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1989 | Buch | 2. Auflage

Experimental facts Kapitel lesen Erstes Kapitel lesen

Mathematical Methods of Classical Mechanics

verfasst von: V. I. Arnold

Verlag: Springer New York

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

In this text, the author constructs the mathematical apparatus of classical mechanics from the beginning, examining all the basic problems in dynamics, including the theory of oscillations, the theory of rigid body motion, and the Hamiltonian formalism. This modern approch, based on the theory of the geometry of manifolds, distinguishes iteself from the traditional approach of standard textbooks. Geometrical considerations are emphasized throughout and include phase spaces and flows, vector fields, and Lie groups. The work includes a detailed discussion of qualitative methods of the theory of dynamical systems and of asymptotic methods like perturbation techniques, averaging, and adiabatic invariance.

Inhaltsverzeichnis

Frontmatter

Newtonian Mechanics

Frontmatter
1. Experimental facts
Abstract
In this chapter we write down the basic experimental facts which lie at the foundation of mechanics: Galileo’s principle of relativity and Newton’s differential equation. We examine constraints on the equation of motion imposed by the relativity principle, and we mention some simple examples.
V. I. Arnold
2. Investigation of the equations of motion
Abstract
In most cases (for example, in the three-body problem) we can neither solve the system of differential equations nor completely describe the behavior of the solutions. In this chapter we consider a few simple but important problems for which Newton’s equations can be solved.
V. I. Arnold

Lagrangian Mechanics

Frontmatter
3. Variational principles
Abstract
In this chapter we show that the motions of a newtonian potential system are extremals of a variational principle, “Hamilton’s principle of least action.”
V. I. Arnold
4. Lagrangian mechanics on manifolds
Abstract
In this chapter we introduce the concepts of a differentiable manifold and its tangent bundle. A lagrangian function, given on the tangent bundle, defines a lagrangian “holonomic system” on a manifold. Systems of point masses with holonomic constraints (e.g., a pendulum or a rigid body) are special cases.
V. I. Arnold
5. Oscillations
Abstract
Because linear equations are easy to solve and study, the theory of linear oscillations is the most highly developed area of mechanics. In many nonlinear problems, linearization produces a satisfactory approximate solution. Even when this is not the case, the study of the linear part of a problem is often a first step, to be followed by the study of the relation between motions in a nonlinear system and in its linear model.
V. I. Arnold
6. Rigid bodies
Abstract
In this chapter we study in detail some very special mechanical problems. These problems are traditionally included in a course on classical mechanics, first because they were solved by Euler and Lagrange, and also because we live in three-dimensional euclidean space, so that most of the mechanical systems with a finite number of degrees of freedom which we are likely to encounter consist of rigid bodies.
V. I. Arnold

Hamiltonian Mechanics

Frontmatter
7. Differential forms
Abstract
Exterior differential forms arise when concepts such as the work of a field along a path and the flux of a fluid through a surface are generalized to higher dimensions.
V. I. Arnold
8. Symplectic manifolds
Abstract
A symplectic structure on a manifold is a closed nondegenerate differential 2-form. The phase space of a mechanical system has a natural symplectic structure.
V. I. Arnold
9. Canonical formalism
Abstract
The coordinate point of view will predominate in this chapter. The technique of generating functions for canonical transformations, developed by Hamilton and Jacobi, is the most powerful method available for integrating the differential equations of dynamics. In addition to this technique, the chapter contains an “odd-dimensional” approach to hamiltonian phase flows.
V. I. Arnold
10. Introduction to perturbation theory
Abstract
Perturbation theory consists of a very useful collection of methods for finding approximate solutions of “perturbed” problems which are close to completely solvable “unperturbed” problems. These methods can be easily justified if we are investigating motion over a small interval of time. Relatively little is known about how far we can trust the conclusions of perturbation theory in investigating motion over large or infinite intervals of time.
V. I. Arnold
Backmatter
Metadaten
Titel
Mathematical Methods of Classical Mechanics
verfasst von
V. I. Arnold
Copyright-Jahr
1989
Verlag
Springer New York
Electronic ISBN
978-1-4757-2063-1
Print ISBN
978-1-4419-3087-3
DOI
https://doi.org/10.1007/978-1-4757-2063-1