The various chapters connect the notions of mechanics of first and second year with the ones which are developed in more specialized subjects as continuum mechanics at first, and fluid-dynamics, quantum mechanics, special relativity, general relativity, electromagnetism, stellar dynamics, celestial mechanics, meteorology, applied differential geometry, and so on.
This book is the ideal mathematical and mechanical preparation for the above-mentioned specialized disciplines. This is a course of Analytical Mechanics which synthesizes the notions of first level mechanics and leads to the various mentioned disciplines by introducing mathematical concepts as tensor and virtual work methods. Analytical mechanics is not only viewed as a self-sufficient mathematical discipline, but as a subject of mechanics preparing for theories of physics and engineering too. One of the author's goals has been to reduce the gap between first and second academic cycles. The intensive use of the tensor calculus contributes to this reduction. The main subjects of mechanics of the first academic cycle are set in this context and let "handle" tensors. Many books relating to the developments of tensor theory are either too abstract since aimed at algebraists only, or too quickly applied to physicists and engineers. The author has found a right compromise which allows to bring closer these points of view; the book chapters are intended for mathematicians, which will find an illustrated presentation of mathematical concepts and solved problems in mechanics, and for physics and engineering students too, since the mathematical foundations are introduced in a practical way. It is the first time that a mechanics course so much develops the tensor calculus by taking into account the two previous sides! Besides the tensors, another mathematical concept is systematically used and largely developed: The virtual work. This notion is clearly introduced from the one of virtual displacements and virtual velocities are also considered.
The introduction of mathematical "tools", namely tensors and virtual works, gives the mechanics treated subjects a great unity; these mathematical notions easily lead to geometrical ways in the frame of differential geometry, as well as to other methods in continuum mechanics for example. In Europe in particular, given its style, this book is aimed to contribute to teaching mechanics European programs, the scientific English language is simple and should be used as a common language for concerned students of all European countries. All definitions and propositions are written with the peculiar strictness proper to mathematicians, but they are always illustrated with numerous examples, remarks and exercises. The concise style of differential geometry books of the author is also found in this mechanics book. When writing this new book, the author had the following assertion fresh in his mind: Pedagogy contributes to Rigor.