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Author: Darryl D Holm Publisher: World Scientific Publishing Company ISBN: 1911299336 Category : Mathematics Languages : en Pages : 311
Book Description
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.
Author: Darryl D Holm Publisher: World Scientific Publishing Company ISBN: 1911299336 Category : Mathematics Languages : en Pages : 311
Book Description
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.
Author: Holm Darryl D Publisher: World Scientific ISBN: 1911298666 Category : Mathematics Languages : en Pages : 412
Book Description
See also GEOMETRIC MECHANICS — Part I: Dynamics and Symmetry (2nd Edition) This textbook introduces modern geometric mechanics to advanced undergraduates and beginning graduate students in mathematics, physics and engineering. In particular, it explains 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.The book uses familiar concrete examples to explain variational calculus on tangent spaces of Lie groups. 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, 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 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. Many worked examples of adjoint and coadjoint actions of Lie groups on smooth manifolds have also been added and the enhanced coursework examples have been expanded. The second edition is ideal for classroom use, student projects and self-study./a
Author: Holm Darryl D Publisher: World Scientific Publishing Company ISBN: 1911298658 Category : Mathematics Languages : en Pages : 468
Book Description
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./a
Author: Darryl D. Holm Publisher: Imperial College Press ISBN: 1848161557 Category : Science Languages : en Pages : 311
Book Description
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.
Author: Jared Maruskin Publisher: Walter de Gruyter GmbH & Co KG ISBN: 3110597802 Category : Science Languages : en Pages : 348
Book Description
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.
Author: Heinz Schade Publisher: Walter de Gruyter GmbH & Co KG ISBN: 3110405490 Category : Mathematics Languages : en Pages : 471
Book Description
Tensor calculus is a prerequisite for many tasks in physics and engineering. This book introduces the symbolic and the index notation side by side and offers easy access to techniques in the field by focusing on algorithms in index notation. It explains the required algebraic tools and contains numerous exercises with answers, making it suitable for self study for students and researchers in areas such as solid mechanics, fluid mechanics, and electrodynamics. Contents Algebraic Tools Tensor Analysis in Symbolic Notation and in Cartesian Coordinates Algebra of Second Order Tensors Tensor Analysis in Curvilinear Coordinates Representation of Tensor Functions Appendices: Solutions to the Problems; Cylindrical Coordinates and Spherical Coordinates
Author: Giovanni Falcone Publisher: Springer ISBN: 3319621815 Category : Mathematics Languages : en Pages : 361
Book Description
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.
Author: Yair Shapira Publisher: Springer Nature ISBN: 3031224221 Category : Mathematics Languages : en Pages : 583
Book Description
This textbook demonstrates the strong interconnections between linear algebra and group theory by presenting them simultaneously, a pedagogical strategy ideal for an interdisciplinary audience. Being approached together at the same time, these two topics complete one another, allowing students to attain a deeper understanding of both subjects. The opening chapters introduce linear algebra with applications to mechanics and statistics, followed by group theory with applications to projective geometry. Then, high-order finite elements are presented to design a regular mesh and assemble the stiffness and mass matrices in advanced applications in quantum chemistry and general relativity. This text is ideal for undergraduates majoring in engineering, physics, chemistry, computer science, or applied mathematics. It is mostly self-contained—readers should only be familiar with elementary calculus. There are numerous exercises, with hints or full solutions provided. A series of roadmaps are also provided to help instructors choose the optimal teaching approach for their discipline. The second edition has been revised and updated throughout and includes new material on the Jordan form, the Hermitian matrix and its eigenbasis, and applications in numerical relativity and electromagnetics.
Author: Darryl D Holm
Author: Darryl D Holm
Author: Holm Darryl D
Author: Holm Darryl D
Author: Darryl D. Holm
Author: Darryl D. Holm
Author: Jared Maruskin
Author: Heinz Schade
Author: Frédéric Barbaresco
Author: Giovanni Falcone
Author: Yair Shapira