Geometric Mechanics On Riemannian Manifolds
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Author | : Ovidiu Calin |
Publisher | : Springer Science & Business Media |
Total Pages | : 285 |
Release | : 2006-03-15 |
Genre | : Mathematics |
ISBN | : 0817644210 |
* A geometric approach to problems in physics, many of which cannot be solved by any other methods * Text is enriched with good examples and exercises at the end of every chapter * Fine for a course or seminar directed at grad and adv. undergrad students interested in elliptic and hyperbolic differential equations, differential geometry, calculus of variations, quantum mechanics, and physics
Author | : Leonor Godinho |
Publisher | : Springer |
Total Pages | : 476 |
Release | : 2014-07-26 |
Genre | : Mathematics |
ISBN | : 3319086669 |
Unlike many other texts on differential geometry, this textbook also offers interesting applications to geometric mechanics and general relativity. The first part is a concise and self-contained introduction to the basics of manifolds, differential forms, metrics and curvature. The second part studies applications to mechanics and relativity including the proofs of the Hawking and Penrose singularity theorems. It can be independently used for one-semester courses in either of these subjects. The main ideas are illustrated and further developed by numerous examples and over 300 exercises. Detailed solutions are provided for many of these exercises, making An Introduction to Riemannian Geometry ideal for self-study.
Author | : Bernhard Riemann |
Publisher | : Birkhäuser |
Total Pages | : 181 |
Release | : 2016-04-19 |
Genre | : Mathematics |
ISBN | : 3319260421 |
This book presents William Clifford’s English translation of Bernhard Riemann’s classic text together with detailed mathematical, historical and philosophical commentary. The basic concepts and ideas, as well as their mathematical background, are provided, putting Riemann’s reasoning into the more general and systematic perspective achieved by later mathematicians and physicists (including Helmholtz, Ricci, Weyl, and Einstein) on the basis of his seminal ideas. Following a historical introduction that positions Riemann’s work in the context of his times, the history of the concept of space in philosophy, physics and mathematics is systematically presented. A subsequent chapter on the reception and influence of the text accompanies the reader from Riemann’s times to contemporary research. Not only mathematicians and historians of the mathematical sciences, but also readers from other disciplines or those with an interest in physics or philosophy will find this work both appealing and insightful.
Author | : Ovidiu Calin |
Publisher | : Cambridge University Press |
Total Pages | : 371 |
Release | : 2009-04-20 |
Genre | : Mathematics |
ISBN | : 0521897300 |
A comprehensive text and reference on sub-Riemannian and Heisenberg manifolds using a novel and robust variational approach.
Author | : Velimir Jurdjevic |
Publisher | : Cambridge University Press |
Total Pages | : 516 |
Release | : 1997 |
Genre | : Mathematics |
ISBN | : 0521495024 |
Geometric control theory is concerned with the evolution of systems subject to physical laws but having some degree of freedom through which motion is to be controlled. This book describes the mathematical theory inspired by the irreversible nature of time evolving events. The first part of the book deals with the issue of being able to steer the system from any point of departure to any desired destination. The second part deals with optimal control, the question of finding the best possible course. An overlap with mathematical physics is demonstrated by the Maximum principle, a fundamental principle of optimality arising from geometric control, which is applied to time-evolving systems governed by physics as well as to man-made systems governed by controls. Applications are drawn from geometry, mechanics, and control of dynamical systems. The geometric language in which the results are expressed allows clear visual interpretations and makes the book accessible to physicists and engineers as well as to mathematicians.
Author | : Darryl D. Holm |
Publisher | : Oxford University Press |
Total Pages | : 537 |
Release | : 2009-07-30 |
Genre | : Mathematics |
ISBN | : 0199212902 |
A graduate level text based partly on lectures in geometry, mechanics, and symmetry given at Imperial College London, this book links traditional classical mechanics texts and advanced modern mathematical treatments of the subject.
Author | : Constantin Udriște |
Publisher | : Springer Science & Business Media |
Total Pages | : 416 |
Release | : 2000 |
Genre | : Mathematics |
ISBN | : 9780792364016 |
The theme of this text is the philosophy that any particle flow generates a particle dynamics, in a suitable geometrical framework. It covers topics that include: geometrical and physical vector fields; field lines; flows; stability of equilibrium points; potential systems and catastrophe geometry; field hypersurfaces; bifurcations; distribution orthogonal to a vector field; extrema with nonholonomic constraints; thermodynamic systems; energies; geometric dynamics induced by a vector field; magnetic fields around piecewise rectilinear electric circuits; geometric magnetic dynamics; and granular materials and their mechanical behaviour. The text should be useful for first-year graduate students in mathematics, mechanics, physics, engineering, biology, chemistry, and economics. It can also be addressed to professors and researchers whose work involves mathematics, mechanics, physics, engineering, biology, chemistry, and economics.
Author | : Edwin J. Beggs |
Publisher | : Springer Nature |
Total Pages | : 826 |
Release | : 2020-01-31 |
Genre | : Science |
ISBN | : 3030302946 |
This book provides a comprehensive account of a modern generalisation of differential geometry in which coordinates need not commute. This requires a reinvention of differential geometry that refers only to the coordinate algebra, now possibly noncommutative, rather than to actual points. Such a theory is needed for the geometry of Hopf algebras or quantum groups, which provide key examples, as well as in physics to model quantum gravity effects in the form of quantum spacetime. The mathematical formalism can be applied to any algebra and includes graph geometry and a Lie theory of finite groups. Even the algebra of 2 x 2 matrices turns out to admit a rich moduli of quantum Riemannian geometries. The approach taken is a `bottom up’ one in which the different layers of geometry are built up in succession, starting from differential forms and proceeding up to the notion of a quantum `Levi-Civita’ bimodule connection, geometric Laplacians and, in some cases, Dirac operators. The book also covers elements of Connes’ approach to the subject coming from cyclic cohomology and spectral triples. Other topics include various other cohomology theories, holomorphic structures and noncommutative D-modules. A unique feature of the book is its constructive approach and its wealth of examples drawn from a large body of literature in mathematical physics, now put on a firm algebraic footing. Including exercises with solutions, it can be used as a textbook for advanced courses as well as a reference for researchers.
Author | : O. Calin |
Publisher | : |
Total Pages | : |
Release | : 2004 |
Genre | : |
ISBN | : 9783764343545 |
Author | : Peter Petersen |
Publisher | : Springer Science & Business Media |
Total Pages | : 443 |
Release | : 2013-06-29 |
Genre | : Mathematics |
ISBN | : 1475764340 |
Intended for a one year course, this volume serves as a single source, introducing students to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialise in Riemannian geometry. Instead of variational techniques, the author uses a unique approach, emphasising distance functions and special co-ordinate systems. He also uses standard calculus with some techniques from differential equations to provide a more elementary route. Many chapters contain material typically found in specialised texts, never before published in a single source. This is one of the few works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory, while also presenting the most up-to-date research - including sections on convergence and compactness of families of manifolds. Thus, this book will appeal to readers with a knowledge of standard manifold theory, including such topics as tensors and Stokes theorem. Various exercises are scattered throughout the text, helping motivate readers to deepen their understanding of the subject.