Mechanics and Dynamical Systems with Mathematica®

Modeling and Applied Mathematics Modeling the behavior of real physical systems by suitable evolution equa­ tions is a relevant, maybe the fundamental, aspect of the interactions be­ tween mathematics and applied sciences. Modeling is, however, only the first step toward the mathematical description and simulation of systems belonging to real world. Indeed, once the evolution equation is proposed, one has to deal with mathematical problems and develop suitable simula­ tions to provide the description of the real system according to the model. Within this framework, one has an evolution equation and the re­ lated mathematical problems obtained by adding all necessary conditions for their solution. Then, a qualitative analysis should be developed: this means proof of existence of solutions and analysis of their qualitative be­ havior. Asymptotic analysis may include a detailed description of stability properties. Quantitative analysis, based upon the application ofsuitable methods and algorithms for the solution of problems, ends up with the simulation that is the representation of the dependent variable versus the independent one. The information obtained by the model has to be compared with those deriving from the experimental observation of the real system. This comparison may finally lead to the validation of the model followed by its application and, maybe, further generalization.

Contenu

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.