This book introduces some aspects of the development of the modern theory of dynamics and simulation to a wide audience of scientifically literate readers. Unlike some other general texts on chaos theory and dynamical systems theory, this book follows the work on a specific problem at the very beginning of the modern era of dynamics, from its inception in 1954 through the early 1970s. It discusses such problems as the nonlinear oscillator simulation carried out by Fermi, Pasta and Ulam at Los Alamos in the 1940s, the seminal discoveries by Lorentz at MIT in the early 1950s, the mathematical…mehr
This book introduces some aspects of the development of the modern theory of dynamics and simulation to a wide audience of scientifically literate readers. Unlike some other general texts on chaos theory and dynamical systems theory, this book follows the work on a specific problem at the very beginning of the modern era of dynamics, from its inception in 1954 through the early 1970s. It discusses such problems as the nonlinear oscillator simulation carried out by Fermi, Pasta and Ulam at Los Alamos in the 1940s, the seminal discoveries by Lorentz at MIT in the early 1950s, the mathematical rediscovery of solitons in the late 1950s and the general problems of computability discussed by Kolmorogov, Arnold and Moser, by Ford, and by many others. In following these developments, one can see the initial development of many of the new and now standard techniques of nonlinear modeling and numerical simulation. No other text focuses so tightly and covers so completely one specific, pernicious problem at the heart of dynamics.
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.
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.
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