Download or read book Variations on a Theme by Kepler written by Victor Guillemin and published by . This book was released on 2006 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Variations on a Theme by Kepler written by Victor Guillemin and published by American Mathematical Soc.. This book was released on 1990 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is based on the Colloquium Lectures presented by Shlomo Sternberg in 1990. The authors delve into the mysterious role that groups, especially Lie groups, play in revealing the laws of nature by focusing on the familiar example of Kepler motion: the motion of a planet under the attraction of the sun according to Kepler's laws. Newton realized that Kepler's second law--that equal areas are swept out in equal times--has to do with the fact that the force is directed radially to the sun. Kepler's second law is really the assertion of the conservation of angular momentum, reflecting the rotational symmetry of the system about the origin of the force. In today's language, we would say that the group $O(3)$ (the orthogonal group in three dimensions) is responsible for Kepler's second law. By the end of the nineteenth century, the inverse square law of attraction was seen to have $O(4)$ symmetry (where $O(4)$ acts on a portion of the six-dimensional phase space of the planet). Even larger groups have since been found to be involved in Kepler motion. In quantum mechanics, the example of Kepler motion manifests itself as the hydrogen atom. Exploring this circle of ideas, the first part of the book was written with the general mathematical reader in mind. The remainder of the book is aimed at specialists. It begins with a demonstration that the Kepler problem and the hydrogen atom exhibit $O(4)$ symmetry and that the form of this symmetry determines the inverse square law in classical mechanics and the spectrum of the hydrogen atom in quantum mechanics. The space of regularized elliptical motions of the Kepler problem (also known as the Kepler manifold) plays a central role in this book. The last portion of the book studies the various cosmological models in this same conformal class (and having varying isometry groups) from the viewpoint of projective geometry. The computation of the hydrogen spectrum provides an illustration of the principle that enlarging the phase space can simplify the equations of motion in the classical setting and aid in the quantization problem in the quantum setting. The authors provide a short summary of the homological quantization of constraints and a list of recent applications to many interesting finite-dimensional settings. The book closes with an outline of Kostant's theory, in which a unitary representation is associated to the minimal nilpotent orbit of $SO(4,4)$ and in which electromagnetism and gravitation are unified in a Kaluza-Klein-type theory in six dimensions.
Download or read book Variations on a Theme by Kepler written by Victor Guillemin and published by American Mathematical Soc.. This book was released on with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is based on the Colloquium Lectures presented by Shlomo Sternberg in 1990. The authors delve into the mysterious role that groups, especially Lie groups, play in revealing the laws of nature by focusing on the familiar example of Kepler motion: the motion of a planet under the attraction of the sun according to Kepler's laws. Newton realized that Kepler's second law--that equal areas are swept out in equal times--has to do with the fact that the force is directed radially to the sun. Kepler's second law is really the assertion of the conservation of angular momentum, reflecting the rotational symmetry of the system about the origin of the force. In today's language, we would say that the group $O(3)$ (the orthogonal group in three dimensions) is responsible for Kepler's second law. By the end of the nineteenth century, the inverse square law of attraction was seen to have $O(4)$ symmetry (where $O(4)$ acts on a portion of the six-dimensional phase space of the planet). Even larger groups h The remainder of the book is aimed at specialists. It begins with a demonstration that the Kepler problem and the hydrogen atom exhibit $O(4)$ symmetry and that the form of this symmetry determines the inverse square law in classical mechanics and the spectrum of the hydrogen atom in quantum mechanics. The space of regularized elliptical motions of the Kepler problem (also known as the Kepler manifold) plays a central role in this book. The last portion of the book studies the various cosmological models in this same conformal class (and having varying isometry groups) from the viewpoint of projective geometry. The computation of the hydrogen spectrum provides an illustration of the principle that enlarging the phase space can simplify the equations of motion in the classical setting and aid in the quantization problem in the quantum setting. The authors provide a short summary of the homological quantization of constraints and a list of recent applications to many interesting finite-dimensional set
Download or read book The Kepler Problem written by Bruno Cordani and published by Birkhäuser. This book was released on 2012-12-06 with total page 454 pages. Available in PDF, EPUB and Kindle. Book excerpt: Because of the correspondences existing among all levels of reality, truths pertaining to a lower level can be considered as symbols of truths at a higher level and can therefore be the "foundation" or support leading by analogy to a knowledge of the latter. This confers to every science a superior or "elevating" meaning, far deeper than its own original one. - R. GUENON, The Crisis of Modern World Having been interested in the Kepler Problem for a long time, I have al ways found it astonishing that no book has been written yet that would address all aspects of the problem. Besides hundreds of articles, at least three books (to my knowledge) have indeed been published al ready on the subject, namely Englefield (1972), Stiefel & Scheifele (1971) and Guillemin & Sternberg (1990). Each of these three books deals only with one or another aspect of the problem, though. For example, En glefield (1972) treats only the quantum aspects, and that in a local way. Similarly, Stiefel & Scheifele (1971) only considers the linearization of the equations of motion with application to the perturbations of celes tial mechanics. Finally, Guillemin & Sternberg (1990) is devoted to the group theoretical and geometrical structure.
Download or read book Integrable Systems and Algebraic Geometry Volume 1 written by Ron Donagi and published by Cambridge University Press. This book was released on 2020-04-02 with total page 421 pages. Available in PDF, EPUB and Kindle. Book excerpt: Created as a celebration of mathematical pioneer Emma Previato, this comprehensive book highlights the connections between algebraic geometry and integrable systems, differential equations, mathematical physics, and many other areas. The authors, many of whom have been at the forefront of research into these topics for the last decades, have all been influenced by Previato's research, as her collaborators, students, or colleagues. The diverse articles in the book demonstrate the wide scope of Previato's work and the inclusion of several survey and introductory articles makes the text accessible to graduate students and non-experts, as well as researchers. This first volume covers a wide range of areas related to integrable systems, often emphasizing the deep connections with algebraic geometry. Common themes include theta functions and Abelian varieties, Lax equations, integrable hierarchies, Hamiltonian flows and difference operators. These powerful tools are applied to spinning top, Hitchin, Painleve and many other notable special equations.
Download or read book Orthogonal Polynomials on the Unit Circle Spectral theory written by Barry Simon and published by American Mathematical Soc.. This book was released on 2005 with total page 608 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents an overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. This book discusses topics such as asymptotics of Toeplitz determinants (Szego's theorems), and limit theorems for the density of the zeros of orthogonal polynomials.
Download or read book Frobenius Manifolds Quantum Cohomology and Moduli Spaces written by I͡U. I. Manin and published by American Mathematical Soc.. This book was released on with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Symmetries and Laplacians written by David Gurarie and published by Courier Corporation. This book was released on 2007-12-21 with total page 466 pages. Available in PDF, EPUB and Kindle. Book excerpt: Designed as an introduction to harmonic analysis and group representations, this book examines concepts, ideas, results, and techniques related to symmetry groups and Laplacians. Its exposition is based largely on examples and applications of general theory, covering a wide range of topics rather than delving deeply into any particular area. Author David Gurarie, a Professor of Mathematics at Case Western Reserve University, focuses on discrete or continuous geometrical objects and structures, such as regular graphs, lattices, and symmetric Riemannian manifolds. Starting with the basics of representation theory, Professor Gurarie discusses commutative harmonic analysis, representations of compact and finite groups, Lie groups, and the Heisenberg group and semidirect products. Among numerous applications included are integrable hamiltonian systems, geodesic flows on symmetric spaces, and the spectral theory of the Hydrogen atom (Schrodinger operator with Coulomb potential) explicated by its Runge-Lenz symmetry. Three helpful appendixes include supplemental information, and the text concludes with references, a list of frequently used notations, and an index.
Download or read book Integrable Systems and Algebraic Geometry written by Ron Donagi and published by Cambridge University Press. This book was released on 2020-04-02 with total page 421 pages. Available in PDF, EPUB and Kindle. Book excerpt: A collection of articles discussing integrable systems and algebraic geometry from leading researchers in the field.
Download or read book Functional Analysis and Semi groups written by Einar Hille and published by American Mathematical Soc.. This book was released on 1996-02-06 with total page 826 pages. Available in PDF, EPUB and Kindle. Book excerpt: Early in 1952 it became obvious that a new printing would be needed, and new advances in the theory called for extensive revision. It has been completely rewritten, mostly by Phillips, and much has been added while keeping the existing framework. Thus, the algebraic tools play a major role, and are introduced early, leading to a more satisfactory operational calculus and spectral theory. The Laplace-Stieltjes transform methods, used by Hille, have not been replaced but rather supplemented by the new tools. - Foreword.
Download or read book Geometric Nonlinear Functional Analysis written by Yoav Benyamini and published by American Mathematical Soc.. This book was released on 2000 with total page 503 pages. Available in PDF, EPUB and Kindle. Book excerpt: A systematic study of geometric nonlinear functional analysis. The main theme is the study of uniformly continuous and Lipschitz functions between Banach spaces. This study leads to the classification of Banach spaces and of their important subsets in the uniform and Lipschitz categories.
Download or read book Fully Nonlinear Elliptic Equations written by Luis A. Caffarelli and published by American Mathematical Soc.. This book was released on 1995 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of the book is to extend classical regularity theorems for solutions of linear elliptic partial differential equations to the context of fully nonlinear elliptic equations. This class of equations often arises in control theory, optimization, and other applications. The authors give a detailed presentation of all the necessary techniques. Instead of treating these techniques in their greatest generality, they outline the key ideas and prove the results needed for developing the subsequent theory. Topics discussed in the book include the theory of viscosity solutions for nonlinear equations, the Alexandroff estimate and Krylov-Safonov Harnack-type inequality for viscosity solutions, uniqueness theory for viscosity solutions, Evans and Krylov regularity theory for convex fully nonlinear equations, and regularity theory for fully nonlinear equations with variable coefficients.
Download or read book Global Solutions of Nonlinear Schrodinger Equations written by Jean Bourgain and published by American Mathematical Soc.. This book was released on 1999 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents recent progress in the theory of nonlinear dispersive equations, primarily the nonlinear Schrodinger (NLS) equation. The Cauchy problem for defocusing NLS with critical nonlinearity is discussed. New techniques and results are described on global existence and properties of solutions with Large Cauchy data. Current research in harmonic analysis around Strichartz's inequalities and its relevance to nonlinear PDE is presented and several topics in NLS theory on bounded domains are reviewed. Using the NLS as an example, the book offers comprehensive insight on current research related to dispersive equations and Hamiltonian PDEs.
Download or read book Longing for the Harmonies Themes and Variations From Modern Physi written by Frank Wilczek and published by W. W. Norton & Company. This book was released on 1989-05-02 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Occasionally, there comes along a popular science book that both scientists and non-scientist can read with pleasure and profit, and this is one."—The New Yorker Devoted to sharing their own delight and awe before the fundamental mysteries of the cosmos, Frank Wilczek (winner of the 2004 Nobel Prize in Physics) and science writer Betsy Devine also have a serious purpose: to reveal to the lay reader how a heightened perception can respond to timeless themes of the physical universe. For example, they show that even the most exotic theories always confirm that physical laws are precisely the same throughout the universe, and they explain how we have learned that the most massive molten stars and the tiniest frozen particles are in physical harmony. In their descriptions of the workings of the half-known universe, Wilczek and Devine bring all of us face to face with the beauty of eternal order and the inevitability of rational ends and beginnings.
Download or read book Orthogonal Polynomials on the Unit Circle written by Barry Simon and published by American Mathematical Soc.. This book was released on 2009-08-05 with total page 498 pages. Available in PDF, EPUB and Kindle. Book excerpt: This two-part book is a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of one-dimensional Schrodinger operators. Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szego's theorems), limit theorems for the density of the zeros of orthogonal polynomials, matrix representations for multiplication by $z$ (CMV matrices), periodic Verblunsky coefficients from the point of view of meromorphic functions on hyperelliptic surfaces, and connections between the theories of orthogonal polynomials on the unit circle and on the real line.
Download or read book Similarity and Symmetry Methods written by Jean-François Ganghoffer and published by Springer. This book was released on 2014-07-19 with total page 380 pages. Available in PDF, EPUB and Kindle. Book excerpt: The principle aim of the book is to present a self-contained, modern account of similarity and symmetry methods, which are important mathematical tools for both physicists, engineers and applied mathematicians. The idea is to provide a balanced presentation of the mathematical techniques and applications of symmetry methods in mathematics, physics and engineering. That is why it includes recent developments and many examples in finding systematically conservation laws, local and nonlocal symmetries for ordinary and partial differential equations. The role of continuous symmetries in classical and quantum field theories is exposed at a technical level accessible even for non specialists. The importance of symmetries in continuum mechanics and mechanics of materials is highlighted through recent developments, such as the construction of constitutive models for various materials combining Lie symmetries with experimental data. As a whole this book is a unique collection of contributions from experts in the field, including specialists in the mathematical treatment of symmetries, researchers using symmetries from a fundamental, applied or numerical viewpoint. The book is a fascinating overview of symmetry methods aimed for graduate students in physics, mathematics and engineering, as well as researchers either willing to enter in the field or to capture recent developments and applications of symmetry methods in different scientific fields.
Download or read book Foundations of Algebraic Geometry written by Andr Weil and published by American Mathematical Soc.. This book was released on 1946-12-31 with total page 386 pages. Available in PDF, EPUB and Kindle. Book excerpt: This classic is one of the cornerstones of modern algebraic geometry. At the same time, it is entirely self-contained, assuming no knowledge whatsoever of algebraic geometry, and no knowledge of modern algebra beyond the simplest facts about abstract fields and their extensions, and the bare rudiments of the theory of ideals.
