Applications of Lie Groups to Differential Equations (Graduate Texts in Mathematics, 107) 2nd Edition by Peter J. Olver (PDF)

Description

A solid introduction to applications of Lie groups to differential equations which have proved to be useful in practice. The computational methods are presented such that graduates and researchers can readily learn to use them. Following an exposition of the applications, the book develops the underlying theory, with many of the topics presented in a novel way, emphasising explicit examples and computations. Further examples, as well as new theoretical developments, appear in the exercises at the end of each chapter.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The book is good. And the shipping is fast.

⭐Probably the definitive work on connecting Lie group theory to Noether’s Theorem. Main complaint is that there are many instances of Olver skipping some important calculational details that require the reader to dig out some key steps in understanding how general principles may be applied.

⭐This book is excellent. Olver presents a vast amount of material in full detail, and does it in an accessible way. Unfortunately the Kindle edition has many typos which are not present in the original book. In one proof (proposition 3.37), several lines are cut out of the middle (mid-sentence) and pasted back at the end of the proof. If you are familiar with this book and the subject matter then you can make sense of things. Otherwise, I would advise you to get a hard copy.

⭐This is a thorough exposition of symmetry methods for ODE’s, PDE’s and Hamiltonian systems. It is a graduate level reference and not an introductory text.

⭐Excellent

⭐A fascinating book. Also, one of my favorites. Yet, I confess, portions of this text are over my head (as the proof of Lemma #7.25, page 448 !). Read the acknowledgements, you will find the names of Garrett Birkhoff, Willard Miller and Davis Sattinger (among others). This is interesting: Why interesting ? Because, in my quest to find accessible literature regarding group theory, those are three names I well remember. The first two (Birkhoff and Miller) provide literature for background to Peter Olver, while the third (Sattinger) provides a companion textbook: that is, the introductory book coauthored with Weaver: Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics. Before study of Olver, read chapter four of Ince’s Ordinary Differential Equations: continuous transformation groups. Allowing for the fact that I have not understood everything Peter Olver has to offer, allow me to review (at least) those portions that I do (believe) I understand !(1) Vector field and its flow, as Peter Olver reminds us, are central to the entire enterprise. Chapter one, we read: “almost all the information in the group is contained in its Lie algebra,” and “it enables us to replace complicated nonlinear conditions of invariance under a group action by relatively simple linear infinitesimal conditions.” (page 43). There is a survey of differential forms, but, Peter Olver reminds the reader that “I have tended to de-emphasize their use.” The first exercise of chapter one (all of the exercises are situated at the end-of-chapter) is a favorite topic: Projective Space (it is a three-part exercise; one out of thirty-seven !). Many of the exercises are multi-part problems.(2) Chapter one was warm-up. Chapter two, symmetry groups. The example (page 90, rotation group) should be quite accessible (actually, chapter two is easier than chapter one !). Prime concepts: prolongation and jet bundles. That is, “various partial derivatives occurring in the system.” An example (page 100) heat equation, is quite accessible ! You get the rotation group, SO(2), again, this time with prolongations (jets). Read: “from this relatively simple example the reader can appreciate the complexity of the operation of prolonging a group of transformations.” (page 102). You will see the rotation group again (page 107). So, a pedagogic strategy: Revisiting an example at higher levels of abstraction as we progress. This is excellent pedagogy ! We are up to chapter two and I am getting excited about this book all over again ! Recall the heat equation (page 100). Revisit it, again (pages 120 and 194). Remember that first exercise in chapter one ? Projective space, it is mentioned again (solvable groups, page 154). Chapter two is a superb exposition and the exercises there are replete with “applied” partial differential equations.(3) Chapter one was warm-up. Chapter two was a real winner. Chapter three will be “practical implementation.”That implementation is (as I have already noted) evident for heat equation (page 194, and again, page 212). However, I do highlight the exposition of quotient manifolds (section #3.4, pages 213-218), about which read: “the geometry underlying these constructions.” Also, the exposition of dimensional analysis (page 218) is excellent material ! We will see “jet-spaces” again: “the real key that unlocks the geometrical insight behind the construction of group-invariant solutions….” (page 228). Remember SO(2) and quotient manifolds ? We met those concepts earlier, they reappear ! How delightful is that ! (see page 234, example # 3.39). Again, pedagogic excellence throughout ! An exercise: “Discuss the group-invariant solutions to the two-dimensional wave equation.” (exercise # 3.4, page 242).(4) A summary: chapter one covered preliminaries. Chapter two introduced jets and prolongations. Chapter three was implementation of the previous chapters. Now: chapter four, conservation laws. That implies variational methods. Much physics reviewed here. In particular, I highlight two examples (gravitational field, page 263) and a discussion of fluid dynamics (page 266). I inform the reader that some of the proofs are tricky to assimilate (at least, they were for me, see page 273). The discussion of Noether’s theorem will consume the better part of ten pages. An exercise informs us that “the heat equation cannot be put into variational form, except through some artificial tricks. Prove, however, that the scale-invariant solution is equivalent to Euler-Lagrange equation.” (hint provided, exercise # 4.14, page 291). Beautiful !(5) Chapter five, a “generalization” of all that came before. an exceptional discussion. We will generalize everything: vector fields (page 296), heat equation (page 297), prolongations (page 306). The new concept is recursion operators: “a mechanism for generating infinite hierarchies of generalized symmetries.” Revisit heat equation by way of recursion operator (example # 5.25, page 315). Meet Frechet derivatives (page 317). Yet, another pass at conservation laws and symmetries (meet the concept of adjoint of differential operator). Yet, another pass at Noether’s theorem (meet the concept of self-adjoint linear systems, page 329) and revisit the Kepler problem (we saw that previously in an example, gravitational field, page 263). The toughest material is ahead: section #5.4, variational complex. Peter Olver reminds us to “become thoroughly familiar with the concepts of ordinary differential forms on manifolds, section #1.5…” before continuing. Excellent advice ! Fascinating material !(6) Let me bring this review to a conclusion. There are two chapters remaining: Hamiltonian systems. If the reader survived through chapter five (in my opinion, the most difficult of all of the previous material), then there is nothing particularly difficult in the remaining two chapters: “the interplay between symmetry groups, conservation laws, and reduction of order for systems in Hamiltonian form.” (page 379). No previous knowledge is assumed, a self-contained introduction ! What we did for finite systems (chapter six) will be generalized for infinite systems (chapter seven). Enjoy and explore ! This textbook is one of my favorites. Not easy-going, but ever-so enriching !

⭐First, let me preface this by saying my review is based on the FIRST EDITION of the book. Also, I have not read the entire thing, but much of it. I had no idea what a Lie Group was before picking this book up and found it to be an excellent introduction to a very fascinating subject.The autor gives a fairly rigorous explication of the fundamentals of manifolds and groups in the first chapter, skipping proofs of harder facts. He then spends the rest of the book focusing on how to find symmetry groups of differential equations and their interpretation. He goes through detailed calculations and provides many helpful examples, without which I would have no chance of understanding the book. He gives very readable and easily applicable formulas for prolongation of group actions and vector fields, and supplies the heavy-handed theorems relating subvarieties of the prolonged group actions to symmetry groups of the DE’s.Algebraists will find the book lacking in details and probably fairly myopic in scope. Applied people such as myself will find it indispensible as a resource for actual computation. The focus of the book is consistent with the original formulations by Lie and Noether and is still relevant and largely untaught in standard courses. Reading this book, I have learned some very helpful TECHNIQUES, and I suspect if that’s what you’re looking for this book will be a Godsend.

⭐Not proper for first contact with the subject ,but as bibliographical resource, this is a realy good one.

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