This is probably the best book in the English language on the tensorial foundation of analytical mechanics. The book presents rigorous derivations of the main concepts of mechanics, in particular integrability and the principles behind various approaches to the derivation of the equations of motion. Beside its analytical merit, the book is a service to the English reader since the best references so far on non-holonomic systems are in German, Russian and French. In addition there are several notations in the classical literature on tensors, i.e., those of Eisenhart, Levi-Civita, Schouten and Synge, with different books use different notations; this book unifies them all.
The first part of the book presents the foundation of tensor calculus, Riemannian geometry and the general idea of integrability. These are stand alone chapters, no other references required. It is worth mentioning that the author avoids the more modern approaches of differential forms and exterior calculus; he does it all with tensors. The book then proceeds into kinematics and kinetics, formulated using strict tensorial properties, such as covariance, contravariance and absolute derivative, and using variational calculus - total displacement vs. virtual displacement, terminology used in deriving the transitivity equation/Hamel coefficients (those coefficients reflect integrability) and the important Frobenius integrability theorem (as opposed to recent approaches that use the concepts of involutive distributions and Lie algebra formulation, this book uses variational "deltas"). The book then presents a formulation of differential geometry on manifolds with application to a particle's motion on a surface. And then a major part is dedicated to different approaches for the derivation of the equations of motion, under constraints, based on Lagrange's principle. There is a comprehensive discussion of constraints, in particular non-holonomic Pfaffians including geometrical considerations/illustrations. The book ends with helpful examples that demonstrate the various methods and the non-integrability concept. There are many more important features in this book, but I'm trying to keep this review short.
Be aware that this book aims for the mathematician and for the analytical minded physicist and engineer. It is demanding reading and requires a solid mathematical background. For the right reader it is a great read and an excellent reference. I tried to give it six stars, but Amazon's limit is five ... .