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.

lgli/Geometric Mechanics 2-Rotating, Translating And Rolling(2e,2011,406p)D.D.Holm_1848167784.pdf

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Geometric mechanics. 2, Rotating, translating and rolling

Holm, Darryl D

United Kingdom and Ireland, United Kingdom

2nd ed., 2011-10-31

lg3092554

{"edition":"2","isbns":["1848167784","9781848167780"],"last_page":410,"publisher":"Icp"}

Contents
Preface
1. gALILEO
1.1 Principle of Galilean relativity
1.2 Galilean transformations
1.2.1 Admissible force laws for an N-particle system
1.3 Subgroups of the Galilean transformations
1.3.1 Matrix representation of SE(3)
1.4 Lie group actions of SE(3)
1.5 Lie group actions of G(3)
1.5.1 Matrix representation of G(3)
1.6 Lie algebra of SE(3)
1.7 Lie algebra of G(3)
2. Newton, Lagrange, Hamilton and the Rigid Body
2.1 Newton
2.1.1 Newtonian form of free rigid rotation
2.1.2 Newtonian form of rigid-body motion
2.2 Lagrange
2.2.1 The principle of stationary action
2.3 Noether's theorem
2.3.1 Lie symmetries and conservation laws
2.3.2 Infinitesimal transformations of a Lie group
2.4 Lagrangian form of rigid-body motion
2.4.1 Hamilton-Pontryagin constrained variations
2.4.2 Manakov's formulation of the SO(n) rigid body
2.4.3 Matrix Euler-Poincare equations
2.4.4 An isospectral eigenvalue problem for the SO(n) rigid body
2.4.5 Manakov's integration of the SO(n) rigid body
2.5 Hamilton
2.5.1 Hamiltonian form of rigid-body motion
2.5.2 Lie-Poisson Hamiltonian rigid-body dynamics
2.5.3 Lie-Poisson bracket
2.5.4 Nambu's R3 Poisson bracket
2.5.5 Clebsch variational principle for the rigid body
2.5.6 Rotating motion with potential energy
3. Quaternions
3.1 Operating with quaternions
3.1.1 Multiplying quaternions using Pauli matrices
3.1.2 Quaternionic conjugate
3.1.3 Decomposition of three-vectors
3.1.4 Alignment dynamics for Newton's second law
3.1.5 Quaternionic dynamics of Kepler's problem
3.2 Quaternionic conjugation
3.2.1 Cayley-Klein parameters
3.2.2 Pure quaternions, Pauli matrices and SU(2)
3.2.3 Tilde map: R3 ≅ su(2) ≅ so(3)
3.2.4 Dual of the tilde map: R3* ≅ su(2)* ≅ so(3)*
3.2.5 Pauli matrices and Poincare's sphere C2→S2
3.2.6 Poincare's sphere and Hopf's fibration
3.2.7 Coquaternions
4. Adjoint and Coadjoint Actions
4.1 Cayley-Klein dynamics for the rigid body
4.1.1 Cayley-Klein parameters, rigid-body dynamics
4.1.2 Body angular frequency
4.1.3 Cayley-Klein parameters
4.2 Actions of quaternions, Lie groups and Lie algebras
4.2.1 AD, Ad, ad, Ad* and ad* actions of quaternions
4.2.2 AD, Ad, and ad for Lie algebras and groups
4.3 Example: The Heisenberg Lie group
4.3.1 Definitions for the Heisenberg group
4.3.2 Adjoint actions: AD, Ad and ad
4.3.3 Coadjoint actions: Ad* and ad*
4.3.4 Coadjoint motion and harmonic oscillations
5. The Special Orthogonal Group SO(3)
5.1 Adjoint and coadjoint actions of SO(3)
5.1.1 Ad and ad operations for the hat map
5.1.2 AD, Ad and ad actions of SO(3)
5.1.3 Dual Lie algebra isomorphism ˇ : so(3)*→R3
6. Adjoint and Coadjoint Semidirect-Product Group Actions
6.1 Special Euclidean group SE(3)
6.2 Adjoint operations for SE(3)
6.3 Adjoint actions of SE(3)'s Lie algebra
6.3.1 The ad action of se(3) on itself
6.3.2 The ad* action of se(3) on its dual se(3)*
6.3.3 Left versus right
6.4 Special Euclidean group SE(2)
6.5 Semidirect-product group SL(2, R)SR2
6.5.1 Definitions for SL(2, R)SR2
6.5.2 AD, Ad, and ad actions
6.5.3 Ad* and ad* actions
6.5.4 Coadjoint motion relation
6.6 Galilean group
6.6.1 Definitions for G(3)
6.6.2 AD, Ad, and ad actions of G(3)
6.7 Iterated semidirect products
7. Euler-Poincare and Lie-Poisson Equations on SE(3)
7.1 Euler-Poincare equations for left-invariant Lagrangians under SE(3)
7.1.1 Legendre transform from se(3) to se(3)*
7.1.2 Lie-Poisson bracket on se(3)*
7.1.3 Coadjoint motion on se(3)*
7.2 Kirchhoff equations on se(3)*
7.2.1 Looks can be deceiving: The heavy top
8. Heavy-Top Equations
8.1 Introduction and definitions
8.2 Heavy-top action principle
8.3 Lie-Poisson brackets
8.3.1 Lie-Poisson brackets and momentum maps
8.3.2 Lie-Poisson brackets for the heavy top
8.4 Clebsch action principle
8.5 Kaluza-Klein construction
9. The Euler-Poincare Theorem
9.1 Action principles on Lie algebras
9.2 Hamilton-Pontryagin principle
9.3 Clebsch approach to Euler-Poincare
9.3.1 Defining the Lie derivative
9.3.2 Clebsch Euler-Poincare principle
9.4 Lie-Poisson Hamiltonian formulation
9.4.1 Cotangent-lift momentum maps
10. Lie-Poisson Hamiltonian Form of a Continuum Spin Chain
10.1 Formulating continuum spin chain equations
10.2 Euler-Poincare equations
10.3 Hamiltonian formulation
11. Momentum Maps
11.1 The momentum map
11.2 Cotangent lift
11.3 Examples of momentum maps
12. Round, Rolling Rigid Bodies
12.1 Introduction
12.1.1 Holonomic versus nonholonomic
12.1.2 The Chaplygin ball
12.2 Nonholonomic Hamilton-Pontryagin variational principle
12.2.1 HP principle for the Chaplygin ball
12.2.2 Circular disk rocking in a vertical plane
12.2.3 Euler's rolling and spinning disk
12.3 Nonholonomic Euler-Poincare reduction
12.3.1 Semidirect-product structure
12.3.2 Euler-Poincare theorem
12.3.3 Constrained reduced Lagrangian
A. Geometrical Structure of Classical Mechanics
A.1 Manifolds
A.2 Motion: Tangent vectors and flows
A.2.1 Vector fields, integral curves and flows
A.2.2 Differentials of functions: The cotangent bundle
A.3 Tangent and cotangent lifts
A.3.1 Summary of derivatives on manifolds
B. Lie Groups and Lie Algebras
B.1 Matrix Lie groups
B.2 Defining matrix Lie algebras
B.3 Examples of matrix Lie groups
B.4 Lie group actions
B.5 Tangent and cotangent lift actions
B.6 Jacobi-Lie bracket
B.7 Lie derivative and Jacobi-Lie bracket
B.7.1 Lie derivative of a vector field
B.7.2 Vector fields in ideal fluid dynamics
C. Enhanced Coursework
C.1 Variations on rigid-body dynamics
C.1.1 Two times
C.1.2 Rotations in complex space
C.1.3 Rotations in four dimensions: SO(4)
C.2 C3 oscillators
C.3 Momentum maps for GL(n, R)
C.4 Motion on the symplectic Lie group Sp(2)
C.5 Two coupled rigid bodies
D. Poincare's 1901 Paper
Bibliography
A
B
C
D・E
F・G
H
I・J・K
L
M
N・O
P・R
S
V・W
Z
Index

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

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.

2021-08-21