The structure of physical modeling

In our article, Overview: Modeling with mathematics, we outlined the basic structure of building and using mathematical models in physics. It involves four kinds of steps as shown in the diagram: creating a model, processing the model mathematically, interpreting the results of the model physically, and evaluating whether the model is good enough for what we want.

Each of these steps is non-trivial and mastering them requires some practice and learning of new skills. But in some sense the first step is the most critical. How are we going to describe parts of a physical systems mathematically? There are three critical steps in deciding on building a mathematical model.

What things do we want to talk about? 

While this looks on the surface as if it should be obvious, it often isn't. Every object that we'll deal with has structure. A mammalian cell has a cell membrane and a nucleus. A jumping kangaroo has a heart, lungs, and a vascular system as well as powerful leg muscles. Even a baseball (a more familiar "just physics" object)  has seams, elastic windings, and a cork core. 

Depending on what characteristics of the system we are trying to describe with our model, we may want to treat it as a "single object" without considering its internal structure at all. Once our kangaroo has left the ground  or our baseball has left the bat with a certain speed, their resulting path does not rely on their internal structure nearly at all. This is true for a first level model. If we want to understanding the forces inside a kangaroo's leg or how the ball interacts with a bat, we might want a finer grained model.

What we will learn, is that at each level of modeling, a useful start is to ask:

What are the objects we ought to consider and what are the interactions between them?

(See the page System Schema Introduction.)

What measurements should we make to describe the characteristics of our system that we are interested in?

In most cases of mathematical modeling in physics, we will be assigning numerical values to quantities in the physical world through measurement. This is a subtle, powerful, and important process. It results in a description that looks like a number but isn't.

Consider for example the true equation: 1 inch = 2.54 cm. The numbers are NOT the same but the lengths represented on the two sides are.

This can cause you considerable trouble if you forget this, since leaving off units that tell what system was used to make the measurement can result in mixing units inappropriately and getting nonsensical results.

We develop a set of tools called Dimensional analysis to help you keep track of the kinds of quantities you are working with. This is an extremely valuable tool to help you catch mis-remembered equations (or help you not be caught by standardized-test writers who try to lead you astray by offering you impossible equations).

What mechanisms make our model run?

The last but perhaps the most critical component of our mathematical model are the mechanisms we choose to drive the model. These are principles — usually expressed in the form of an equation — that relates our measurements to each other and may tell how they change.

In physics, often our principle will be a fundamental physical principle — something like Newton's 2nd law or The conservation of mechanical energy that hold for very large numbers of systems and situations. Other times, they will be principles that we have developed from more general principles for specific situations, such as Fick's law or the Hagen-Poiseuille equation. In still other cases, we will have phenomenology — heuristic laws that have been created to fit experimental observations. See the page on phenomenology and mechanism. 

These three ideas show the subtlety and complexity of mathematical modeling in science. One of our primary goals for this class is to help you to learn to think more deeply and incisively about what is going on whenever there is a mathematical model in science. It is rarely as straightforward as it looks on the surface!

Joe Redish 3/25/19