perturbed by other forces. With both analytic and numerical methods, the book studies several of these systems—including for example the hydrogen atom or the solar system, with the associated Arnold web—through modern tools such as the frequency modified fourier transform, wavelets, and the frequency modulation indicator. Meanwhile, it draws heavily on the more standard KAM and Nekhoroshev theorems.

Geography of Order and Chaos in Mechanics will be a valuable resource for professional researchers and certain advanced undergraduate students in mathematics and physics, but mostly will be an exceptional reference for Ph.D. students with an interest in perturbation theory.



Chapter 1. Introductory Survey

Order and chaos, invariant tori, KAM theory, resonances, Arnold web, diffusion …, these are “keywords” in the theory of dynamical systems. But for mathematicians who are not directly involved in this area they may sound a bit vague. To grasp what they are about, consider the following question: Given a conservative mechanical system, thus without dissipative forces, what is its ultimate fate? Without further information the answer probably cannot be given in general. However, if we restrict ourselves to the special, but very important, case of a

slightly perturbed integrable system

, we can claim that crucial progresses have been achieved. An example is our solar system, in which the perturbations are the interactions between planets. In this book we will suggest how to guess an answer, and on the way all these keywords will come into play.

Bruno Cordani

Chapter 2. Analytical Mechanics and Integrable Systems

Analytical mechanics is the basic tool we will utilize through the whole book. The aim of this chapter is to succinctly introduce and define some basic concepts, such as Hamilton equations, symplectic (i.e., canonical) transformations, symmetry and reduction, integrable systems, and action-angle variables. This will be done in the third section of the chapter. In the first two sections we recall the main ideas of differential geometry and Lie groups, which are the natural language of analytical mechanics.

Bruno Cordani

Chapter 3. Perturbation Theory

Given the Hamiltonian



of a completely integrable system, the

perturbed problem

is described by the Hamiltonian







, where


is a function whose numerical value is of the same order of



, and


<< 1. The perturbed problem thus differs slightly from the unperturbed one, but unfortunately the same is not true for the solution: a small perturbation can give rise to


effects, i.e., to a slow but progressive wandering from the unperturbed, and known for infinite time, solution.

Bruno Cordani

Chapter 4. Numerical Tools I: ODE Integration

In the following chapters we will study some concrete quasi-integrable Hamiltonian systems. Their analytical approximate normal form will be deduced and compared with the “true” motion, obtained from numerical integration. Moreover, the geography of the resonances will be detected thanks to the tools described in the next chapter, which again require a numerical integration.

Bruno Cordani

Chapter 5. Numerical Tools II: Detecting Order, Chaos, and Resonances

Detecting and studying how order and chaos are distributed in quasi-integrable Hamiltonian systems and, in particular, exploring the geography of the resonances is surely a major task in the dynamical systems area.

Bruno Cordani

Chapter 6. The Kepler Problem

In this chapter, we will study the group-geometrical structure of the Kepler problem and point out how this structure also turns out to be useful in the study of the perturbed case.

Bruno Cordani

Chapter 7. The KEPLER Program

In this chapter we will describe how to use the KEPLER program, trying to be self-contained and without referring excessively to the underlying mathematical structure. The command statements are gathered in four windows, which one selects in the pop-up menu in the left-top corner. See the Appendix A for snapshots of the MATLAB programs in the CD.

Bruno Cordani

Chapter 8. Some Perturbed Keplerian Systems

We now possess all the necessary tools to study some interesting Keplerian perturbed systems: the Stark–Quadratic–Zeeman problem, the circular restricted three–body problem, and the motion of a satellite around an oblate primary. In all three cases we will first find the normal integrable form, comparing the relative motion with the “true” one obtained by numerical integration. Several concrete examples will be given, showing in general a very good agreement between the analytical and numerical results. What the normal integrable form is not able to show is the presence of resonances, which are just the indicators of nonintegrability. Then, with the Frequency Modulation Indicator (FMI) we will analyze how order, chaos, and resonances are localized in action space, thus completing the study of the three quasi-integrable systems.

Bruno Cordani

Chapter 9. The Multi-Body Gravitational Problem

Deducing the motion of bodies interacting gravitationally is probably the most important mechanical problem but also the most difficult. The threebody problem is not integrable, even if the masses are very small but of comparable size, and this fact prevents in general the use of perturbative methods.

Bruno Cordani

Chapter 10. Final Remarks and Perspectives

The focus of this book is on theoretical and numerical investigations of quasi-integrable Hamiltonian systems. The goal is to understand the qualitative and quantitative features of the relative dynamics, even for systems with three or more degrees of freedom. By combining analytical, numerical, and geometrical methods, in effect one can also grasp the geography of the resonances, and hence the distribution of order and chaos.

Bruno Cordani


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