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Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds

Taeyoung Lee, Melvin Leok, and N. Harris McClamroch
Publication Date: 
Number of Pages: 
Interaction of Mechanics and Mathematics
[Reviewed by
Michael Berg
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This is not a book on quantum mechanics or quantum field theory. I state this because my own exposure to Lagrangian and Hamiltonian dynamics comes from the comparatively narrow domain of quantum physics, where the standard approach to quantum mechanics via the Schrödinger wave equation is based on the stipulation of a Hamiltonian and then, in contrast, there is the famous reformulation of quantum mechanics in Lagrangian terms as given in Richard Feynman’s Princeton PhD thesis. The latter introduces the “integral” that is named after him; the Feynman path integral and its associated Feynman diagrams are now part and parcel of a most fecund approach to quantum electrodynamics and, subsequently, quantum field theory (into which framework both quantum mechanics and quantum electrodynamics can be fitted). Perhaps most interestingly, in the famous work on knot theory by Edward Witten, in which a knot’s Jones polynomial’s values are produced through a formalism of Feynman integrals (on a suitable low-dimensional manifold in which the knot or link is imbedded), the approach is Lagrangian: it is all built on a Chern-Simons Lagrangian constructed from a connection of a principal bundle whose fibre is a specially chosen Lie group. On the other hand, if the same knotty subject is approached through so-called axiomatic topological quantum field theory, the approach is Hamiltonian: see the gorgeous if dense book by Sir Michael Atiyah, The Geometry and Physics of Knots, for an unsurpassed (high-level) introduction to this material. In any case, the interplay between Hamiltonian and Lagrangian dynamics is at the heart of all this modern low-dimensional topology and knot theory, and this certainly augurs for the relevance and importance of these themes from physics — even for pure mathematics. It is of course already clear that it is important to applied mathematics.

That said, it is apposite that the book under review should appear in Springer Verlag’s series titled, “Interaction of Mechanics and Mathematics.” (I am reminded of “Mech-Math,” the very famous department of Mechanics and Mathematics at Lomonosov Moscow State University: this is probably the most overt paean paid to this juxtaposition in history: their claim is that you can’t have one without the other. Of course this is false, at least in one direction, but the joining of the two disciplines has certainly been fecund, and the Soviets chiseled this in architectural marble.) The book is aimed at “mathematicians, engineers, and physicists with a basic knowledge of mechanics.” I think I might barely qualify on this count, despite the fact that my exposure to the latter was gained on quite an ad hoc basis. But there’s more: we also read on the book’s back cover that “[s]ome basic background in differential geometry is helpful but not essential, as the relevant concepts are introduced in the book.” Given that I have a great deal more experience with differential geometry than with mechanics, this might suggest an easier time for me, at least in this area.

So, why so much about how the book might hit me? Well, I think that many of us mathematicians are, like me, more comfortable with differential geometry than with mechanics, and maybe much more so. Is this book a good fit for us? I think the answer is yes. One indication of this good fit is the authors’ very thorough and detailed treatment of the Euler-Lagrange equations in the middle of the book. In point of fact Euler and Lagrange don’t appear till almost three hundred pages into the book, and this is testimony to the thoroughness of the authors’ approach: the first five chapters are devoted to extensive discussions of manifolds and vector fields, kinematics, classical Lagrangian and Hamiltonian dynamics, and then, specializing, Lagrangian and Hamiltonian dynamics on Cartesian products of circles and Cartesian products of spheres. The Euler-Lagrange equations make their appearance in the context of dynamics on Lie groups, specifically \(SO(3)\) and \(SE(3)\), the latter being the special Euclidean group, realized as a submanifold of \(GL(3)\times\mathbf{R}^3\). After this work on specific Lie groups, the authors go on to the more general theme of dynamics (as always of the Lagrangian and Hamiltonian type) on manifolds and the special cases where we have a Lie group structure on the configuration manifold. The book’s last two chapters deal with rigid and multi-body systems and deformable multi-body systems. There is supplementary material on the calculus of variations (rightly so, of course) and linearization.

This well-written and expansive book is ambitious in its scope in that it aims at sound and thorough pedagogy as far as its subject matter is concerned, and it also aims at preparing the reader for computational work: note the subtitle, viz., “A geometric approach to modeling and analysis.” In that sense it is a particularly timely book, seeing that we have computing power at our disposal like never before. There are many good examples accompanying or even guiding the text, as well as extensive problem sets for the properly serious student. This book hits its targets square and should prove very valuable to its readership, be they mathematicians, engineers, or physicists. 

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

See the table of contents in the publisher's webpage.