Geometric Mechanics - Part Ii: Rotating, Translating And Rolling (2nd Edition)

by Darryl D Holm

Overview

Featured in the recommended literature list for the International Society of Nonlinear Mathematical Physics: https://isnmp.de/Book-Reviews-and-Recommendations/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.

Product Details

ISBN-13:	9781848167773
Publisher:	Imperial College Press
Publication date:	11/02/2011
Pages:	412
Product dimensions:	6.20(w) x 9.00(h) x 0.90(d)

Table of Contents

Preface     xiii
Galileo     1
Principle of Galilean relativity     1
Galilean transformations     2
Lie group actions of SE(3) & G(3)     8
Newton, Lagrange, Hamilton     13
Newton     13
Newtonian form of free rigid rotation     13
Newtonian form of rigid-body motion     23
Lagrange     28
The principle of stationary action     28
Noether's theorem     30
Lie symmetries & conservation laws     30
Infinitesimal transformations of a Lie group     31
Lagrangian form of rigid-body motion     39
Hamilton-Pontryagin variations     43
Manakov's formulation of the SO(n) rigid body     45
Matrix Euler-Poincare equations     46
Manakov's integration of the SO(n) rigid body     47
Hamilton     49
Hamiltonian form of rigid-body motion     50
Lie-Poisson Hamiltonian rigid-body dynamics     51
Nambu's R[superscript 3] Poisson bracket     53
Clebsch variational principle for the rigid body     56
Quaternions     61
Operating with quaternions     62
Quaternion multiplication using Paulimatrices     62
Quaternionic conjugate     64
Decomposition of three-vectors     67
Alignment dynamics for Newton's 2nd Law     68
Quaternionic dynamics of Kepler's problem     71
Quaternionic conjugation     74
Quaternionic conjugation in CK terms     74
Pure quaternions, Pauli matrices & SU (2)     79
Tilde map: R[superscript 3] [bsime] su(2) [bsime] so(3)     81
Pauli matrices and Poincare's sphere C[superscript 2] to S[superscript 2]     82
Poincare's sphere and Hopf's fibration     83
Quaternionic conjugacy     87
Cayley-Klein dynamics     87
Cayley-Klein parameters, rigid-body dynamics     87
Body angular frequency     89
Hamilton's principle in CK parameters     90
Actions of quaternions     91
AD, Ad, ad, Ad* & ad* actions of quaternions     92
AD-, Ad- & ad- for Lie algebras and groups     92
Special orthogonal group     101
Adjoint and coadjoint actions of SO(3)     101
Ad and ad operations for the hat map     101
AD, Ad & ad actions of SO(3)     103
Dual Lie algebra isomorphism [characters not reproducible]: so(3)* to R[superscript 3]      104
The special Euclidean group     109
Introduction to SE(3)     109
Adjoint operations for SE(3)     111
AD operation for SE(3)     111
Ad-operation for SE(3)     111
Ad*-operation for SE(3)     112
Adjoint actions of se(3)     114
The ad action of se(3) on itself     114
The ad* action of se(3) on its dual se(3)*     116
Left versus Right     118
Left-invariant tangent vectors     118
The special Euclidean group SE(2)     120
Geometric Mechanics on SE(3)     123
Left-invariant Lagrangians     123
Legendre transform from se(3) to se(3)*     125
Lie-Poisson bracket on se(3)*     126
Coadjoint motion on se(3)*     127
Kirchhoff equations     129
Looks can be deceiving: The heavy top     131
Heavy top equations     133
Introduction and definitions     133
Heavy top action principle     134
Lie-Poisson brackets     136
Lie-Poisson brackets and momentum maps     136
The heavy top Lie-Poisson brackets     137
Clebsch action principle     139
Kaluza-Klein construction      140
The Euler-Poincare theorem     143
Action principles on Lie algebras     143
Hamilton-Pontryagin principle     146
Clebsch approach to Euler-Poincare     147
Defining the Lie derivative     149
Clebsch Euler-Poincare principle     150
Lie-Poisson Hamiltonian formulation     152
Cotangent-lift momentum maps     153
Lie-Poisson Hamiltonian form     155
Hamiltonian continuum spin chain     156
Momentum maps     165
The standard momentum map     165
Cotangent lift     167
Examples     169
Round rolling rigid bodies     181
Introduction     182
Holonomic versus nonholonomic     182
Chaplygin's top     183
Hamilton-Pontryagin principle     187
HP principle for Chaplygin's top     190
Circular disk rocking in a vertical plane     194
Euler's rolling and spinning disk     197
Nonholonomic symmetry reduction     203
Semidirect-product structure     203
Euler-Poincare theorem     206
Geometrical structure     213
Manifolds      213
Motion: Tangent vectors and flows     221
Vector fields, integral curves and flows     222
Differentials of functions: The cotangent bundle     224
Tangent and cotangent lifts     225
Summary of derivatives on manifolds     226
Lie groups and Lie algebras     229
Matrix Lie groups     229
Defining matrix Lie algebras     234
Examples of matrix Lie groups     235
Lie group actions     237
Left and right translations on a Lie group     239
Tangent and cotangent lift actions     240
Jacobi-Lie bracket     243
Lie derivative and Jacobi-Lie bracket     245
Lie derivative of a vector field     245
Vector fields in ideal fluid dynamics     247
Enhanced coursework     249
Variations on rigid-body dynamics     249
Two times     249
Rotations in complex space     254
Rotations in four dimensions: SO(4)     258
C[superscript 3] oscillators     262
GL(n, R) symmetry     267
Bibliography     273
Index     289

Editorial Reviews

These volumes are a valuable addition to the literature on geometric mechanics. They are very well written, can be used for various courses independently or in conjunction with other texts, both more elementary and advanced, and can also serve as a good starting point for someone interested in the subject and willing to learn the principles and techniques of geometric mechanics. -- MathSciNet "MathSciNet"

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