GEOMETRIC MECHANICS: Dynamics and Symmetry (Pt. I) |  | Author: Darryl D. Holm
Publisher: Imperial College Press
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Media: Hardcover
Pages: 376
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Dimensions (in): 9.1 x 6.1 x 0.9
ISBN: 1848161956
Dewey Decimal Number: 515
EAN: 9781848161955
ASIN: 1848161956
Publication Date: May 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 ray optics, resonant oscillators and the elastic spherical pendulum from a unified geometric viewpoint, by formulating their solutions using reduction by Lie-group symmetries. The only prerequisites are linear algebra, calculus and some familiarity with the Euler Lagrange variational principles 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 aid 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 book's many worked exercises make it ideal for both classroom use and self-study. In particular, a substantial appendix containing both introductory examples and enhanced coursework problems with worked answers is included to help the student develop proficiency in using the powerful methods of geometric mechanics. Contents: Fermat s Ray Optics; Newton Lagrange, Hamilton; Differential Forms; Resonances and S1 Reduction; Elastic Spherical Pendulum; Maxwell Bloch Equations.
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| Customer Reviews:
Discovery in Geometric Mechanics Part I September 16, 2008
J. Jones (London)
1 out of 1 found this review helpful
This book is a page-turner. Reading it doesn't feel like text-book learning. Instead, the book conveys a sense of story-telling and discovery. The story of each example is told efficiently, but lovingly. And each example reveals yet another interesting facet of geometric mechanics.
The first chapter spends 80 pages telling the thrilling story of Fermat's Principle (FP) for ray optics. Along the way, it reveals the essentials of geometric mechanics without missing a beat. The first page defines light rays via FP. The second page uses it to explain mirages. The third page derives the eikonal equation from FP and a moment later the Euler-Lagrange equations appear. Huygens wavelets are explained by complementarity with light rays, and presto the scalar Hamilton-Jacobi equation appears. We get this far in only 9 pages, but everything is complete and interwoven, with no gaps. The remainder of Chapter 1 develops the key concepts of geometric mechanics flowing naturally from FP. Of course, geodesic flows appear and the laws of refraction discovered by Ibn-Sahl (984) and rediscovered much later by Snell (1621). However, did you know that ray optics has a geometric phase? Did you know that it has two different kinds of momentum maps, one of which was known to Lagrange? You might have guessed its Poisson bracket. But what about its Nambu bracket?
Armed with most of the essential concepts of geometric mechanics by the FP example in the first chapter, the rest of the book gets to the modern concepts just as quickly and efficiently, through a series of illuminating examples. These examples include: the free rigid body; ideal fluid dynamics; resonantly-coupled nonlinear oscillators; bifurcation sequences of nonlinear optical traveling-wave pulses; the remarkable step-wise precession of the swing plane of the elastic spherical pendulum; and the many geometric reductions of the Maxwell-Bloch equations for self-induced transparency in laser-matter interaction.
Review of Geometric Mechanics (Part I: dynamics and symmetry) August 29, 2008
Matthew F. Dixon (Stanford, CA)
5 out of 5 found this review helpful
Although Chapter 1 is a gentle entry point for anyone who has completed a first year mechanics course, it introduces a great deal of essential concepts. Using the ray optics example, this Chapter introduces Hamilton's principle in canonical variables, the canonical Poisson Bracket, the Hamiltonian optics equations, momentum maps and the conserved skewness. The exercises instill intuition and help develop an understanding of why the skewness is conserved in addition to helping the student get to grips with the material. At this stage, no knowledge of linear algebra, differential geometry or group theory is required - so the first three Sections really set the reader up with a solid intuition and command of the basics in preparation for the more technical material that follows.
I particularly felt at ease with the 'hands-on' approach that is often lacking in graduate applied mathematics texts. By referring to the index, I was quickly able to structure my understanding through the recurring themes of the book, which time after time revealed fundamental concepts, such as geometric phase, in simple examples of reduction.
The following four Sections prepare the reader with the language of modern geometric mechanics and take us from canonical phase space to phase planes for radial position and momentum. In the axisymmetric invariant coordinates, Section 1.4 and 1.5 introduce the reader to orbit manifolds and flows on Hamiltonian vector fields. At this point, I think that readers will really start to appreciate the simplicity of the example in getting to grips with the geometry of the solution as many interesting properties are revealed just from the S^1 symmetries. The reader is exposed to the actions defined by the canonical Poisson brackets associated with various phase space functions which is conveniently written in matrix form. The next Sections fit together effortlessly. In Section 1.6, we learn that the matrices giving the finite transformations are symplectic. Not only that, but we are introduced to Lie transformation groups through the symplectic group and before long we are given all the machinery to understand the non-canonical R^3 Poisson bracket for ray optics and how it relates to the canonical one. Staying with the symplectic group, we are introduced to arguably the most fundamental concept in the Chapter - momentum maps.
I found the summary of the properties of momentum maps a useful motivation for getting to grips with the material. I also found myself returning to the earlier material in the Section and asking myself how the canonical flow and properties such as 'Skewness' are mapped to the reduced phase space through the momentum map. This is really where the text starts to pay off - I found that I quickly had obtained a sounder understanding of how the various concepts fit together than ever before.
I also found the final Section on 'ten geometrical features of ray optics' very helpful in assimilating the various concepts introduced in the Chapter. So overall, a very workable and tightly interwoven sequence of Sections with plenty of milestones and examples to retreat to if the material doesn't stick on a first read.
Chapter 2 turns to particle mechanics - the first few Sections prepare the reader for describing geodesic motion on Riemannian manifolds. In doing so, the rigid body motion that follows is perfectly set up. We see the role of the inertia matrix as a metric and familiarize ourselves with the notion of a Lie group as a smooth manifold. The remaining Sections of the Chapter provide an opportunity to take models studied in standard mechanics classes and cast them into the geometric framework. I found the comparison essential, as this is where the reader can test how well they understand the concepts in the first Chapter. The spherical pendulum is a useful starting point for the material on the elastic spherical pendulum in Chapter 5 and the bead-on-the-hoop model is helpful in understanding the optical traveling wave model in Chapter 4.
I found all the material in the first part of Chapter 3 clear enough to be able to jump straight into Section 3.5 on Euler's fluid equations. The material presented in this Chapter should be familiar to anyone whose taken courses on the mathematical description of fluid dynamics or who has a fluid mechanics background. The material in Section 3.5 is especially fundamental and sits along side the few other great texts on topological fluid dynamics and the mathematical description of fluid dynamics. It's especially useful to anyone interested in geophysical fluids. We are also taken right to the forefront of geometric fluid dynamics - inconclusive issues are indicated and modern techniques for determining stability of equilibria such as the Energy-Casimir method are applied to the non-trivial Euler-Boussinesq equations. Through the combined coverage of the Euler and the Euler-Boussinesq methods we learn the geometric effect of the additional buoyancy term. We also learn under what conditions the exterior derivative of the Euler Fluid equation gives a Stokes theorem and the role that helicity plays in the geometric description of Euler flows. The first few Sections on differential forms and exterior calculus comprise an excellent reference in their own right.
I was glad to finally see a unified treatment of resonances appearing in a geometric mechanics textbook. It's also apparent for the range of applications presented, just how widespread geometric mechanics has become. Admittedly, I read Chapter 4 before 3 as it's more accessible than the material on ideal fluid dynamics in Chapter 3, which requires familiarity with differential forms. This Chapter also makes some essential references to the first Chapter which really helps the reader to relate the material such as the correspondence of orbit manifolds, quotient maps, Poincare spheres, Nambu and R^3 bracket for 1:1 and then n:m resonances to the material in Chapter 1 on optical ray transmission. Many of these properties follow effortlessly having worked hard on Chapter 1 - so I found this material to serve as an excellent example of the power of geometric mechanics. The remainder of the Chapter considers optical traveling-wave pulses and shows us how to reduce the dynamics to the Poincare sphere. In doing so we are reminded of the similarity of the reduced traveling-wave equations with the bead on the hoop model in Chapter 2. The Chapter ends with analysis of the bifurcation of the
reduced dynamics (for a non-parity invariant material). Like all the visualizations in this book, I found that the visualization of the bifurcation choreography served to develop my intuition on the qualitative features of the reduced dynamics and motivate independent enquiries.
In Chapter 5 we learn how to craft models for describing multi-scale wave propagation phenomena by approximating the Lagrangian for small excitations and then averaging over its oscillating phases. I was pleased to find material on three-wave interaction and precession of the swing plane as these are rarely presented in a mathematical context in graduate texts. In Chapter 6, we are led through another application of the Averaged Lagrangian method in deriving phase-averaged equations of motion, referred to here as the Maxwell-Schrodinger envelope (MSE) equations. The example serves to remind us that by studying a seemingly over-simplified model, geometric mechanics reveals the understanding necessary to describe challenging scientific problems. We are swiftly shown that the three-wave equations for the 1:1:2 resonance of the elastic spherical pendulum are identical to the MSE equations for describing self-induced transparency of an optical laser pulse. The remaining material treats the reduced dynamics of the MSE equations by presenting the Lie-Poisson system and systematically dealing with different classes of Casimir functions, each one revealing unique and entirely different classes of level sets. This material demonstrates the power of the geometric approach in systematically reducing a challenging problem into a series of more workable ones.
Overall this book equips an advanced undergraduate or graduate student in applied mathematics, physics or engineering with both the mathematical tools and orientation to be able to confidently undertake research in the field of modern geometric mechanics. The Sections in the first Chapter are cleverly woven together so that the less mathematically advanced student can concentrate on understanding the principles rather than wrestle with the technicalities. The examples serve as portals from standard classical mechanics to modern geometric mechanic and help the reader become acquainted with the language and various more advanced aspects of geometric mechanics. They quickly prove the merits of geometric mechanics over its classical counterpart by revealing beautiful phase portraits and stability properties which develop a qualitative intuition for the geometric description. Some of these examples, despite their simplicity, bear all the essential material with which to tackle many challenging modern engineering and physics problems. For example, the Chapter on the elastic spherical pendulum is a very useful entry point for describing the powerful concept of averaged Lagrangians, a critical step in understanding the dynamics of three-wave interactions. Time and time again, we see throughout the book fundamental concepts revealed through recurring themes which the index is set up to help the reader pursue.
At the same time this book is even more compelling through its demonstration of geometric mechanics as a framework to solve challenging problems by linking different subjects in applied mathematics such as dynamical systems, fluid dynamics and the linear and non-linear theory of waves. We are shown throughout the book how the geometric description is a central tenet of discoveries in fluid stability, fluid conservation laws, bifurcation analysis and wave interaction. Throughout the book we see how often dispensing of unnecessary coordinate systems, degrees of freedom and approximations and instead adopting the geometric approach provides a systematic basis for classifying and simplifying problems to reveal hidden mysteries.
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