FUNDAMENTAL PRINCIPLES OF CLASSICAL MECHANICS: A GEOMETRICAL PERSPECTIVE

This is a fantastic mix between Differential Geometry/Manifold Theory/Differential Forms and Mechanics. Relatively advanced concepts like the KAM theorem, Principal Bundles, the Darbeaux Theorem, Symplectic Geometry and restricted Three-Body may seemingly intimidate the reader, but the book starts out gently with the fundamental problem of electrodynamics (classical atomic modeling), basic Lagrangian and Hamiltonian formalism and even Newtonian mechanics located after a thorough introduction that at least makes this mathematician proud.It should be made clear, however, that this is primarily a Mathematical Physics book intending to construct a modeling of mechanics with modern differential geometric concepts. Much like Sternberg did with circuit theory, electrodynamics, optics and thermodynamics in his two-volume Mathematical Physics set and Callahan did with General Relativity, here Lam finally takes upper undergraduate/beginning graduate geometric mathematical physics to task on its orginal home field of Classical and Modern Mechanics. It is a much welcome sophomore volume linking Singer's introduction to Arnold's great work on the subject, and if you're at all interested in applications of differential geometry, you would do well in picking it up.If you need a thorough, graduate- or upper undergradute-level approach to the subject, however, this volume is not for you; you should focus on getting Goldstein, Taylor or Morin.

This has become my favorite Classical Mechanics book, mainly because it does such a good job of teaching the subject in a differential geometry framework. There are several chapters on moving frames, connections, fiber bundles, etc that are just pure gold. Usually this material is presented in 'higher-level' contexts, where it's much more abstract and really tough to develop an intuition for how it relates to physics. Lam does a great job illustrating the use of these higher-level concepts in familiar contexts that make them appear very natural and intuitive.

This is a modern text emphasizing the algebra of forms and geometric point of view. The exercises and examples encourage mastery of this new language, which began to become more current with Harley Flanders' text in the 70s and is now even dominant in theoretical circles. However, in attempting to apply these concepts in concrete settings, I have encountered great difficulty. More details in methodology emerge from less enlightened, clunkier and much older texts. If there were a text somewhere that works out examples of problems in at 4D (2 space and 2 momentum) in all their glory, I would immediately buy it. Elementary examples in 2D leave out the geometry; 6D is good but often too much, but 4D is sorely needed. 4D is the language of accelerator physics. The anticipation of such a classical mechanics book therefore lives on.

I appreciate Kai Lam for trying to bridge the texts between strong mathematical style of the Springer Verlag Yellow book series (GTM) and that of Physics texts, but eventually I realized that the trend of mathematician texts like GTM is just terrible for learning for physics. Physicists need to be able to have a strong visual idea of the mathematical ideas in order to apply them to problems. The book starts out at an easy level with vectors, but then very rapidly progresses to very abstract methods of exterior forms, manifolds, which needs a stronger motivation for physics than he provides. I am still not convinced that one needs to introduce all the formal differential geometric structure of Cartan and his followers to do physics. I have looked at a great deal of Physics, Mathematics, and Mathematical Physics text that try to use the bundle structure and exterior forms, and am still not sold on it. I prefer the old tensor calculus of Ricci as developed in Kreyszig's Differential Geometry, and then for gauge theory simply writing down the gauge fields without even talking about bundles. It is good to know they possess the same mathematical structure as the Levi-Civita connection used in GR, but I don't think you need to develop an abstract formalism to unify the two. Too much set theory, yuck. Please let us calculate. Yes they are similar, but unfortunately the formal structure developed I think adds way too much structure without giving strong visual intuition. If authors insist on developing the bundle structure they need to do a better job of motivating it for physics. I have yet to see any theorems that need this structure at this level of a Fundamental Principles of Classical Mechanics book.I bought this book in hopes that it could allow me to understand the Symplectic Geometry and Bundle Differential Geometry that is being developed to apply to theoretical physics problems, but at this point after studying many texts on the subject, I am doubtful that they will have very much importance to the practical development of physics, as well as completing any sort of unification. If you want to study classical mechanics at the advanced level I recommend Classical Dynamics by Sudarshan and Mukunda, although it is a bit heavier than you need I think, it is a book to study for a life time and has a pretty complete development of the mechanics essential for Quantum Theory. Many Classical Mechanics books can be lacking in sophisticated discussion of Hamiltonian mechanics, which is central to Quantum Theory. That is why I recommend Sudarshan and Mukunda. I think It is also worth taking a look at Whittaker's Analytical Mechanics for reference, as that seems to be the other treatise on the subject that is written at the level of physicists that provides a thorough discussion of the foundations and more. The other book to take a look at is Analytical and Canonical Formalism in Physics by Mercier.