Mathematical models : mechanical vibrations, population dynamics, and traffic flow : an introduction to applied mathematics

Foreword Preface to the classics edition Preface Part I. Mechanical Vibrations: Introduction to Mathematical Models in the Physical Sciences Newton's Law Newton's Law as Applied to a Spring-Mass System Gravity Oscillation of a Spring-Mass System Dimensions and Units Qualitative and Quantitative Behavior of a Spring-Mass System Initial Value Problem A Two-Mass Oscillator Friction Oscillations of a Damped System Underdamped Oscillations Overdamped and Critically Damped Oscillations A Pendulum How Small is Small? A Dimensionless Time Variable Nonlinear Frictionless Systems Linearized Stability Analysis of an Equilibrium Solution Conservation of Energy Energy Curves Phase Plane of a Linear Oscillator Phase Plane of a Nonlinear Pendulum Can a Pendulum Stop? What Happens if a Pendulum is Pushed Too Hard? Period of a Nonlinear Pendulum Nonlinear Oscillations with Damping Equilibrium Positions and Linearized Stability Nonlinear Pendulum with Damping Further Readings in Mechanical Vibrations Part II. Population Dynamics-Mathematical Ecology. Introduction to Mathematical Models in Biology Population Models A Discrete One-Species Model Constant Coefficient First-Order Difference Equations Exponential Growth Discrete Once-Species Models with an Age Distribution Stochastic Birth Processes Density-Dependent Growth Phase Plane Solution of the Logistic Equation Explicit Solution of the Logistic Equation Growth Models with Time Delays Linear Constant Coefficient Difference Equations Destabilizing Influence of Delays Introduction to Two-Species Models Phase Plane, Equilibrium, and linearization System of Two Constant Coefficient First-Order Differential Equations, Stability of Two-Species Equilibrium Populations Phase Plane of Linear Systems Predator-Prey Models Derivation of the Lotka-Volterra Equations Qualitative Solution of the Lotka- Volterra Equations Average Populations of Predators and Preys Man's Influence on Predator-Prey Ecosystems Limitations of the Lotka-Volterra Equation Two Competing Species Further Reading in Mathematical Ecology Part III. Traffic Flow. Introduction to Traffic Flow Automobile Velocities and a Velocity Field Traffic Flow and Traffic Density Flow Equals Density Times Velocity Conservation of the Number of Cars A Velocity-Density Relationship Experimental Observations Traffic Flow Steady-State Car-Following Models Partial Differential Equations Linearization A Linear Partial Differential Equation Traffic Density Waves An Interpretation of Traffic Waves A Nearly Uniform Traffic Flow Example Nonuniform Traffic - The Method of Characteristics After a Traffic Light Turns Green A Linear Velocity-Density Relationship An Example Wave Propagation of Automobile Brake Lights Congestion Ahead Discontinuous Traffic Uniform Traffic Stopped by a Red Light A Stationary Shock Wave The Earliest Shock Validity of Linearization Effect of a Red Light or an Accident Exits and Entrances Constantly Entering Cars A Highway Entrance Further reading in traffic flow Index.