Antonio Romano - Böcker

Providing a unified treatment of the subject, this text addresses modelling, qualitative analysis and simulation of real physical systems using ordinary differential equations. The scientific computational aspects are presented using the computer program, Mathematica, both for plentiful exercises and examples. Special attention is given to classical mechanics in light of new computational methods and concept.

Many interesting behaviors of real physical, biological, economical, and chemical systems can be described by ordinary differential equations (ODEs). Scientific Computing with Mathematica for Ordinary Differential Equations provides a general framework useful for the applications, on the conceptual aspects of the theory of ODEs, as well as a sophisticated use of Mathematica software for the solutions of problems related to ODEs. In particular, a chapter is devoted to the use ODEs and Mathematica in the Dynamics of rigid bodies. Mathematical methods and scientific computation are dealt with jointly to supply a unified presentation. The main problems of ordinary differential equations such as, phase portrait, approximate solutions, periodic orbits, stability, bifurcation, and boundary problems are covered in an integrated fashion with numerous worked examples and computer program demonstrations using Mathematica.

R In the companion book (Continuum Mechanics Using Mathematica )to this volume, we explained the foundations of continuum mechanics and described some basic applications of ?uid dynamics and linear elasticity. However, deciding on the approach and content of this book, Continuum Mechanics: Advanced Topics and Research Trends, proved to be a more di?culttask.Afteralongperiodofre?ection,wemadethedecisiontodirect our e?orts into drafting a book that demonstrates the ?exibility and great potential of continuum physics to describe the wide range of macroscopic phenomena that we can observe. It is the opinion of the authors that this is the most stimulating way to learn continuum mechanics. However, it is also quite evident that this aim cannot be fully realized in a single book. Consequently,inthis book wechoseto presentonly thebasicsofinteresting continuum mechanics models, along with some important applications of them. We assume that the reader is familiar with all of the basic principles of continuum mechanics: the general balance laws, constitutive equations, isotropygroupsfor materials,the laws of thermodynamics, ordinarywaves, etc. All of these concepts can be found in Continuum Mechanics Using Mathematica and many other books. We believe that this book gives the reader a su?ciently wide view of the “boundless forest” of continuum mechanics, before focusing his or her attention on the beauty and complex structure of single trees within it (- deed,wecouldsaythatContinuumMechanics UsingMathematica provides only the fertile humus on which the trees of this forest take root!).

Many interesting behaviors of real physical, biological, economical, and chemical systems can be described by ordinary differential equations (ODEs). Scientific Computing with Mathematica for Ordinary Differential Equations provides a general framework useful for the applications, on the conceptual aspects of the theory of ODEs, as well as a sophisticated use of Mathematica software for the solutions of problems related to ODEs. In particular, a chapter is devoted to the use ODEs and Mathematica in the Dynamics of rigid bodies. Mathematical methods and scientific computation are dealt with jointly to supply a unified presentation. The main problems of ordinary differential equations such as, phase portrait, approximate solutions, periodic orbits, stability, bifurcation, and boundary problems are covered in an integrated fashion with numerous worked examples and computer program demonstrations using Mathematica. Topics and Features:*Explains how to use the Mathematica package ODE.m to support qualitative and quantitative problem solving *End-of- chapter exercise sets incorporating the use of Mathematica programs *Detailed description and explanation of the mathematical procedures underlying the programs written in Mathematica *Appendix describing the use of ten notebooks to guide the reader through all the exercises. This book is an essential text/reference for students, graduates and practitioners in applied mathematics and engineering interested in ODE's problems in both the qualitative and quantitative description of solutions with the Mathematica program. It is also suitable as a self-

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.

Continuum Mechanics using Mathematica(R)

This textbook's methodological approach familiarizes readers with the mathematical tools required to correctly define and solve problems in continuum mechanics. Covering essential principles and fundamental applications, this second edition of Continuum Mechanics using Mathematica(R) provides a solid basis for a deeper study of more challenging and specialized problems related to nonlinear elasticity, polar continua, mixtures, piezoelectricity, ferroelectricity, magneto-fluid mechanics and state changes (see A. Romano, A. Marasco, Continuum Mechanics: Advanced Topics and Research Trends, Springer (Birkhauser), 2010, ISBN 978-0-8176-4869-5). Key topics and features: * Concise presentation strikes a balance between fundamentals and applications * Requisite mathematical background carefully collected in two introductory chapters and one appendix * Recent developments highlighted through coverage of more significant applications to areas such as wave propagation, fluid mechanics, porous media, linear elasticity.This second edition expands the key topics and features to include: * Two new applications of fluid dynamics: meteorology and navigation * New exercises at the end of the existing chapters * The packages are rewritten for Mathematica 9 Continuum Mechanics using Mathematica(R): Fundamentals, Applications and Scientific Computing is aimed at advanced undergraduates, graduate students and researchers in applied mathematics, mathematical physics and engineering. It may serve as a course textbook or self-study reference for anyone seeking a solid foundation in continuum mechanics.

This textbook takes a broad yet thorough approach to mechanics, aimed at bridging the gap between classical analytic and modern differential geometric approaches to the subject.

This unique textbook offers a mathematically rigorous presentation of the theory of relativity, emphasizing the need for a critical analysis of the foundations of general relativity in order to best study the theory and its implications.  The transitions from classical mechanics to special relativity and then to general relativity are explored in detail as well, helping readers to gain a more profound and nuanced understanding of the theory as a whole.After reviewing the fundamentals of differential geometry and classical mechanics, the text introduces special relativity, first using the physical approach proposed by Einstein and then via Minkowski’s mathematical model. The authors then address the relativistic thermodynamics of continua and electromagnetic fields in matter – topics which are normally covered only very briefly in other treatments – in the next two chapters. The text then turns to a discussion of general relativity by means of the authors’ unique criticalapproach, underlining the difficulty of recognizing the physical meaning of some statements, such as the physical meaning of coordinates and the derivation of physical quantities from those of space-time. Chapters in this section cover the model of space-time proposed by Schwarzschild; black holes; the Friedman equations and the different cosmological models they describe; and the Fermi-Walker derivative.Well-suited for graduate students in physics and mathematics who have a strong foundation in real analysis, classical mechanics, and general physics, this textbook is appropriate for a variety of graduate-level courses that cover topics in relativity. Additionally, it will interest physicists and other researchers who wish to further study the subtleties of these theories and understand the contemporary scholarly discussions surrounding them.

This unique textbook offers a mathematically rigorous presentation of the theory of relativity, emphasizing the need for a critical analysis of the foundations of general relativity in order to best study the theory and its implications.

This unique textbook offers a mathematically rigorous presentation of the theory of relativity, emphasizing the need for a critical analysis of the foundations of general relativity in order to best study the theory and its implications. The transitions from classical mechanics to special relativity and then to general relativity are explored in detail as well, helping readers to gain a more profound and nuanced understanding of the theory as a whole.After reviewing the fundamentals of differential geometry and classical mechanics, the text introduces special relativity, first using the physical approach proposed by Einstein and then via Minkowski’s mathematical model. The authors then address the relativistic thermodynamics of continua and electromagnetic fields in matter – topics which are normally covered only very briefly in other treatments – in the next two chapters. The text then turns to a discussion of general relativity by means of the authors’ unique critical approach, underlining the difficulty of recognizing the physical meaning of some statements, such as the physical meaning of coordinates and the derivation of physical quantities from those of space-time. Chapters in this section cover the model of space-time proposed by Schwarzschild; black holes; the Friedman equations and the different cosmological models they describe; and the Fermi-Walker derivative.This Second Edition includes numerous additions to topics covered in the First Edition, as well as new chapters on the five-dimensional Kaluza-Klein theory, physical coordinate systems and GPS, and a detailed discussion of post-Netwonian theories.Well-suited for graduate students in physics and mathematics who have a strong foundation in real analysis, classical mechanics, and general physics, this textbook is appropriate for a variety of graduate-level courses that cover topics in relativity. Additionally, it will interest physicists and other researchers who wish to further study the subtleties of these theories and understand the contemporary scholarly discussions surrounding them.

This book offers a new approach to third-order aberrations theory. It presents the basic elements of third-order aberrations of optical telescopes, along with the necessary mathematical background, to design almost all optical combinations used in astronomy.

This textbook takes a broad yet thorough approach to mechanics, aimed at bridging the gap between classical analytic and modern differential geometric approaches to the subject.