The interaction between mathematics and mechanics is a never ending source of new developments. This present textbook includes a wide –ranging spectrum of topics from the three body problem and gyroscope theory to bifurcation theory, optimization, control and continuum mechanics of elastic bodies and fluids. For each of the covered topics the reader can practice mathematical experiments by using a large assortment of Matlab-programs which are available on the author’s homepage. The self-contained and clear presentation including Matlab is often useful to teach technical details by the program itself (learning by doing), which was not possible by just presenting the formula in the past. The reader will be able to produce each picture or diagram from the book by themselves and to arbitrarily alter the data or algorithms.