Simple mechanical control systems

Abstract

The word "simple" as used in the title of this chapter has a technical meaning that we shall provide during the course of the chapter. In this context, "simple" does not mean "easy." The preceding two chapters were preparatory material, and were presented fairly hastily, with the reader being directed to references for further details. In this chapter, we begin the presentation of the material that can be considered the core of the book. The aim in this chapter is to provide the methodology for going from a physical problem to a mathematical model. What is more, the mathematical model we consider uses the tools described in Chapter 3, particularly the affine connection formalism of Section 3.8. We are extremely systematic in our presentation, far more so than is the norm. This has the disadvantage of making the presentation lengthy. To this end, we should mention that it is not normal, when modeling a mechanical system, to go systematically through the steps we outline in this chapter. Indeed, someone experienced with the process can perform many of the steps we describe "just by looking at the system." The systematic presentation we provide has two advantages to counterbalance its length. 1. For complicated systems, one cannot always describe the mathematical model "just by looking at the system." In these cases, even for experienced researchers, it is useful to have a procedure to fall back on, and to prevent the making of careless modeling errors. 2. By making all constructions perfectly explicit, we draw very clear lines between physical concepts and mathematical objects (more precisely, differential geometric objects). Indeed, it is our (admittedly somewhat Pla-tonic) belief that the mathematical models we arrive at are, along with equivalent models, e.g., Hamiltonian models, the unique correct mathematical models. That is to say, other ways of thinking about these models are approximations to what we present. Some approximations are still useful , and perhaps easier to understand, but are still approximations, and so subject to limitations as such.