Geometric Mechanics and Its Applications

Geometric Mechanics and Its Applications
Author: Weipeng Hu
Publisher: Springer Nature
Total Pages: 540
Release: 2023-01-01
Genre: Technology & Engineering
ISBN: 9811974357

To make the content of the book more systematic, this book mainly briefs some related basic knowledge reported by other monographs and papers about geometric mechanics. The main content of this book is based on the last 20 years’ jobs of the authors. All physical processes can be formulated as the Hamiltonian form with the energy conservation law as well as the symplectic structure if all dissipative effects are ignored. On the one hand, the important status of the Hamiltonian mechanics is emphasized. On the other hand, a higher requirement is proposed for the numerical analysis on the Hamiltonian system, namely the results of the numerical analysis on the Hamiltonian system should reproduce the geometric properties of which, including the first integral, the symplectic structure as well as the energy conservation law.

Geometric Mechanics on Riemannian Manifolds

Geometric Mechanics on Riemannian Manifolds
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

Geometric Mechanics and Symmetry

Geometric Mechanics and Symmetry
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.

Symplectic Geometry and Quantum Mechanics

Symplectic Geometry and Quantum Mechanics
Author: Maurice A. de Gosson
Publisher: Springer Science & Business Media
Total Pages: 375
Release: 2006-08-06
Genre: Mathematics
ISBN: 3764375752

This book offers a complete discussion of techniques and topics intervening in the mathematical treatment of quantum and semi-classical mechanics. It starts with a very readable introduction to symplectic geometry. Many topics are also of genuine interest for pure mathematicians working in geometry and topology.

Hamiltonian Mechanical Systems and Geometric Quantization

Hamiltonian Mechanical Systems and Geometric Quantization
Author: Mircea Puta
Publisher: Springer Science & Business Media
Total Pages: 289
Release: 2012-12-06
Genre: Mathematics
ISBN: 9401119929

This volume presents various aspects of the geometry of symplectic and Poisson manifolds, and applications in Hamiltonian mechanics and geometric quantization are indicated. Chapter 1 presents some general facts about symplectic vector space, symplectic manifolds and symplectic reduction. Chapter 2 deals with the study of Hamiltonian mechanics. Chapter 3 considers some standard facts concerning Lie groups and algebras which lead to the theory of momentum mappings and the Marsden--Weinstein reduction. Chapters 4 and 5 consider the theory and the stability of equilibrium solutions of Hamilton--Poisson mechanical systems. Chapters 6 and 7 are devoted to the theory of geometric quantization. This leads, in Chapter 8, to topics such as foliated cohomology, the theory of the Dolbeault--Kostant complex, and their applications. A discussion of the relation between geometric quantization and the Marsden--Weinstein reduction is presented in Chapter 9. The final chapter considers extending the theory of geometric quantization to Poisson manifolds, via the theory of symplectic groupoids. Each chapter concludes with problems and solutions, many of which present significant applications and, in some cases, major theorems. For graduate students and researchers whose interests and work involve symplectic geometry and Hamiltonian mechanics.

Geometry, Mechanics, and Dynamics

Geometry, Mechanics, and Dynamics
Author: Dong Eui Chang
Publisher: Springer
Total Pages: 506
Release: 2015-04-16
Genre: Mathematics
ISBN: 1493924419

This book illustrates the broad range of Jerry Marsden’s mathematical legacy in areas of geometry, mechanics, and dynamics, from very pure mathematics to very applied, but always with a geometric perspective. Each contribution develops its material from the viewpoint of geometric mechanics beginning at the very foundations, introducing readers to modern issues via illustrations in a wide range of topics. The twenty refereed papers contained in this volume are based on lectures and research performed during the month of July 2012 at the Fields Institute for Research in Mathematical Sciences, in a program in honor of Marsden's legacy. The unified treatment of the wide breadth of topics treated in this book will be of interest to both experts and novices in geometric mechanics. Experts will recognize applications of their own familiar concepts and methods in a wide variety of fields, some of which they may never have approached from a geometric viewpoint. Novices may choose topics that interest them among the various fields and learn about geometric approaches and perspectives toward those topics that will be new for them as well.

Geometric Control of Mechanical Systems

Geometric Control of Mechanical Systems
Author: Francesco Bullo
Publisher: Springer
Total Pages: 727
Release: 2019-06-12
Genre: Science
ISBN: 1489972765

The area of analysis and control of mechanical systems using differential geometry is flourishing. This book collects many results over the last decade and provides a comprehensive introduction to the area.

Introduction to Mechanics and Symmetry

Introduction to Mechanics and Symmetry
Author: Jerrold E. Marsden
Publisher: Springer Science & Business Media
Total Pages: 593
Release: 2013-03-19
Genre: Science
ISBN: 0387217924

A development of the basic theory and applications of mechanics with an emphasis on the role of symmetry. The book includes numerous specific applications, making it beneficial to physicists and engineers. Specific examples and applications show how the theory works, backed by up-to-date techniques, all of which make the text accessible to a wide variety of readers, especially senior undergraduates and graduates in mathematics, physics and engineering. This second edition has been rewritten and updated for clarity throughout, with a major revamping and expansion of the exercises. Internet supplements containing additional material are also available.

Geometric Continuum Mechanics

Geometric Continuum Mechanics
Author: Reuven Segev
Publisher: Springer Nature
Total Pages: 416
Release: 2020-05-13
Genre: Mathematics
ISBN: 3030426831

This contributed volume explores the applications of various topics in modern differential geometry to the foundations of continuum mechanics. In particular, the contributors use notions from areas such as global analysis, algebraic topology, and geometric measure theory. Chapter authors are experts in their respective areas, and provide important insights from the most recent research. Organized into two parts, the book first covers kinematics, forces, and stress theory, and then addresses defects, uniformity, and homogeneity. Specific topics covered include: Global stress and hyper-stress theories Applications of de Rham currents to singular dislocations Manifolds of mappings for continuum mechanics Kinematics of defects in solid crystals Geometric Continuum Mechanics will appeal to graduate students and researchers in the fields of mechanics, physics, and engineering who seek a more rigorous mathematical understanding of the area. Mathematicians interested in applications of analysis and geometry will also find the topics covered here of interest.

Geometric Dynamics

Geometric Dynamics
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.