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| Author | : John M. Charap |
| Publisher | : Cambridge University Press |
| Total Pages | : 354 |
| Release | : 1995-01-05 |
| Genre | : Mathematics |
| ISBN | : 9780521482714 |
A lively, varied and topical presentation of this branch of theoretical physics.
| Author | : Heinz J. Rothe |
| Publisher | : World Scientific |
| Total Pages | : 317 |
| Release | : 2010 |
| Genre | : Science |
| ISBN | : 9814299642 |
This book is an introduction to the field of constrained Hamiltonian systems and their quantization, a topic which is of central interest to theoretical physicists who wish to obtain a deeper understanding of the quantization of gauge theories, such as describing the fundamental interactions in nature. Beginning with the early work of Dirac, the book covers the main developments in the field up to more recent topics, such as the field?antifield formalism of Batalin and Vilkovisky, including a short discussion of how gauge anomalies may be incorporated into this formalism. All topics are well illustrated with examples emphasizing points of central interest. The book should enable graduate students to follow the literature on this subject without much problems, and to perform research in this field.
| Author | : Richard H. Cushman |
| Publisher | : World Scientific |
| Total Pages | : 421 |
| Release | : 2010 |
| Genre | : Mathematics |
| ISBN | : 9814289493 |
1. Nonholonomically constrained motions. 1.1. Newton's equations. 1.2. Constraints. 1.3. Lagrange-d'Alembert equations. 1.4. Lagrange derivative. 1.5. Hamilton-d'Alembert equations. 1.6. Distributional Hamiltonian formulation. 1.7. Almost Poisson brackets. 1.8. Momenta and momentum equation. 1.9. Projection principle. 1.10. Accessible sets. 1.11. Constants of motion. 1.12. Notes -- 2. Group actions and orbit spaces. 2.1. Group actions. 2.2. Orbit spaces. 2.3. Isotropy and orbit types. 2.4. Smooth structure on an orbit space. 2.5. Subcartesian spaces. 2.6. Stratification of the orbit space by orbit types. 2.7. Derivations and vector fields on a differential space. 2.8. Vector fields on a stratified differential space. 2.9. Vector fields on an orbit space. 2.10. Tangent objects to an orbit space. 2.11. Notes -- 3. Symmetry and reduction. 3.1. Dynamical systems with symmetry. 3.2. Nonholonomic singular reduction. 3.3. Nonholonomic regular reduction. 3.4. Chaplygin systems. 3.5. Orbit types and reduction. 3.6. Conservation laws. 3.7. Lifted actions and the momentum equation. 3.8. Notes -- 4. Reconstruction, relative equilibria and relative periodic orbits. 4.1. Reconstruction. 4.2. Relative equilibria. 4.3. Relative periodic orbits. 4.4. Notes -- 5. Carathéodory's sleigh. 5.1. Basic set up. 5.2. Equations of motion. 5.3. Reduction of the E(2) symmetry. 5.4. Motion on the E(2) reduced phase space. 5.5. Reconstruction. 5.6. Notes -- 6. Convex rolling rigid body. 6.1. Basic set up. 6.2. Unconstrained motion. 6.3. Constraint distribution. 6.4. Constrained equations of motion. 6.5. Reduction of the translational [symbol] symmetry. 6.6. Reduction of E(2) symmetry. 6.7. Body of revolution. 6.8. Notes -- 7. The rolling disk. 7.1. General set up. 7.2. Reduction of the E(2) x S[symbol] symmetry. 7.3. Reconstruction. 7.4. Relative equilibria. 7.5. A potential function on an interval. 7.6. Scaling. 7.7. Solutions of the rescaled Chaplygin equations. 7.8. Bifurcations of a vertical disk. 7.9. The global geometry of the degeneracy locus. 7.10. Falling flat. 7.11. Near falling flat. 7.12. The bifurcation diagram. 7.13. The integral map. 7.14. Constant energy slices. 7.15. The spatial rotational shift. 7.16. Notes.
| Author | : Meera Sitharam |
| Publisher | : CRC Press |
| Total Pages | : 711 |
| Release | : 2018-07-20 |
| Genre | : Mathematics |
| ISBN | : 1351647431 |
The Handbook of Geometric Constraint Systems Principles is an entry point to the currently used principal mathematical and computational tools and techniques of the geometric constraint system (GCS). It functions as a single source containing the core principles and results, accessible to both beginners and experts. The handbook provides a guide for students learning basic concepts, as well as experts looking to pinpoint specific results or approaches in the broad landscape. As such, the editors created this handbook to serve as a useful tool for navigating the varied concepts, approaches and results found in GCS research. Key Features: A comprehensive reference handbook authored by top researchers Includes fundamentals and techniques from multiple perspectives that span several research communities Provides recent results and a graded program of open problems and conjectures Can be used for senior undergraduate or graduate topics course introduction to the area Detailed list of figures and tables About the Editors: Meera Sitharam is currently an Associate Professor at the University of Florida’s Department of Computer & Information Science and Engineering. She received her Ph.D. at the University of Wisconsin, Madison. Audrey St. John is an Associate Professor of Computer Science at Mount Holyoke College, who received her Ph. D. from UMass Amherst. Jessica Sidman is a Professor of Mathematics on the John S. Kennedy Foundation at Mount Holyoke College. She received her Ph.D. from the University of Michigan.
| Author | : Simon Foucart |
| Publisher | : Springer Nature |
| Total Pages | : 294 |
| Release | : |
| Genre | : |
| ISBN | : 3031664973 |
| Author | : Frédéric Barbaresco |
| Publisher | : Springer Nature |
| Total Pages | : 466 |
| Release | : 2021-06-27 |
| Genre | : Mathematics |
| ISBN | : 3030779572 |
Machine learning and artificial intelligence increasingly use methodological tools rooted in statistical physics. Conversely, limitations and pitfalls encountered in AI question the very foundations of statistical physics. This interplay between AI and statistical physics has been attested since the birth of AI, and principles underpinning statistical physics can shed new light on the conceptual basis of AI. During the last fifty years, statistical physics has been investigated through new geometric structures allowing covariant formalization of the thermodynamics. Inference methods in machine learning have begun to adapt these new geometric structures to process data in more abstract representation spaces. This volume collects selected contributions on the interplay of statistical physics and artificial intelligence. The aim is to provide a constructive dialogue around a common foundation to allow the establishment of new principles and laws governing these two disciplines in a unified manner. The contributions were presented at the workshop on the Joint Structures and Common Foundation of Statistical Physics, Information Geometry and Inference for Learning which was held in Les Houches in July 2020. The various theoretical approaches are discussed in the context of potential applications in cognitive systems, machine learning, signal processing.
| Author | : Liang-Yee Cheng |
| Publisher | : Springer Nature |
| Total Pages | : 1080 |
| Release | : 2022-08-12 |
| Genre | : Technology & Engineering |
| ISBN | : 3031135881 |
This book covers recent achievements on the ever-expanding field of Geometry and Graphics on both analogical and digital fronts, from theoretical investigations to a broad range of applications, new teaching methodologies, and historical aspects. It is from 20th International Conference on Geometry and Graphics (ICGG2022), a series of conference that started in 1978 and promoted by International Society for Geometry and Graphics, which aims to foster international collaboration and stimulate the scientific research and teaching innovations in the multidisciplinary field. The contents of the book are organized in: Theoretical Geometry and Graphics; Applied Geometry and Graphics; Engineering Computer Graphics; Graphics Education; Geometry and Graphics in History, and are intent for the academics, researchers, and professionals in architecture, engineering, industrial design, mathematics, and arts.
| Author | : Satyajit Jena |
| Publisher | : Springer Nature |
| Total Pages | : 1353 |
| Release | : |
| Genre | : |
| ISBN | : 9819702895 |
| Author | : Olga Krupkova |
| Publisher | : Springer |
| Total Pages | : 261 |
| Release | : 2006-11-14 |
| Genre | : Mathematics |
| ISBN | : 3540696571 |
The book provides a comprehensive theory of ODE which come as Euler-Lagrange equations from generally higher-order Lagrangians. Emphasis is laid on applying methods from differential geometry (fibered manifolds and their jet-prolongations) and global analysis (distributions and exterior differential systems). Lagrangian and Hamiltonian dynamics, Hamilton-Jacobi theory, etc., for any Lagrangian system of any order are presented. The key idea - to build up these theories as related with the class of equivalent Lagrangians - distinguishes this book from other texts on higher-order mechanics. The reader should be familiar with elements of differential geometry, global analysis and the calculus of variations.
| Author | : Philipp Grohs |
| Publisher | : Springer Nature |
| Total Pages | : 703 |
| Release | : 2020-04-03 |
| Genre | : Mathematics |
| ISBN | : 3030313514 |
This book covers different, current research directions in the context of variational methods for non-linear geometric data. Each chapter is authored by leading experts in the respective discipline and provides an introduction, an overview and a description of the current state of the art. Non-linear geometric data arises in various applications in science and engineering. Examples of nonlinear data spaces are diverse and include, for instance, nonlinear spaces of matrices, spaces of curves, shapes as well as manifolds of probability measures. Applications can be found in biology, medicine, product engineering, geography and computer vision for instance. Variational methods on the other hand have evolved to being amongst the most powerful tools for applied mathematics. They involve techniques from various branches of mathematics such as statistics, modeling, optimization, numerical mathematics and analysis. The vast majority of research on variational methods, however, is focused on data in linear spaces. Variational methods for non-linear data is currently an emerging research topic. As a result, and since such methods involve various branches of mathematics, there is a plethora of different, recent approaches dealing with different aspects of variational methods for nonlinear geometric data. Research results are rather scattered and appear in journals of different mathematical communities. The main purpose of the book is to account for that by providing, for the first time, a comprehensive collection of different research directions and existing approaches in this context. It is organized in a way that leading researchers from the different fields provide an introductory overview of recent research directions in their respective discipline. As such, the book is a unique reference work for both newcomers in the field of variational methods for non-linear geometric data, as well as for established experts that aim at to exploit new research directions or collaborations. Chapter 9 of this book is available open access under a CC BY 4.0 license at link.springer.com.
