The Genesis of Simulation in Dynamics: Pursuing the Fermi-Pasta-Ulam Problem

ISBN-13: 9780387982373 / Angielski / Miękka / 1997 / 176 str.

Thomas P. Weissert

Opis

We hear a lot about dynamical systems theory these days-"chaos dy namics" is the popular idiom-but very little of this abstract discipline is taught in undergraduate or even in graduate school, unless that happens to be one's own field. People may have heard some of the terms, but they may not understand what a dynamicist does. Just what does a dynamicist do? Although I've been through the training, I spend much more time talk ing and thinking about dynamics than actually doing it, which is a circum stance that affords me an interesting perspective. Some time after defending my dissertation, when I stopped back by my alma mater on a visit through Boulder, I ran into one of my advisors, a practicing dynamicist, and he was excited to show me his new workstation and its resident software. Hope fully, by relating what I experienced then, I may in part help to answer my opening question. He sat down in front of a very large screen, at least 19 inches across, so closely, that it filled his field of view completely. At his fingertips were the usual mouse and keyboard. He fired up the software and chose a model; I think we were out to explore "just the standard map. " As the program began, the screen remained black, except for a small blinking cursor at the middle."

Spis treści

I: History.- 1. The FPU Model and Simulation: “A Little Discovery”.- 1.1. Development.- 1.2. Dynamics to Statistical Mechanics.- 1.3. Surfaces of Constraint.- 1.4. Global Versus Local Analysis.- 1.5. Simulation.- 1.6. Loading the Nonlinear String.- 1.7. Modal Representation.- 1.8. Model Considerations.- 1.9. Results.- 1.10. Discussion Post Hoc.- 2. The FPU Research Program: Echoes on a String.- 2.1. The Threads of a Research Program.- 2.2. The Nonlinear Discrete Lattice.- 2.3. Ford, 1961.- 2.4. Jackson, 1963.- 2.5. Ford and Waters, 1963.- 2.6. The Continuous String.- 2.7. In the Continuous Limit.- 2.8. Discreteness as Viscosity.- 2.9. The First Soliton Paper.- 3. The Kolmogorov—Arnold—Moser Theorem: “Here Comes the Surprise”.- 3.1. A Brief History of Dynamics.- 3.2. The Fundamental Problem of Dynamics.- 3.3. The Small Divisors Problem.- 3.4. Poincaré to Kolmogorov.- 3.5. The Conjecture.- 3.6. Beyond the Blaze.- 3.7. The Hénon and Heiles Simulation, 1964.- 4. Research Threads Come Together: Harmonic Convergence.- 4.1. The Story Continues.- 4.2. Izrailev and Chirikov, 1966.- 4.3. Zabusky and Deem, 1967.- 4.4. Walker and Ford, 1969: Physical Review.- 4.5. Ford and Lunsford, 1970.- 4.6. Lunsford and Ford, 1972.- 4.7. The Toda Lattice Is Integrable.- II: Philosophy.- 5. Steps to an Epistemology of Simulation.- 5.1. Introduction.- 5.2. Hierarchy of Modeling.- 5.3. Historical Significance.- 5.4. Experiment.- 5.5. Epistemology.- 5.6. Preconceptions.- 5.7. Strategies for Belief and Pursuit.- 5.8. Case Study I: Fermi—Pasta—Ulam.- 5.9. Case Study II: Hénon and Heiles.- 5.10. Methodology.- 5.11. Irreversibility.- 5.12. Proof.- 5.13. Proof and Simulation.- Append.- A. Hamiltonian Dynamics: Language of Abstraction.- A.1. Topology and Phase-Space Trajectories.- A.2. Canonical Transformations.- A.3. Transforming the Unperturbed String.- A.4. Cyclic Coordinates.- A.5. Liouville Integrability.- A.6. The Action-Angle Variables.- A.7. Dynamics on a Torus.- A.8. Commensurability: Two Types of Motion.- A.9. Digital Representation.- A.10.Physical Reality and the Continuum.- A.11.Perturbing the String.- References.