Geometric Mechanics: Rotating, translating and rolling

Author: Darryl D. Holm

Publisher: Imperial College Press

ISBN: 1848161557

Category: Science

Page: 294

View: 6530

Introduces the tools and language of modern geometric mechanics to advanced undergraduate and beginning graduate students in mathematics, physics, and engineering. This book treats the dynamics of rotating, spinning and rolling rigid bodies from a geometric viewpoint, by formulating their solutions as coadjoint motions generated by Lie groups.

Geometric Mechanics

Part I: Dynamics and Symmetry

Author: Darryl D Holm

Publisher: World Scientific Publishing Company

ISBN: 1911298658

Category: Technology & Engineering

Page: 468

View: 6776

See also GEOMETRIC MECHANICS — Part II: Rotating, Translating and Rolling (2nd Edition) This textbook introduces the tools and language of modern geometric mechanics to advanced undergraduates and beginning graduate students in mathematics, physics and engineering. It treats the fundamental problems of dynamical systems from the viewpoint of Lie group symmetry in variational principles. The only prerequisites are linear algebra, calculus and some familiarity with Hamilton's principle and canonical Poisson brackets in classical mechanics at the beginning undergraduate level. The ideas and concepts of geometric mechanics are explained in the context of explicit examples. Through these examples, the student develops skills in performing computational manipulations, starting from Fermat's principle, working through the theory of differential forms on manifolds and transferring these ideas to the applications of reduction by symmetry to reveal Lie–Poisson Hamiltonian formulations and momentum maps in physical applications. The many Exercises and Worked Answers in the text enable the student to grasp the essential aspects of the subject. In addition, the modern language and application of differential forms is explained in the context of geometric mechanics, so that the importance of Lie derivatives and their flows is clear. All theorems are stated and proved explicitly. The organisation of the first edition has been preserved in the second edition. However, the substance of the text has been rewritten throughout to improve the flow and to enrich the development of the material. In particular, the role of Noether's theorem about the implications of Lie group symmetries for conservation laws of dynamical systems has been emphasised throughout, with many applications. Contents: Fermat's Ray Optics:Fermat's principleHamiltonian formulation of axial ray opticsHamiltonian form of optical transmissionAxisymmetric invariant coordinatesGeometry of invariant coordinatesSymplectic matricesLie algebrasEquilibrium solutionsMomentum mapsLie–Poisson bracketsDivergenceless vector fieldsGeometry of solution behaviourGeometric ray optics in anisotropic mediaTen geometrical features of ray opticsNewton, Lagrange, Hamilton and the Rigid Body:NewtonLagrangeHamiltonRigid-body motionSpherical pendulumLie, Poincaré, Cartan: Differential Forms:Poincaré and symplectic manifoldsPreliminaries for exterior calculusDifferential forms and Lie derivativesLie derivativeFormulations of ideal fluid dynamicsHodge star operator on ℝ3Poincaré's lemma: Closed vs exact differential formsEuler's equations in Maxwell formEuler's equations in Hodge-star form in ℝ4Resonances and S1 Reduction:Dynamics of two coupled oscillators on ℂ2The action of SU(2) on ℂ2Geometric and dynamic S1 phasesKummer shapes for n:m resonancesOptical travelling-wave pulsesElastic Spherical Pendulum:Introduction and problem formulationEquations of motionReduction and reconstruction of solutionsMaxwell-Bloch Laser-Matter Equations:Self-induced transparencyClassifying Lie–Poisson Hamiltonian structures for real-valued Maxwell–Bloch systemReductions to the two-dimensional level sets of the distinguished functionsRemarks on geometric phasesEnhanced Coursework:Problem formulations and selected solutionsIntroduction to oscillatory motionPlanar isotropic simple harmonic oscillator (PISHO)Complex phase space for two oscillatorsTwo-dimensional resonant oscillatorsA quadratically nonlinear oscillatorLie derivatives and differential formsExercises for Review and Further Study:The reduced Kepler problem: Newton (1686)Hamiltonian reduction by stagesℝ3 bracket for the spherical pendulumMaxwell–Bloch equationsModulation equationsThe Hopf map2:1 resonant oscillatorsA steady Euler fluid flowDynamics of vorticity gradientThe C Neumann problem (1859) Readership: Advanced undergraduate and graduate students in mathematics, physics and engineering; non-experts interested in geometric mechanics, dynamics and symmetry.

Dynamical Systems and Geometric Mechanics

An Introduction

Author: Jared Maruskin

Publisher: de Gruyter

ISBN: 3110597802

Category: Mathematics

Page: 348

View: 4800

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.

Introduction to Dynamical Systems and Geometric Mechanics

Author: Jared M. Maruskin

Publisher: Solar Crest Publishing LLC

ISBN: 0985062711

Category: Mathematics

Page: 360

View: 6636

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 explores similar systems that instead evolve on differentiable manifolds. In the study of geometric mechanics, however, additional geometric structures are often present, since such systems arise from the laws of nature that govern the motions of particles, bodies, and even galaxies. In the first part of the text, we discuss linearization and stability of trajectories and fixed points, invariant manifold theory, periodic orbits, Poincar maps, Floquet theory, the Poincar -Bendixson theorem, bifurcations, and chaos. The second part of the text begins with a self-contained chapter on differential geometry that introduces notions of manifolds, mappings, vector fields, the Jacobi-Lie bracket, and differential forms. The final chapters cover Lagrangian and Hamiltonian mechanics from a modern geometric perspective, mechanics on Lie groups, and nonholonomic mechanics via both moving frames and fiber bundle decompositions. The text can be reasonably digested in a single-semester introductory graduate-level course. Each chapter concludes with an application that can serve as a springboard project for further investigation or in-class discussion.

Differential Geometrical Theory of Statistics

Author: Frédéric Barbaresco,Frank Nielsen

Publisher: MDPI

ISBN: 3038424242

Category: Computers

Page: 472

View: 8902

This book is a printed edition of the Special Issue "Differential Geometrical Theory of Statistics" that was published in Entropy

Lie Groups, Differential Equations, and Geometry

Advances and Surveys

Author: Giovanni Falcone

Publisher: Springer

ISBN: 3319621815

Category: Mathematics

Page: 361

View: 2779

This book collects a series of contributions addressing the various contexts in which the theory of Lie groups is applied. A preliminary chapter serves the reader both as a basic reference source and as an ongoing thread that runs through the subsequent chapters. From representation theory and Gerstenhaber algebras to control theory, from differential equations to Finsler geometry and Lepage manifolds, the book introduces young researchers in Mathematics to a wealth of different topics, encouraging a multidisciplinary approach to research. As such, it is suitable for students in doctoral courses, and will also benefit researchers who want to expand their field of interest.

Geometric Mechanics

Part II: Rotating, Translating and Rolling

Author: Darryl D Holm

Publisher: World Scientific Publishing Company

ISBN: 1911299336

Category: Mathematics

Page: 312

View: 4727

This textbook introduces the tools and language of modern geometric mechanics to advanced undergraduate and beginning graduate students in mathematics, physics, and engineering. It treats the dynamics of rotating, spinning and rolling rigid bodies from a geometric viewpoint, by formulating their solutions as coadjoint motions generated by Lie groups. The only prerequisites are linear algebra, multivariable calculus and some familiarity with Euler-Lagrange variational principles and canonical Poisson brackets in classical mechanics at the beginning undergraduate level. Variational calculus on tangent spaces of Lie groups is explained in the context of familiar concrete examples. Through these examples, the student develops skills in performing computational manipulations, starting from vectors and matrices, working through the theory of quaternions to understand rotations, and then transferring these skills to the computation of more abstract adjoint and coadjoint motions, Lie-Poisson Hamiltonian formulations, momentum maps and finally dynamics with nonholonomic constraints. The 120 Exercises and 55 Worked Answers help the student to grasp the essential aspects of the subject, and to develop proficiency in using the powerful methods of geometric mechanics. In addition, all theorems are stated and proved explicitly. The book's many examples and worked exercises make it ideal for both classroom use and self-study. Contents: GalileoNewton, Lagrange, HamiltonQuaternionsQuaternionic ConjugacySpecial Orthogonal GroupThe Special Euclidean GroupGeometric Mechanics on SE(3)Heavy Top EquationsThe Euler–Poincaré TheoremLie–Poisson Hamiltonian FormMomentum MapsRound Rolling Rigid Bodies Readership: Advanced undergraduate and graduate students in mathematics, physics and engineering; researchers interested in learning the basic ideas in the fields; non-experts interested in geometric mechanics, dynamics and symmetry.

Cartan for Beginners

Differential Geometry Via Moving Frames and Exterior Differential Systems

Author: Thomas Andrew Ivey,J. M. Landsberg

Publisher: American Mathematical Soc.

ISBN: 0821833758

Category: Mathematics

Page: 378

View: 8390

This book is an introduction to Cartan's approach to differential geometry. Two central methods in Cartan's geometry are the theory of exterior differential systems and the method of moving frames. This book presents thorough and modern treatments of both subjects, including their applications to both classic and contemporary problems. It begins with the classical geometry of surfaces and basic Riemannian geometry in the language of moving frames, along with an elementary introduction to exterior differential systems. Key concepts are developed incrementally with motivating examples leading to definitions, theorems, and proofs. Once the basics of the methods are established, the authors develop applications and advanced topics.One notable application is to complex algebraic geometry, where they expand and update important results from projective differential geometry. The book features an introduction to $G$-structures and a treatment of the theory of connections. The Cartan machinery is also applied to obtain explicit solutions of PDEs via Darboux's method, the method of characteristics, and Cartan's method of equivalence. This text is suitable for a one-year graduate course in differential geometry, and parts of it can be used for a one-semester course. It has numerous exercises and examples throughout. It will also be useful to experts in areas such as PDEs and algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields.

Geometric Mechanics and Symmetry

From Finite to Infinite Dimensions

Author: Darryl D. Holm,Tanya Schmah,Cristina Stoica

Publisher: Oxford University Press

ISBN: 0199212902

Category: Mathematics

Page: 515

View: 944

Geometric Mechanics and Symmetry is a friendly and fast-paced introduction to the geometric approach to classical mechanics, suitable for a one- or two- semester course for beginning graduate students or advanced undergraduates. It fills a gap between traditional classical mechanics texts and advanced modern mathematical treatments of the subject.The modern geometric approach illuminates and unifies manyseemingly disparate mechanical problems from several areas of science and engineering. In particular, the book concentrates on the similarities between finite-dimensional rigid body motion and infinite-dimensional systems such asfluid flow. The illustrations and examples, together with a large number of exercises, both solved and unsolved, make the book particularly useful.

Modern Robotics

Author: Kevin M. Lynch,Frank C. Park

Publisher: Cambridge University Press

ISBN: 1107156300

Category: Computers

Page: 544

View: 9209

A modern and unified treatment of the mechanics, planning, and control of robots, suitable for a first course in robotics.

Geometric Mechanics: Dynamics and symmetry

Author: Darryl D. Holm

Publisher: Imperial College Press

ISBN: 1848161956

Category: Mathematics

Page: 354

View: 5926

Advanced undergraduate and graduate students in mathematics, physics and engineering.

Introduction to Mechanics and Symmetry

A Basic Exposition of Classical Mechanical Systems

Author: J.E. Marsden,Tudor Ratiu

Publisher: Springer Science & Business Media

ISBN: 0387217924

Category: Science

Page: 586

View: 9844

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.

Mathematical Methods of Classical Mechanics

Author: V.I. Arnol'd

Publisher: Springer Science & Business Media

ISBN: 1475720637

Category: Mathematics

Page: 520

View: 7055

This book constructs the mathematical apparatus of classical mechanics from the beginning, examining basic problems in dynamics like the theory of oscillations and the Hamiltonian formalism. The author emphasizes geometrical considerations and includes phase spaces and flows, vector fields, and Lie groups. Discussion includes qualitative methods of the theory of dynamical systems and of asymptotic methods like averaging and adiabatic invariance.

An Introduction to Mechanics

Author: Daniel Kleppner,Robert Kolenkow

Publisher: Cambridge University Press

ISBN: 0521198119

Category: Science

Page: 566

View: 5469

This second edition is ideal for classical mechanics courses for first- and second-year undergraduates with foundation skills in mathematics.

Classical Dynamics

Author: Donald T. Greenwood

Publisher: Courier Corporation

ISBN: 0486138798

Category: Science

Page: 368

View: 6366

Graduate-level text provides strong background in more abstract areas of dynamical theory. Hamilton's equations, d'Alembert's principle, Hamilton-Jacobi theory, other topics. Problems and references. 1977 edition.

Geometric Mechanics: Rotating, translating and rolling

Author: Darryl D. Holm

Publisher: N.A

ISBN: 9781848167759

Category: Geometry, Differential

Page: N.A

View: 3815

Advanced undergraduate and graduate students in mathematics, physics and engineering.

Finite Element Procedures

Author: Klaus-Jürgen Bathe

Publisher: Klaus-Jurgen Bathe

ISBN: 9780979004902

Category: Engineering mathematics

Page: 1037

View: 1921

Theory of Applied Robotics

Kinematics, Dynamics, and Control (2nd Edition)

Author: Reza N. Jazar

Publisher: Springer Science & Business Media

ISBN: 1441917500

Category: Technology & Engineering

Page: 883

View: 829

The second edition of this book would not have been possible without the comments and suggestions from students, especially those at Columbia University. Many of the new topics introduced here are a direct result of student feedback that helped refine and clarify the material. The intention of this book was to develop material that the author would have liked to have had available as a student. Theory of Applied Robotics: Kinematics, Dynamics, and Control (2nd Edition) explains robotics concepts in detail, concentrating on their practical use. Related theorems and formal proofs are provided, as are real-life applications. The second edition includes updated and expanded exercise sets and problems. New coverage includes: components and mechanisms of a robotic system with actuators, sensors and controllers, along with updated and expanded material on kinematics. New coverage is also provided in sensing and control including position sensors, speed sensors and acceleration sensors. Students, researchers, and practicing engineers alike will appreciate this user-friendly presentation of a wealth of robotics topics, most notably orientation, velocity, and forward kinematics.

Introduction to Sports Biomechanics

Analysing Human Movement Patterns

Author: Roger Bartlett

Publisher: Routledge

ISBN: 1135818185

Category: Science

Page: 304

View: 1191

First published in 1996. Routledge is an imprint of Taylor & Francis, an informa company.

Nonholonomic Mechanics and Control

Author: A.M. Bloch

Publisher: Springer

ISBN: 1493930176

Category: Science

Page: 565

View: 9429

This book explores connections between control theory and geometric mechanics. The author links control theory with a geometric view of classical mechanics in both its Lagrangian and Hamiltonian formulations, and in particular with the theory of mechanical systems subject to motion constraints. The synthesis is appropriate as there is a rich connection between mechanics and nonlinear control theory. The book provides a unified treatment of nonlinear control theory and constrained mechanical systems that incorporates material not available in other recent texts. The book benefits graduate students and researchers in the area who want to enhance their understanding and enhance their techniques.