Tensor Calculus and Analytical Dynamics (Library of Engineering Mathematics) [Englisch] [Gebundene Ausgabe]

This book is not a text book. It is, in some sense, the final word on tensor formalism in finite degree of freedom (analytical) mechanics. It is one of the most scholarly books I have come across. The list of references is very exhaustive and the author is well read in the literature on the subject, not just in english, but also in russian, french, and german. The style is clear and concise, the notation is carefully chosen and summarized in a useful section where conventions, notation, and basic formulae are listed.

Papastavridis is an author with a unique attitude towards mathematics. He avoids the coordinate free formulation of tensors on manifolds. In his view, the exterior differential calculus is an esoteric abstraction which is hard to grasp by many and thus has the danger of turning down able people from embarking on doing research in analytical mechanics. This is basically a recapitulation of his views which have been explicitly stated within the book in a broader context.

With that said, don`t expect to find anything pertaining to modern differential geometric view of mechanics. However, this book presents one of the most extensive survey of tensor analysis with indices. The bibliography is indeed comprehensive, and a welcome feature in such a monograph.

Personally, I benefitted alot from this book both in terms of physical aspects of mechanics and in terms of classical tensor analysis. However, I still believe in the power of mathematical abstractions in grasping of the holistic image of a physical and/or mathematical entity. In this respect, the language of differential forms is rather important and allows further useful topological generalizations like cohomology. It is true that the current engineering/science curricula does not leave much space for the modern view, but this is ultimately where it will be heading to. Despite his dislike of exterior calculus, Papastavridis inevitably builds a strong basis for delving into tensor analysis on manifolds. For the latter Bishop and Goldberg is still the best choice with an unbeatable price.

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 ... .