Overview

There are many ways of introducing the concept of probability in classical, deterministic physics. This volume is concerned with one approach, known as 'the method of arbitrary functions', which was first considered by Poincare. Essentially, the method proceeds by associating some uncertainty to our knowledge of both the initial conditions and the values of the physical constants that characterize the evolution of a physical system. By modeling this uncertainty by a probability density distribution, it is then possible to analyze how the state of the system evolves through time. This approach may be applied to a wide variety of classical problems and the author considers here examples as diverse as bouncing balls, simple and coupled harmonic oscillators, integrable systems (such as spinning tops), planetary motion, and billiards. An important aspect of this account is to study the speed of convergence for solutions in order to determine the practical relevance of the method of arbitrary functions for specific examples. Consequently, both new results on convergence, and tractable upper bounds are derived and applied.

Full Product Details

Author:   Eduardo M.R.A. Engel
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   Softcover reprint of the original 1st ed. 1992
Volume:   71
Dimensions:   Width: 17.00cm , Height: 0.90cm , Length: 24.40cm
Weight:   0.322kg
ISBN:  

9780387977409

ISBN 10:   0387977406
Pages:   155
Publication Date:   21 February 1992
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Out of stock  
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Table of Contents

1 Introduction.- 1.1 The Simple Harmonic Oscillator.- 1.2 Philosophical Interpretations.- 1.3 Coupled Harmonic Oscillators.- 1.4 Mathematical Results.- 1.5 Calculating Rates of Convergence.- 1.6 Hopf's Approach.- 1.7 Physical and Statistical Independence.- 1.8 Statistical Regularity of a Dynamical System.- 1.9 More Applications.- 2 Preliminaries.- 2.1 Basic Notation.- 2.2 Weak-star Convergence.- 2.3 Variation Distance.- 2.4 Sup Distance.- 2.5 Some Concepts from Number Theory.- 3 One Dimensional Case.- 3.1 Mathematical Results.- 3.1.1 Weak-star Convergence.- 3.1.2 Bounds on the Rate of Convergence.- 3.1.3 Exact Rates of Convergence.- 3.1.4 Fastest Rate of Convergence.- 3.2 Applications.- 3.2.1 A Bouncing Ball.- 3.2.2 Coin Tossing.- 3.2.3 Throwing a Dart at a Wall.- 3.2.4 Poincare's Roulette Argument.- 3.2.5 Poincare's Law of Small Planets.- 3.2.6 An Example from the Dynamical Systems Literature.- 4 Higher Dimensions.- 4.1 Mathematical Results.- 4.1.1 Weak-star Convergence.- 4.1.2 Bounds on the Rate of Convergence.- 4.1.3 Exact Rates of Convergence.- 4.2 Applications.- 4.2.1 Lagrange's Top and Integrable Systems.- 4.2.2 Coupled Harmonic Oscillators.- 4.2.3 Billiards.- 4.2.4 Gas Molecules in a Room.- 4.2.5 Random Number Generators.- 4.2.6 Repeated Observations.- 5 Hopf's Approach.- 5.1 Force as a Function of Only Velocity: One Dimensional case.- 5.2 Force as a Function of Only Velocity: Higher Dimensions.- 5.3 The Force also Depends on the Position.- 5.4 Statistical Regularity of a Dynamical System.- 5.5 Physical and Statistical Independence.- 5.6 The Method of Arbitrary Functions and Ergodic Theory.- 5.7 Partial Statistical Regularity.- 6 Non Diagonal Case.- 6.1 Mathematical Results.- 6.1.1 Convergence in the Variation Distance.- 6.1.2 Weak-star Convergence.- 6.1.3 Rates of Convergence.- 6.2 Linear Differential Equations.- 6.3 Automorphisms of the n-dimensional Torus.- References.

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