Computational algorithms are used to communicate precisely some of the methods used in the analysis of dynamical phenomena. Expressing the methods of variational mechanics in a computer language forces them to be unambiguous and computationally effective. Computation requires us to be precise about the representation of mechanical and geometric notions as computational objects and permits us to represent explicitly the algorithms for manipulating these objects. Also, once formalized as a procedure, a mathematical idea becomes a tool that can be used directly to compute results.

Active exploration on the part of the student is an essential part of the learning experience. Our focus is on understanding the motion of systems; to learn about motion the student must actively explore the motion of systems through simulation and experiment. The exercises and projects are an integral part of the presentation.

That the mathematics is precise enough to be interpreted automatically allows active exploration to be extended to it. The requirement that the computer be able to interpret any expression provides strict and immediate feedback as to whether the expression is correctly formulated. Experience demonstrates that interaction with the computer in this way uncovers and corrects many deficiencies in understanding.

In this book we express computational methods in Scheme, a dialect of the Lisp family of programming languages that we also use in our introductory computer science subject at MIT. There are many good expositions of Scheme. We provide a short introduction to Scheme in an appendix.