It is well known and in this repect I fully agree with the author, the so-called “elementary mechanics” is far from being an elementary subject. Even “trivial” facts must be carefully treated. This is no new finding and has already been documented in the literature because prominent scientists who wrote seminal books on “non-elementary mechanics” [for example, G. Hamel, Theoretische Mechanik. – Eine einheitliche Einführung in die gesamte Mechanik. (German) (Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. LVII) Berlin, Göttingen, Heidelberg: Springer-Verlag XVI (1949; Zbl 0036.24301)] also wrote books on elementary mechanics [G. Hamel, Elementare Mechanik. Ein Lehrbuch, enthaltend: eine Begründung der allgemeinen Mechanik; die Mechanik der Systeme starrer Körper; die synthetischen und die Elemente der analytischen Methoden, sowie eine Einführung in die Prinzipien der Mechanik deformierbarer Systeme. (German) Leipzig und Berlin: B. G. Teubner. XVIII (1912; JFM 43.0764.01)]). Also V. I. Arnold’s book [V. I. Arnold, Mathematical methods of classical mechanics. Graduate Texts in Mathematics. 60. New York-Heidelberg-Berlin: Springer-Verlag. X (1978; Zbl 0386.70001)], deals in the first chapters with material considered as elementary mechanics.
Hence, starting to read Professor Spivak’s book and having these two mentioned books in mind, I was curious what a differential geometer’s special viewpoint will be.
One thing can be asserted. His viewpoint is quite different from the mainstream but nevertheless original, even if, at some points, surprising. To mention one point regarding the latter, it is difficult to understand what a great problem the concept of the rigid body made to him. Taking into account that everything in real life we want to analyse from a mechanics viewpoint has to be cast into a proper mechanical model, then in order to achieve simplicity, these models consist of objects not existing in reality, such as, for example, mass points, rigid bodies, elastic bodies, inextensible strings, membrans without bending stiffness etc. Although the introduction of the rigid body which covers many pages is interesting to read, in the reviewer’s opinion there is a better way to introduce the rigid body as it is done in Arnold’s or Hamel’s books, where it is defined in few lines. Such a definition is easily understood.
For example, it is intuitively clear that even a strongly deformable sponge of rectangular shape with three different principal moments of inertia can be modeled as rigid body if it is used for the class room experiment to show its stable or unstable rotations about its major axes of inertia. The motion of the sponge which is thrown into the air with a strong rotation about one of its axes is basically a rigid body motion because the angular velocity will not significantly influence its shape although it can be strongly deformed by the hand of the person throwing it into the air.
For somebody working in mechanics the first half of this first book (a second book with more advanced material is in preparation) perhaps will not be really relevant whereas the second half where connections of geometric concepts with, for example, holonomic and nonholonomic constraints are made, is of great interest and is a pleasure to read. Very nice is also that some, not completely trivial, examples are worked out in detail. Also the final chapter on statically indeterminate structures is interesting and correct in the conclusions even if the author still has some doubts concerning the results of his experiments.
The resume of this book is absolutely positive, and I can only encourage the author to write the second book, which as he mentions will deal with less elementary subjects like symplectic structures where his expertise in differential geometry certainly will be very beneficial.