GEOMETRIC MECHANICS: Part 2, Rotating, Translating and Rolling (Pt. II) |  | Author: Darryl D. Holm
Publisher: Imperial College Press
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Pages: 312
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Dimensions (in): 9.1 x 6.1 x 0.8
ISBN: 1848161557
Dewey Decimal Number: 531
EAN: 9781848161559
ASIN: 1848161557
Publication Date: April 14, 2008
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Product 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: Galileo; Newton, Lagrange, Hamilton; Quaternions; Quaternionic Conjugacy; Special Orthogonal Group; The Special Euclidean Group; Geometric Mechanics on SE(3); Heavy Top Equations; The Euler Poincaré Theorem; Lie Poisson Hamiltonian Form; Momentum Maps; Round Rolling Rigid Bodies.
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| Customer Reviews:
Postdoc's review of part II October 8, 2008
Matthew F. Dixon (Stanford, CA)
2 out of 2 found this review helpful
I bought both volumes of geometric mechanics and I am glad that I did.
The first volume provides a road map through geometric mechanics and
cleverly serves to equip the advanced undergraduate or graduate student
with powerful methodology, intuition and knowledge of the research
literature. The second volume is based on lecture notes for a geometric
mechanics class given by the author in the mathematics department at
Imperial College. I was fortunate enough to take this class and volume
II, for me personally, is a 'keep-for-life' copy of lecture notes which
were the single most useful material that I encountered as a graduate
student.
Following in the format of the first volume, this book concentrates on
the most fundamental aspects of modern geometric mechanics - many of
which are covered in a text book for the first time here. The Chapters
on Quaternions, the Euler-Poincare theorem and the Hamilton-Pontryagin
principle are the devices through which cutting edge science is being
done as we speak. Similarly, the book also quickly establishes a
conceptual model through which to structure the material. In contrast to
volume I, however, this is done less by example and more by theory.
In Chapter 2, we are introduced to the body and spatial representations
of rigid body dynamics. The body and the spatial representation of rigid
body dynamics correspond to the convective and spatial representation of
continuum dynamics respectively. The spatial representation of the rigid
body dynamics is also key to understanding the geometric description of
continuum dynamics, in which quantities are advected by the continuum.
We also learn of the relative merits of the Lagrangian or Hamiltonian
description of dynamics. In fact, Chapter 2 goes so far as to prepare
the reader for Chapter 9, clearly introducing the Euler-Poincare
theorem, the Hamilton Pontryagin principle and the Clebsch variational
principle. These are the pegs that leading researchers in this field
hang their hats on.
Chapters 3 and 4 are exclusive in their coverage of quaternions from a
geomechanics perspective. We learn of the Hopf fibration and how to
represent unit quaternions as SU(2) elements. The exposition is both
technical and motivating. If the alignment dynamics of particle
trajectories, the vorticity vector in an incompressible fluid can be
conveniently reworked into the quaternionic formulation, what other
classical models have yet to be considered?
Chapters 5 introduces the reader to co-adjoint and co-Adjoint orbits on
so(3)*, adjoint operations on SE(3), semi-direct products and the
diamond operator. These are technicalities essential for Chapter 9 and
there are several exercises to become conversant with this material.
Chapter 6 distinguishes left and right invariance, a concept which is
crucial to the distinction between the body and spatial representations
of rigid body dynamics. The payoff from investing time in these Chapters
comes in Chapter 7.
Through the Euler Poincare equation for SE(3), we are introduced to the
powerful Kelvin-Noether theorem, which is associated to the Kelvin
circulation theorem of ideal fluid motion. It is remarkable that a
vector and an element of SO(3) can be, through the language of geometric
mechanics, used to describe ideal fluid circulation. Staying with SE(3),
as the configuration space, Chapter 8 challenges readers to assimilate
their understanding of geometric mechanics with the physical laws of
heavy top motion. We are further introduced to the Clebsch action
principle for the heavy top and the Kaluza-Klein construction which is
best known for describing the dynamics of a charged particle in a
magnetic field. Section 8.3.1. on Lie-Poisson brackets and momentum maps
is particularly helpful in seating the material introduced in further
Chapters.
For the student that has carefully worked through the earlier Chapters,
Chapter 9 is where the formalism starts to give away to the profound.
Through the Clebsch Euler-Poincare principle, we see how the definition
of the diamond operator, the ad and ad* operators and (later in Chapter
11) the cotangent lift momentum maps are all systematically given just
by taking variations of the Clebsch action principle. Chapter 10 takes
us from rigid body dynamics to the continuum through the continuum spin
chain. This is a bridging chapter for any student embarking on research
in geometric continuum dynamics and supplements the material presented
in Volume I on continuum dynamics with a more detailed exposition.
Chapter 12 both serves to fortify the material presented in Chapter 9
and demonstrate the utility of modern geometric mechanics to the study
of dynamical systems. Through casting the problem of Chaplygin's top and
Euler's disk into the framework of Hamilton Pontryagin, powerful results
are systematically derived which arm the student with the ability to
pursue a deeper and broader analysis.
Overall, this book serves as an essential text for independent and class
study of modern geometric mechanics. Whilst the material is
self-contained and an excellent reference, the text carefully
substantiates many of the expositions and exercises and complements the
format of Volume I by presenting a more axiomatic discourse. The book is
dedicated to ensuring that the student is able to leverage the power of
modern geometric mechanics through the application of the Clebsch
Euler-Poincare and Hamilton Pontryagin principles.
Discovery in Geometric Mechanics Part II September 16, 2008
J. Jones (London)
2 out of 2 found this review helpful
This text reveals the concepts of geometric mechanics that are discovered in the course of solving its key finite-dimensional problems.
Galilean relativity and the idea of inertial reference frames are explained in Chapter 1. Rotating motion is then treated in Chapters 2, 3 and 4, first by reviewing Newton's and Lagrange's approaches, then by following Hamilton's approach via quaternions and Cayley-Klein parameters, not Euler angles.
Hamilton's rules for multiplication of quaternions introduce the adjoint and coadjoint actions that lie at the heart of geometric mechanics. For the rotations and translations in three dimensions studied in Chapters 5 and 6, the adjoint and coadjoint actions are both equivalent to the vector cross product. Poincaré [1901] opened the field of geometric mechanics by noticing that the coadjoint action of a Lie algebra on its dual space defines the motion generated by any Lie group.
When applied to Hamilton's principle defined on the tangent space of an arbitrary Lie group, the adjoint and coadjoint actions studied in Chapter 6 result in the Euler-Poincaré equations derived in Chapter 7. Legendre transforming the Lagrangian in Hamilton's principle summons the Lie-Poisson Hamiltonian formulation of dynamics on a Lie group.
The Euler-Poincaré equations provide the framework for all of the applications treated in this text. These applications include finite dimensional dynamics of three-dimensional rotations and translations in the special Euclidean group SE(3). The Euler-Poincaré problem on SE(3) recovers Kirchhoff's classic treatment in modern form of the dynamics of an ellipsoidal body moving in an incompressible fluid flow without vorticity.
The Euler-Poincaré formulation of Kirchhoff's problem on SE(3) in Chapter 7 couples rotations and translations, but it does not yet introduce potential energy. The semidirect-product structure of SE(3), however, introduces the key idea of semidirect-production reduction for incorporating potential energy. Namely, the same semidirect-product structure is also invoked in passing from rotations of a free rigid body to rotations of a heavy top with a fixed point of support under gravity.
The heavy top treated in Chapter 8 is a key example, because it introduces the dual representation of the action of a Lie algebra on a vector space. This is the diamond operation, by which the forces and torques produced by potential energy gradients are represented in the Euler-Poincaré framework in Chapters 9 and 10. The diamond operation is then found in Chapter 11 to lie at the heart of the standard (cotangent-lift) momentum map.
This observation reveals the relation between the results of reduction by Lie symmetry on the Lagrangian and Hamiltonian sides. Namely,
Lie symmetry reduction on the Lagrangian side produces the
Euler-Poincaré equation, whose formulation on the Hamiltonian side as a
Lie-Poisson equation governs the dynamics of the momentum map
associated with the cotangent lift of the Lie-algebra action of that
Lie symmetry on the configuration manifold.
The primary purpose of this book is to explain that statement, so that it may be understood by undergraduate students in mathematics, physics and engineering.
In the Euler-Poincaré framework, the adjoint and coadjoint actions combine with the diamond operation to provide a powerful tool for investigating other applications of geometric mechanics, including nonholonomic constraints discussed in Chapter 12. In this chapter, nonholonomic mechanics is discussed in the context of two classic problems, known as Chaplygin's top (a rolling ball whose mass distribution is not symmetric) and Euler's disk (a spinning, falling, rolling, flat coin). In these classic examples, the semidirect-product structure couples rotations, translations and potential energy together with the rolling constraint.
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