Most of the text comes from this level courses that the author taught at universities and engineering schools. In the particular case where such a course cannot be taught to engineers, a lot of introduced matters constitute the mathematical and mechanical bases of applied engineering mechanics. 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.