Table of Contents

I Mathematical Methods for Differential Equations.- 1 Models and Differential Equations.- 1.1 Introduction.- 1.2 Mathematical Models and Computation.- 1.3 Examples of Mathematical Models.- 1.4 Validation, Determinism, and Stochasticity.- 2 Models and Mathematical Problems.- 2.1 Introduction.- 2.2 Classification of Models.- 2.3 Statement of Problems.- 2.4 Solution of Initial-Value Problems.- 2.5 Representation of the Dynamic Response.- 2.6 On the Solution of Boundary-Value Problems.- 2.7 Problems.- 3 Stability and Perturbation Methods.- 3.1 Introduction.- 3.2 Stability Definitions.- 3.3 Linear Stability Methods.- 3.4 Nonlinear Stability.- 3.5 Regular Perturbation Methods.- 3.6 Problems.- II Mathematical Methods of Classical Mechanics.- 4 Newtonian Dynamics.- 4.1 Introduction.- 4.2 Principles of Newtonian Mechanics.- 4.3 Balance Laws for Systems of Point Masses.- 4.4 Active and Reactive Forces.- 4.4.1 Constraints and reactive forces.- 4.4.2 Active forces and force fields.- 4.5 Applications.- 4.5.1 Dynamics of simple pendulum.- 4.5.2 Particle subject to a central force.- 4.5.3 Heavy particle falling in air.- 4.5.4 Three-point masses subject to elastic forces.- 4.6 Problems.- 5 Rigid Body Dynamics.- 5.1 Introduction.- 5.2 Rigid Body Models.- 5.3 Active and Reactive Forces in Rigid Body Dynamics.- 5.4 Constrained Rigid Body Models.- 5.5 Articulated Systems.- 5.6 Applications.- 5.6.1 Rigid body model of a vehicle and plane dynamics.- 5.6.2 Compound pendulum.- 5.6.3 Uniform rotations.- 5.6.4 Free rotations of a gyroscope.- 5.6.5 Ball on an inclined plane.- 5.7 Problems.- 6 Energy Methods and Lagrangian Mechanics.- 6.1 Introduction.- 6.2 Elementary and Virtual Work.- 6.3 Energy Theorems.- 6.4 The Method of Lagrange Equations.- 6.5 Potential and First Integrals.- 6.6 Energy Methods and Stability.- 6.7 Applications.- 6.7.1 Three body articulated system.- 6.7.2 Stability of Duffing’s model.- 6.7.3 Free rotations or Poinsot’s motion.- 6.7.4 Heavy gyroscope.- 6.7.5 The rolling coin.- 6.8 Problems.- III Bifurcations, Chaotic Dynamics, Stochastic Models, and Discretization of Continuous Models.- 7 Deterministic and Stochastic Models in Applied Sciences.- 7.1 Introduction.- 7.2 Mathematical Modeling in Applied Sciences.- 7.3 Examples of Mathematical Models.- 7.4 Further Remarks on Modeling.- 7.5 Mathematical Modeling and Stochasticity.- 7.5.1 Random variables and stochastic calculus.- 7.5.2 Moment representation of the dynamic response.- 7.5.3 Statistical representation of large systems..- 7.6 Problems.- 8 Chaotic Dynamics, Stability, and Bifurcations.- 8.1 Introduction.- 8.2 Stability Diagrams.- 8.3 Stability Diagrams and Potential Energy.- 8.4 Limit Cycles.- 8.5 Hopf Bifurcation.- 8.6 Chaotic Motions.- 8.7 Applications.- 8.7.1 Ring on a rotating wire.- 8.7.2 Metallic meter.- 8.7.3 Line galloping model.- 8.7.4 Flutter instability model.- 8.7.5 Models presenting transition to chaos.- 8.8 Problems.- 9 Discrete Models of Continuous Systems.- 9.1 Introduction.- 9.2 Diffusion Models.- 9.3 Mathematical Models of Traffic Flow.- 9.4 Mathematical Statement of Problems.- 9.5 Discretization of Continuous Models.- 9.6 Problems.- Appendix I. Numerical Methods for Ordinary Differential Equations.- 1 Introduction.- 2 Numerical Methods for Initial-Value Problems.- 3 Numerical Methods for Boundary-Value Problems.- Appendix II. Kinematics, Applied Forces, Momentum and Mechanical Energy.- 1 Introduction.- 2 Systems of Applied Forces.- 3 Fundamental of Kinematics.- 4 Center of Mass.- 5 Tensor of Inertia.- 6 Linear Momentum.- 7 Angular Momentum.- 8 Kinetic Energy.- Appendix III. Scientific Programs.- 1 Introduction to Programming.- 2 Scientific Programs.- References.