Mathematica as a Tool: An introduction with practical examples

More than ten years ago, I wanted to carry out coordinate transformations for Hamiltonian systems, in order to discuss the stability of certain equilibrium posi­ tions. Basically, the calculations only involved rational expressions, but they turned out to be extremely complicated, because the third and fourth order terms had to be included. After several months of filling whole blocks of paper with for­ mulas, I was close to resignation. But, by a lucky incident, I met a colleague who showed me the computer algebra package Reduce. It still required a lot of patience and tricks, but Reduce finally did produce the desired results. After this experience, I wondered, why only a few engineers and scientists were aware of the strengths of such computer algebra programs. The mathematical treatment of scientific problems often leads to calculations which can only be solved "by hand" with a considerable investment of time, while a suitable com­ puter algebra program produces the solution within a couple of seconds or min­ utes. Even if a closed symbolic solution is not possible, such programs can often simplify a problem, before the cruder tool of numerical simulations is applied.