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2 comments about GEOMETRIC MECHANICS: Part 2, Rotating, Translating and Rolling.

1. 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.



2. 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.

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5 comments about Fluid Mechanics.

1. This book is not good at all FOR STUDENTS. The organization is confusing and mathematics used to derive the equation does not show any physical meaning. Examples given are confusing and misleading. Avoid this book at any cost.



2. Streeter wrote this book to impress other professors only. The target of this book is for lecturer, not for college students.



3. This book does seem to cover material in reasonable detail which makes for a better understanding of the basic concepts. Perhaps it could be written more lucidly but I don't think it's as bad as one star. I might change my mind after I read a few more chapters....!



4. Okay, I am an engineer who suffered this book through three- count 'em- fluid mehcanics subjects (Fluids I, II and yes, you guessed it, III! ah the pretentions of roman numerals should have warned me...alas I digress...)

   Okay the fact is with this book you need three things to make it work: 1) a damn good lecturer, 2) Some effort to do the problems and 3)Schaum's solved problems in fluid mechanics or something similiar.

   I do disagree that it is impossible. Or REALLY hard or written to impress his professor friends...it just requires some thought...Like anything of a less common nature...less people are going to understand it. That's the nature of the beast! Sure there are many places where it falls down...particularly in the first three chapters. But later on it becomes quite good. From chapter 3 onwards it is a useful work, particularly in Part 2 on applications of fluid mechanics.

   Covers: fluid propoerties (very briefly!), Fluid statics (need the ability to do vectors to be able to do this chapter!), Fluid flow (calculus required!). Dimensional Similitude (this is cool stuff...), Fluid resistance, Compressible flow and Ideal fluid flow...comprise part 1. Part 2 is Fluid Measurement (ie. Weirs, Turbulence, viscosity, pressure, orifices...etc), Turbomachinery, Closed conduit flow, Open channel flow (not bad either), unsteady flow (good old waterhammer! How to stop water hammer when you turn on a tap/faucet...turn on another one at the same time of the same heat- hot or cold!) and some appendices on mathematical techniques: SImpsons rule, Bisection method, Newton-Raphson, Runge-Kutta and physical propoerties of some fluids. Answers (not worked) are provided from even numbered questions.

   Put it this way: if you can work through this book then you KNOW you understand fluid mechanics, so in that respect it gets a third star: it's a good yardstick of your ability. But for all that it is very disheartening, as it always is, to learn that one may not be a clever as one things one is and this is a book to expose shortcomings such as these. COnsider a challenge book: like everest...see the hill take the hill and then jeer it because you've bested it. Guess this book just takes that kind of person: pesistant, dogged and damn sure not to let it beat them.

   If you're a casual learner or not so keen on challenger yourself with something so tragic as fluid mechanics...avoid this book like the plague because you'll waste your money and probably damage something when you hurl the book with frustration!

   Hope that helps...:)




5. The Book is quite ok for introductory courses as well as applications in Hydraulics. It is biased towards civil engineering and in my opinion is a book worthy to purchase.