# Mathematical Concepts in Modeling

Modeling is a broad term with many meanings. Would it not be more exciting if this were a book about fashion models and a collection of pretty pictures? Well, a model is something uncommon or unreal. It is the copy of an object. The objects that I will focus on are inert, but nevertheless exciting. We are dealing with aircraft, spacecraft, and missiles. However, instead of building scaled replicas of these ve­hicles, we construct mathematical models of their dynamic behavior. Launching models is always more fun than just having them sitting on your shelf. I will teach you how to make them soar on your computer. But first we have to lay the founda­tion. Classical mechanics, a branch of physics, will be our cornerstone. Digging deep into the past, I found an interesting axiomatic treatment of the principles of mechanics. It will serve us well when we lay out the canon of modeling. Partic­ularly useful is the principle of material indifference, which we will employ for several proofs.

The mathematical language we use consists of tensors and matrices. That may get you excited, but calm down—the bare essentials of Cartesian tensors will suffice. We will talk about frames, coordinate systems, transformation matrices, and so on, in a systematic order. If you are rusty in matrix algebra, brush up with Appendix A.

Of course, all theory is only as good as it is able to solve practical problems; at least that is the opinion of most engineers. I subscribe to that philosophy also and will show you in this chapter just how well tensors model geometrical problems. Throughout this book they will be our companions. Our motto is “from tensor modeling to matrix coding.” Thus, expand your mind and go back to explore the future!

2.1 Classical Mechanics

At the turn of the last century, physicists thought that all of the laws of the physical universe were known. Over three centuries, Galileo, Newton, Bernoulli, D’Alembert, Euler, and Lagrange built the structure of the branch of physics that we call mechanics. Today, after another century of breathtaking progress in the physical sciences, we fondly remember that fully developed branch as classical mechanics. Although physicists have turned their back on it, engineers have ex­plored it through many adventures, from first flight to a visit to the moon.