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Rating: 
Summary: OK, but old-fashioned, few examples, and not many diagrams
Review: .
This was probably a good book in its day (1950-1970), but
it's really old-fashioned now. A lot has happened in the
field of mechanics since Lanczos wrote it. For example:- Computers are now used extensively to analyze and
simulate mechanical systems. - The modern language of mechanics is much more geometric
and independent of any particular choice of coordinates.
If readers stop at Lanczos, they will have trouble
understanding the modern literature. He doesn't even
distinguish between vectors and one forms. - Dynamical systems theory / qualitative dynamics has
contributed a lot to the understanding of mechanics
in the past 30 years. You won't read anything about
stable/unstable manifolds or strange attractors in
Lanczos. The "problems" are so easy that they border on the
ridiculous. And don't try finding them at the end
of each chapter --- this book predates modern textbook
format. Lanczos hides his problems like Easter eggs. In conclusion, this book is of historical interest only.
If you want to learn about modern mechanics, read
something that was published recently. (I should add that the book is well-written, but that
doesn't fix the fact that it is dated.)
Rating: 
Summary: Lucid and elegant -- a true classic
Review: Lanczos' book is a compelling analysis of the principles of Lagrangian and Hamiltonian mechanics. It reminds me a bit of Feyman's Lectures on Physics because it focuses on the motivating principles behind advanced mechanics. In an elegant and flowing style, Lanczos guides the reader through a walking tour of the principles of mechanics, peppered with historical footnotes. If you understand how to use mechanics, but want to understand how the underlying principles are developed, this is an excellent choice.
Rating:
Summary: A Joy to Read
Review: Lanzcos has his name attached to many area of mathematical physics. Besides that, he is an excellent writer. This book on variationnal principle of mechanics is the most beautifull and profound one I've read on the subject. This book is well suited for undergrad or grad students, but even for reasearcher who uses the tools of analytical mechanics, it's worth reading it.
Rating:
Summary: A Treasure
Review: Surely Lanzcos is one of the few educators that compare to Feynman. This book presents an exceptionally lucid and engaging development of the main variational principles of mechanics with the discussion of Noethers theorem being the most cogent I've seen. While the problems may not be exceedingly difficult, this is because the problems are intended to illuminate the ideas under discussion. Lanzcos divides mechanics very roughly into two camps, the vectorial/one-form viewpoint versus variational/analytical view. Since Lanzcos aspires to the variational view, the criticism concerning the lack of discussion of vectorial/one-form is rather moot and explains why there are not a great number of drawings. While this may be a problem for some, in my view figures can convey a false sense of understanding. This is not a work that one can skim, it expects considerable reflection on the contents. We all differ in our needs and wants but in my view, this exposition exudes a truly infectious sense of wonder that such simple ideas could be so powerful and beautiful. But since no one book in this field may suffice for all one might also consider "Mechanics: From Newton's Laws to Deterministic Chaos" by Florian Scheck which admittedly covers mathematical methods that Lanzcos does not yet fails to capture the essence of the ideas as well as the Lanzcos text does.
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