George E. Hrabovsky, President, MAST

Hello and welcome to our fourth year. It has been strongly suggested, not unreasonably, that I reduce the time I take to cover a topic in no more than three columns. To that end I propose the following: Each column will address an entire project, or at least a significant part of such a project. I will include very little background material within the body of the column. Instead I will supply links to existing background material that can be found elsewhere.

In a previous column, "The Return of the Theorist," I described how to develop a theoretical physics library (including many freely available resources). Keep these in mind as you proceed through these columns.

If, for any particular column, you can think of another way of doing it, send that to me, too. It is likely that I will include your solution in a column, or have you submit it to the editor and include a link to it from my column. It is possible that fundamental results could be discovered in this way; if such happens then I might instead suggest that you publish in a peer-reviewed journal. In this way significant collaborations could be developed.

Of course, there is a great temptation to dive into the open problems at the cutting edge of theory. After all, that's where the great contributions are hiding. It is always a great desire to attempt to discover the answer to one of these. Of course, it is extremely unlikely that an amateur will ever solve one of these. But it could happen. Even if we are unable to crack any of these nuts, an intermediate result could be found. Even if that does not happen, we can revel in the attempts and learn a lot of math at the same time.

Almost all theory involves equations of one sort or another. There are essentially three activities in theoretical physics: Derivation of new equations, solving equations, and understanding the equations and their solutions.

Derivation is the process of arriving at an equation from existing principles. Often this is done after observational or experimental data is analyzed and some pattern emerges. Frequently such patterns emerge as empirical formulas, which are equations pulled directly from the data. The formal derivation from fundamental principles tells us that the pattern in the data is completely consistent with what we already understand. This is a powerful argument for accepting the new equation.

Once we have a new equation, we need to make predictions that can be tested in a lab somewhere. This requires us to solve an equation for variables and parameters appropriate to the situation being studied. In short, we must make calculations whose results can be compared to new experiments of observations.

In either case we can attempt to understand the ramifications of what we are doing. In deriving an equation we are developing new principles of physics. Understanding the derivation means that we understand the physical principles. In a very real sense, the equation is the embodiment of the physical principles synthesized into an understandable form. It is also true that an equation by itself makes no predictions! The presence of an equation cannot tell us anything about specific cases where reality might be involved. It is only by solving the equation and understanding the behavior of the solution that we can learn about reality. There are always many more solutions that are not realistic than those that are. This is the difference between mathematics and science.

Background

One very old method of deriving relationships between variables is a technique of analyzing the units of the related variables. This method is called dimensional analysis.

The method works like this:

Step 1: Choose a phenomenon to analyze.

Step 2: Write an equation relating the variable under study to suspected parameters and variables.

Step 3: Determine the suitable units for all parameters and variables.

Step 4: Multiply by any necessary constants having units required to make the units on each side of the equal sign come out the same.

Example

You have done an experiment where you have determined that the force exerted on your car by air resistance is proportional to the speed of the car in some way. You decide to use dimensional analysis to determine what form this resistive force must have.

Necessary Principles

Definition 1: . This is Newton's second law of motion and it tells us the units to use for force. Here is the force we are guessing at, is the mass with dimensional units and represents acceleration with dimensional units of length per time squared, . This force must have dimensional units of .

Definition 2: The dimensional form for velocity is .

Dimensional Analysis

We begin by stating our premise,

We write this in dimensional form

The dimensional forms must be equal. We assign an arbitrary constant to the power , and we raise the velocity to the power . This gives us,

We can use this to compare units and their powers on each side. The powers of on each side form the equation,

or

We also acquire the equation

or,

This gives us the dimensional form,

The time drop out,

We cancel the distances,

So we have

Or,

So we can write an expression for the actual force to within some constant of proportionality, , and dependent upon the distance traveled . We take our expression in dimensional quantities,

We then apply the constant of proportionality, then we rewrite the symbols to get the final form,

We are now done.

Created by Mathematica  (February 28, 2005)