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Mathematical Aspects of Classical and Celestial Mechanics

Third Edition

verfasst von: Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt

Verlag: Springer Berlin Heidelberg

Buchreihe : Encyclopaedia of Mathematical Sciences

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

In this book we describe the basic principles, problems, and methods of cl- sical mechanics. Our main attention is devoted to the mathematical side of the subject. Although the physical background of the models considered here and the applied aspects of the phenomena studied in this book are explored to a considerably lesser extent, we have tried to set forth ?rst and foremost the “working” apparatus of classical mechanics. This apparatus is contained mainly in Chapters 1, 3, 5, 6, and 8. Chapter 1 is devoted to the basic mathematical models of classical - chanics that are usually used for describing the motion of real mechanical systems. Special attention is given to the study of motion with constraints and to the problems of realization of constraints in dynamics. In Chapter 3 we discuss symmetry groups of mechanical systems and the corresponding conservation laws. We also expound various aspects of ord- reduction theory for systems with symmetries, which is often used in appli- tions. Chapter 4 is devoted to variational principles and methods of classical mechanics. They allow one, in particular, to obtain non-trivial results on the existence of periodic trajectories. Special attention is given to the case where the region of possible motion has a non-empty boundary. Applications of the variational methods to the theory of stability of motion are indicated.

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Inhaltsverzeichnis

Frontmatter

1. Basic Principles of Classical Mechanics

Abstract

For describing the motion of a mechanical system various mathematical models are used based on different “principles” - laws of motion. In this chapter we list the basic objects and principles of classical dynamics. The simplest and most important model of the motion of real bodies is Newtonian mechanics, which describes the motion of a free system of interacting points in threedimensional Euclidean space. In § 1.6 we discuss the suitability of applying Newtonian mechanics when dealing with complicated models of motion.

Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt

2. The n-Body Problem

Abstract

Suppose that two points (r₁, m₁) and (r₂, m₂) interact with each other with potential energy U(|r₁ - r₂|), so that the equations of motion have the form

Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt

3. Symmetry Groups and Order Reduction

Abstract

Let (M, L) be a Lagrangian system and v a smooth field on M. The field v gives rise to the one-parameter group g of diffeomorphisms g^α : M → M defined by the differential equation

Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt

4. Variational Principles and Methods

Abstract

One of the fundamental objects of classical mechanics is a Lagrangian system - a pair (M, L), where M is a smooth manifold (the configuration space of the mechanical system), and L a smooth function on the tangent bundle TM (the Lagrange function or Lagrangian).

Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt

5. Integrable Systems and Integration Methods

Abstract

Di.erential equations, including Hamilton’s equations, are customarily divided into integrable and non-integrable ones. “When, however, one attempts to formulate a precise de.nition of integrability, many possibilities appear, each with a certain intrinsic theoretic interest.”¹

Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt

6. Perturbation Theory for Integrable Systems

Abstract

In nature we often encounter systems that differ from integrable ones by small perturbations. For example, the problem of the motion of the planets around the Sun can be regarded as a perturbation of the integrable problem of the motion of non-interacting points around a fixed attracting centre. Methods developed for studying such problems are united under the general name perturbation theory. These methods are usually simple and effective.

Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt

7. Non-Integrable Systems

Abstract

A common feature of various approaches to the problem of integrating Hamiltonian systems considered in Chapter 5 is the existence of sufficiently many independent first integrals - “conservation laws”.

Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt

8. Theory of Small Oscillations

Abstract

The study of the oscillations of a system in a neighbourhood of an equilibrium position or a periodic motion usually begins with linearization. The linearized system can be integrated. After this is done, the main properties of the oscillations in the original system can often be determined by using the theory of Poincaré-Birkhoff normal forms.

Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt

9. Tensor Invariants of Equations of Dynamics

Abstract

A tensor invariant is a tensor field in the phase space that is invariant under the action of the phase flow. The most frequently occurring invariants are first integrals, symmetry fields, invariant differential forms. Closely related to them there are objects of more general nature: frozen-in direction fields and integral invariants. Tensor invariants play an essential role both in the theory of exact integration of equations of dynamics and in their qualitative analysis.

Vladimir I. Arnold, Valery V. Kozlov, Anatoly I. Neishtadt

Backmatter

Titel: Mathematical Aspects of Classical and Celestial Mechanics
verfasst von: Vladimir I. Arnold
Valery V. Kozlov
Anatoly I. Neishtadt
Copyright-Jahr: 2006
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-48926-9
Print ISBN: 978-3-540-28246-4
DOI: https://doi.org/10.1007/978-3-540-48926-9

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