Mathematical models : mechanical vibrations, population dynamics, and traffic flow : an introduction to applied mathematics. - "This SIAM edition is an unabridged republication of the work first published by Prentice-Hall, Inc., Englewood Cliffs, New Jers

Mathematics is a grand subject in the way it can be applied to various problems in science and engineering. To use mathematics, one needs to understand the physical context. The author uses mathematical techniques along with observations and experiments to give an in-depth look at models for mechanical vibrations, population dynamics, and traffic flow. Equal emphasis is placed on the mathematical formulation of the problem and the interpretation of the results. In the sections on mechanical vibrations and population dynamics, the author emphasizes the nonlinear aspects of ordinary differential equations and develops the concepts of equilibrium solutions and their stability. He introduces phase plane methods for the nonlinear pendulum and for predator-prey and competing species models. Title Table of Contents Foreword Preface Mechanical Vibrations 1. Introduction to Mathematical Models in the Physical Sciences 2. Newton's Law 3. Newton's Law as Applied to a Sping-Mass System 4. Gravity Exercises 5. Oscillation of a Spring-Mass System Exercises 6. Dimension and Units 7. Qualitative and Quantitative Behavior of a Spring-Mass System Exercises 8. Initial Value Problem Exercises 9. A Two-Mass Oscillator Exercises 10. Friction Exercises 11. Oscillations of a Damped System Exercises 12. Underdamped Oscillations Exercises 13. Overdamped and Critically Damped Oscillations Exercises 14. A Pendulum Exercises 15. How Small is Small? Exercises 16. A Dimensionless Time Variable Exercises 17. Nonlinear Frictionless Systems 18. Linearized Stability Analysis of an Equilibrium Solution Exercises 19. Conservation of Energy Exercises 20. Energy Curves 21. Phase Plane of a Linear Oscillator Exrcises 22. Phase Plane of a Nonlinear Pendulum Exercises 23. Can a Pendulum Stop? Exercises 24. What Happens If a Pendulum Is Pushed Too Hard? Exercises 25. Period of a Nonlinear Pendulum Exercises 26. Nonlinear Oscillations with Damping Exercises 27. Equilibrium Positions and Linearized Stability Exercises 28. Nonlinear Pendulum with Damping Exercises 29. Further Readings in Mechanical Vibrations Population Dynamics-Mathematical Ecology Traffic Flow Index The author uses mathematical techniques along with observations and experiments to give an in-depth look at models for mechanical vibrations, population dynamics, and traffic flow. Equal emphasis is placed on the mathematical formulation of the problem and the interpretation of the results. In the sections on mechanical vibrations and population dynamics, the author emphasizes the nonlinear aspects of ordinary differential equations and develops the concepts of equilibrium solutions and their stability. He introduces phase plane methods for the nonlinear pendulum and for predator-prey and competing species models. Haberman develops the method of characteristics to analyze the nonlinear partial differential equations that describe traffic flow. Fan-shaped characteristics describe the traffic situation that occurs when a traffic light turns green and shock waves describe the effects of a red light or traffic accident. Although it was written over 20 years ago, this book is still relevant. It is intended as an introduction to applied mathematics, but can be used for undergraduate courses in mathematical modeling or nonlinear dynamical systems or to supplement courses in ordinary or partial differential equations Science attempts to establish an understanding of all types of phenomena.