Download or read book Dynamics on Differential One Forms written by Troy L. Story and published by iUniverse. This book was released on 2002 with total page 127 pages. Available in PDF, EPUB and Kindle. Book excerpt: Dynamics on Differential One-Forms proposes a unifying principle for mathematical models of dynamic systems. In "Thermodynamics on One-Forms (chapter I)", the long-standing problem of deriving irreversibility in thermodynamics from reversibility in Hamiltonian mechanics, is solved. Differential geometric analysis shows thermodynamics and Hamiltonian mechanics are both irreversible on representative extended phase spaces. "Dynamics on Differential One-Forms (II)" generalizes (I) to Hamiltonian mechanics, geometric optics, thermodynamics, black holes, electromagnetic fields and string fields. Mathematical models for these systems are revealed as representations of a unifying principle; namely, description of a dynamic system with a characteristic differential one-form on an odd-dimensional differentiable manifold leads, by analysis with exterior calculus, to a set of differential equations and a tangent vector defining system transformations. Relationships between models using exterior calculus and conventional calculus imply a technical definition of dynamic equilibrium. "Global Analysis of Composite Particles (III)" uses differential topology to develop the theory of large vibration-rotation interactions for composite particles. A global classical Hamiltonian and corresponding quantum Hamiltonian operator are derived, then applied to the molecular vibration-rotation problem. "Characteristic Electromagnetic and Yang-Mills Gauge (IV)" uses differential geometry to remove some of the arbitrariness in the gauge, and shows how gauge functions for electromagnetic and Yang-Mills fields follow the same differential equation.
Download or read book Electricity and Magnetism for Mathematicians written by Thomas A. Garrity and published by Cambridge University Press. This book was released on 2015-01-19 with total page 297 pages. Available in PDF, EPUB and Kindle. Book excerpt: Maxwell's equations have led to many important mathematical discoveries. This text introduces mathematics students to some of their wonders.
Download or read book Differential Geometry Applied To Dynamical Systems With Cd rom written by Jean-marc Ginoux and published by World Scientific. This book was released on 2009-04-03 with total page 341 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book aims to present a new approach called Flow Curvature Method that applies Differential Geometry to Dynamical Systems. Hence, for a trajectory curve, an integral of any n-dimensional dynamical system as a curve in Euclidean n-space, the curvature of the trajectory — or the flow — may be analytically computed. Then, the location of the points where the curvature of the flow vanishes defines a manifold called flow curvature manifold. Such a manifold being defined from the time derivatives of the velocity vector field, contains information about the dynamics of the system, hence identifying the main features of the system such as fixed points and their stability, local bifurcations of codimension one, center manifold equation, normal forms, linear invariant manifolds (straight lines, planes, hyperplanes).In the case of singularly perturbed systems or slow-fast dynamical systems, the flow curvature manifold directly provides the slow invariant manifold analytical equation associated with such systems. Also, starting from the flow curvature manifold, it will be demonstrated how to find again the corresponding dynamical system, thus solving the inverse problem.
Download or read book Differential Dynamical Systems Revised Edition written by James D. Meiss and published by SIAM. This book was released on 2017-01-24 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential equations are the basis for models of any physical systems that exhibit smooth change. This book combines much of the material found in a traditional course on ordinary differential equations with an introduction to the more modern theory of dynamical systems. Applications of this theory to physics, biology, chemistry, and engineering are shown through examples in such areas as population modeling, fluid dynamics, electronics, and mechanics. Differential Dynamical Systems begins with coverage of linear systems, including matrix algebra; the focus then shifts to foundational material on nonlinear differential equations, making heavy use of the contraction-mapping theorem. Subsequent chapters deal specifically with dynamical systems concepts?flow, stability, invariant manifolds, the phase plane, bifurcation, chaos, and Hamiltonian dynamics. This new edition contains several important updates and revisions throughout the book. Throughout the book, the author includes exercises to help students develop an analytical and geometrical understanding of dynamics. Many of the exercises and examples are based on applications and some involve computation; an appendix offers simple codes written in Maple, Mathematica, and MATLAB software to give students practice with computation applied to dynamical systems problems.
Download or read book Introduction to Differential Geometry with Applications to Navier Stokes Dynamics written by Troy L Story and published by iUniverse. This book was released on 2005 with total page 165 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduction to Differential Geometry with applications to Navier-Stokes Dynamics is an invaluable manuscript for anyone who wants to understand and use exterior calculus and differential geometry, the modern approach to calculus and geometry. Author Troy Story makes use of over thirty years of research experience to provide a smooth transition from conventional calculus to exterior calculus and differential geometry, assuming only a knowledge of conventional calculus. Introduction to Differential Geometry with applications to Navier-Stokes Dynamics includes the topics: Geometry, Exterior calculus, Homology and co-homology, Applications of differential geometry and exterior calculus to: Hamiltonian mechanics, geometric optics, irreversible thermodynamics, black hole dynamics, electromagnetism, classical string fields, and Navier-Stokes dynamics.
Download or read book Galileo Unbound written by David D. Nolte and published by Oxford University Press. This book was released on 2018-07-12 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: Galileo Unbound traces the journey that brought us from Galileo's law of free fall to today's geneticists measuring evolutionary drift, entangled quantum particles moving among many worlds, and our lives as trajectories traversing a health space with thousands of dimensions. Remarkably, common themes persist that predict the evolution of species as readily as the orbits of planets or the collapse of stars into black holes. This book tells the history of spaces of expanding dimension and increasing abstraction and how they continue today to give new insight into the physics of complex systems. Galileo published the first modern law of motion, the Law of Fall, that was ideal and simple, laying the foundation upon which Newton built the first theory of dynamics. Early in the twentieth century, geometry became the cause of motion rather than the result when Einstein envisioned the fabric of space-time warped by mass and energy, forcing light rays to bend past the Sun. Possibly more radical was Feynman's dilemma of quantum particles taking all paths at once — setting the stage for the modern fields of quantum field theory and quantum computing. Yet as concepts of motion have evolved, one thing has remained constant, the need to track ever more complex changes and to capture their essence, to find patterns in the chaos as we try to predict and control our world.
Download or read book A Visual Introduction to Differential Forms and Calculus on Manifolds written by Jon Pierre Fortney and published by Springer. This book was released on 2018-11-03 with total page 470 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explains and helps readers to develop geometric intuition as it relates to differential forms. It includes over 250 figures to aid understanding and enable readers to visualize the concepts being discussed. The author gradually builds up to the basic ideas and concepts so that definitions, when made, do not appear out of nowhere, and both the importance and role that theorems play is evident as or before they are presented. With a clear writing style and easy-to- understand motivations for each topic, this book is primarily aimed at second- or third-year undergraduate math and physics students with a basic knowledge of vector calculus and linear algebra.
Download or read book Visual Differential Geometry and Forms written by Tristan Needham and published by Princeton University Press. This book was released on 2021-07-13 with total page 530 pages. Available in PDF, EPUB and Kindle. Book excerpt: An inviting, intuitive, and visual exploration of differential geometry and forms Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newton’s geometrical methods to provide geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner. Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss’s famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einstein’s field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology. The final act elucidates such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell’s equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan’s method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book. Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught.
Download or read book Differential Forms in Mathematical Physics written by and published by Elsevier. This book was released on 2009-06-17 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential Forms in Mathematical Physics
Download or read book Differential Forms and the Geometry of General Relativity written by Tevian Dray and published by CRC Press. This book was released on 2014-10-20 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential Forms and the Geometry of General Relativity provides readers with a coherent path to understanding relativity. Requiring little more than calculus and some linear algebra, it helps readers learn just enough differential geometry to grasp the basics of general relativity. The book contains two intertwined but distinct halves. Designed for advanced undergraduate or beginning graduate students in mathematics or physics, most of the text requires little more than familiarity with calculus and linear algebra. The first half presents an introduction to general relativity that describes some of the surprising implications of relativity without introducing more formalism than necessary. This nonstandard approach uses differential forms rather than tensor calculus and minimizes the use of "index gymnastics" as much as possible. The second half of the book takes a more detailed look at the mathematics of differential forms. It covers the theory behind the mathematics used in the first half by emphasizing a conceptual understanding instead of formal proofs. The book provides a language to describe curvature, the key geometric idea in general relativity.
Download or read book Dynamical Systems and Geometric Mechanics written by Jared Maruskin and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-08-21 with total page 417 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduction to Dynamical Systems and Geometric Mechanics provides a comprehensive tour of two fields that are intimately entwined: dynamical systems is the study of the behavior of physical systems that may be described by a set of nonlinear first-order ordinary differential equations in Euclidean space, whereas geometric mechanics explore similar systems that instead evolve on differentiable manifolds. The first part discusses the linearization and stability of trajectories and fixed points, invariant manifold theory, periodic orbits, Poincaré maps, Floquet theory, the Poincaré-Bendixson theorem, bifurcations, and chaos. The second part of the book begins with a self-contained chapter on differential geometry that introduces notions of manifolds, mappings, vector fields, the Jacobi-Lie bracket, and differential forms.
Download or read book Advanced Calculus written by Harold M. Edwards and published by Springer Science & Business Media. This book was released on 1994-01-05 with total page 532 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a high-level introduction to vector calculus based solidly on differential forms. Informal but sophisticated, it is geometrically and physically intuitive yet mathematically rigorous. It offers remarkably diverse applications, physical and mathematical, and provides a firm foundation for further studies.
Download or read book Dynamics written by Carlos M. Roithmayr and published by Cambridge University Press. This book was released on 2016-03-09 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is ideal for teaching students in engineering or physics the skills necessary to analyze motions of complex mechanical systems such as spacecraft, robotic manipulators, and articulated scientific instruments. Kane's method, which emerged recently, reduces the labor needed to derive equations of motion and leads to equations that are simpler and more readily solved by computer, in comparison to earlier, classical approaches. Moreover, the method is highly systematic and thus easy to teach. This book is a revision of Dynamics: Theory and Applications (1985), by T. R. Kane and D. A. Levinson, and presents the method for forming equations of motion by constructing generalized active forces and generalized inertia forces. Important additional topics include approaches for dealing with finite rotation, an updated treatment of constraint forces and constraint torques, an extension of Kane's method to deal with a broader class of nonholonomic constraint equations, and other recent advances.
Download or read book Applied Differential Geometry written by William L. Burke and published by Cambridge University Press. This book was released on 1985-05-31 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a self-contained introductory textbook on the calculus of differential forms and modern differential geometry. The intended audience is physicists, so the author emphasises applications and geometrical reasoning in order to give results and concepts a precise but intuitive meaning without getting bogged down in analysis. The large number of diagrams helps elucidate the fundamental ideas. Mathematical topics covered include differentiable manifolds, differential forms and twisted forms, the Hodge star operator, exterior differential systems and symplectic geometry. All of the mathematics is motivated and illustrated by useful physical examples.
Download or read book Typical Singularities of Differential 1 forms and Pfaffian Equations written by Mikhail Zhitomirskiĭ and published by American Mathematical Soc.. This book was released on 1992 with total page 194 pages. Available in PDF, EPUB and Kindle. Book excerpt: Singularities and the classification of 1-forms and Pfaffian equations are interesting not only as classical problems, but also because of their applications in contact geometry, partial differential equations, control theory, nonholonomic dynamics, and variational problems. In addition to collecting results on the geometry of singularities and classification of differential forms and Pfaffian equations, this monograph discusses applications and closely related classification problems. Zhitomirskii presents proofs with all results and ends each chapter with a summary of the main results, a tabulation of the singularities, and a list of the normal forms. The main results of the book are also collected together in the introduction.
Download or read book Topology of Closed One Forms written by Michael Farber and published by American Mathematical Soc.. This book was released on 2004 with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: Farber examines the geometrical, topological, and dynamical properties of closed one-forms, highlighting the relations between their global and local features. He describes the Novikov numbers and inequalities, the universal complex and its construction, Bott-type inequalities and those with Von Neumann Betti numbers, equivariant theory, the exactness of Novikov inequalities, the Morse theory of harmonic forms, and Lusternick-Schnirelman theory. Annotation : 2004 Book News, Inc., Portland, OR (booknews.com).
Download or read book Differential Geometry and Its Applications written by Old?ich Kowalski and published by World Scientific. This book was released on 2008 with total page 732 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains invited lectures and selected research papers in the fields of classical and modern differential geometry, global analysis, and geometric methods in physics, presented at the 10th International Conference on Differential Geometry and its Applications (DGA2007), held in Olomouc, Czech Republic.The book covers recent developments and the latest results in the following fields: Riemannian geometry, connections, jets, differential invariants, the calculus of variations on manifolds, differential equations, Finsler structures, and geometric methods in physics. It is also a celebration of the 300th anniversary of the birth of one of the greatest mathematicians, Leonhard Euler, and includes the Euler lecture ?Leonhard Euler ? 300 years on? by R Wilson. Notable contributors include J F Cari¤ena, M Castrill¢n L¢pez, J Erichhorn, J-H Eschenburg, I Kol ?, A P Kopylov, J Korba?, O Kowalski, B Kruglikov, D Krupka, O Krupkov , R Landre, Haizhong Li, S Maeda, M A Malakhaltsev, O I Mokhov, J Mu¤oz Masqu, S Preston, V Rovenski, D J Saunders, M Sekizawa, J Slov k, J Szilasi, L Tam ssy, P Walczak, and others.
