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2018 | Buch

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Mathematical Physics: Classical Mechanics

verfasst von: Prof. Dr. Andreas Knauf

Verlag: Springer Berlin Heidelberg

Buchreihe : UNITEXT

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

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Über dieses Buch

As a limit theory of quantum mechanics, classical dynamics comprises a large variety of phenomena, from computable (integrable) to chaotic (mixing) behavior. This book presents the KAM (Kolmogorov-Arnold-Moser) theory and asymptotic completeness in classical scattering. Including a wealth of fascinating examples in physics, it offers not only an excellent selection of basic topics, but also an introduction to a number of current areas of research in the field of classical mechanics. Thanks to the didactic structure and concise appendices, the presentation is self-contained and requires only knowledge of the basic courses in mathematics.

The book addresses the needs of graduate and senior undergraduate students in mathematics and physics, and of researchers interested in approaching classical mechanics from a modern point of view.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

“And he was told to tell the truth, otherwise one would have recourse to torture. [He replied:] I am here to obey, but I have not held this opinion after the determination was made, as I said.”

Andreas Knauf

Chapter 2. Dynamical Systems

Abstract

Dynamics can be viewed under different aspects, and with a variety of additional structures; accordingly, there are different definitions of dynamical systems.

Andreas Knauf

Chapter 3. Ordinary Differential Equations

Abstract

Differential equations are as varied as the phenomena of nature described by them.

Andreas Knauf

Chapter 4. Linear Dynamics

Abstract

A particularly important class of differential equations consists of the linear ones; by Theorem 3.29, we may assume that we have a system of first order.

Andreas Knauf

Chapter 5. Classification of Linear Flows

Abstract

We know the flow on the phase space \({\mathbb R}^n\) that is generated by a linear differential equation.

Andreas Knauf

Chapter 6. Hamiltonian Equations and Symplectic Group

Abstract

“And he was told to tell the truth, otherwise one would have recourse to torture. [He replied:] I am here to obey, but I have not held this opinion after the determination was made, as I said”.

Andreas Knauf

Chapter 7. Stability Theory

Abstract

The notions of stability introduced in Definition 2.21 have so far mainly been applied to linear systems. As we analyze nonlinear systems in the present chapter, asymptotic stability will turn out to be robust. An equilibrium is asymptotically stable if the linearization of the vector field at this point is asymptotically stable (Theorem 7.6).

Andreas Knauf

Chapter 8. Variational Principles

Abstract

The Lagrange equations arising from a Lagrange function are second order differential equations. With this formalism, it is possible to realize constraints (such as occur in applications when objects are affixed to an axle or connected by rods) by simply restricting the Lagrange function.

Andreas Knauf

Chapter 9. Ergodic Theory

Abstract

The studies about the foundations of geometry suggest to us the problem of treating, according to this paradigm, those disciplines of physics in which mathematics is already today playing a prominent role; primarily these are probability and mechanics’.

Andreas Knauf

Chapter 10. Symplectic Geometry

Abstract

When one leaves the special case of linear Hamiltonian differential equations behind, the symplectic bilinear form studied in Chapter 6 becomes a symplectic form, and Lagrangian subspaces become Lagrangian submanifolds.

Andreas Knauf

Chapter 11. Motion in a Potential

Abstract

This class of Hamiltonian motion is the most important one.

Andreas Knauf

Chapter 12. Scattering Theory

Abstract

The major part of our knowledge about molecules, atoms, and elementary particles is obtained by scattering experiments, in which particles of a specific initial velocity collide with each other or with a fixed target.

Andreas Knauf

Chapter 13. Integrable Systems and Symmetries

Abstract

“When, however, one attempts to formulate a precise definition of integrability, many possibilities appear, each with a certain intrinsic theoretic interest.”

Andreas Knauf

Chapter 14. Rigid and Non-Rigid Bodies

Abstract

Until now, we have mostly studied the motion of point masses.

Andreas Knauf

Chapter 15. Perturbation Theory

Andreas Knauf

Chapter 16. Relativistic Mechanics

Abstract

The principle of relativity states that in the laws of physics, only relative velocities occur, so that it is in particular meaningless to postulate a state of absolute rest.

Andreas Knauf

Chapter 17. Symplectic Topology

Abstract

In the theory of dynamical systems, topological methods are often employed when the dynamics is too complicated to answer questions like the one about the existence of periodic orbits directly.

Andreas Knauf

Backmatter

Titel: Mathematical Physics: Classical Mechanics
verfasst von: Prof. Dr. Andreas Knauf
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-662-55774-7
Print ISBN: 978-3-662-55772-3
DOI: https://doi.org/10.1007/978-3-662-55774-7