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Name: P. M.
Status: student
Age: 19
Location: N/A
Country: N/A
Date: Thursday, August 22, 2002


Question:
Overlooking some Physics journals, I have noticed that it contains more math than a mere mortal would be willing to shake a stick at. Could someone please explain how mathematics models physics?


Replies:
Your question is very good!

It is not always easy to construct a model for a physical system, and it will usually involve mathematical formulas. That is O.K. if you understand where/how the formulas evolved. In a single journal this is frequently not apparent, since those publications are directed to a readership who already know that. This is frustrating for any new-comer to a subject. You do not know "jargon" most areas use.

The second aspect of how models are constructed is, "How to devise my own model for a problem I am interested in?" And this is a different issue. Allow me to get on my "soap box" for a moment. Part of the problem in model--making is the reality that we do not teach, even starting at first grade, how to solve the dreaded "word problems". We know how to manipulate numbers-- from counting in first grade through calculus and beyond, but when faced with a verbal statement that requires "translation" into a mathematical statement, students often draw a mental blank. That is part of our teaching at all levels that needs to be addressed by educators -- our teachers do not know how to approach solving word problems either! Now I am off my "soap box".

There are several levels of modeling (What follows is not complete.).

1. FITTING THE DATA. Presented with a fairly simple set of data one can often observe that: The data seem to: fall on a straight line, be a parabola, be linear if I plot the response vs. the logarithm of the independent variable -- and so on. This observation is based solely upon inspecting the data, nothing more. This is the weakest type of model because it gives no "reason" for the fit; it only allows a way of estimating (guessing) and interpolating the data. The latter is dangerous because with data outside the range of the measurements we don't know what the system is going to do.

At the other end there may be a "theoretical" model for the experimental results. As an example we know from quantum mechanics that the frequency of emission lines for a hydrogen atom is predicted to be of the form: f = constant /(n1^2 - n2^2), where "n1" and "n2" are integers. So if we are measuring the spectrum of hydrogen atoms, it would be prudent to use that theoretical model to treat our data.

In between, are "semi-empirical" models that involve some theoretical framework, but not a complete analysis of the problem and system we are studying. An example is the measuring the rate of a chemical reaction. We might have some intuition or information to suggest that the rate should be a certain function of time, but we don't have a complete picture. An example where a good model exists (although the "reason" for the model is not well understood) is radioactive decay. In that case the rate of decay: dC/dt = k*C, that is the rate of decay is proportional to the concentration of radioactive nuclei.

A third approach is to actually construct a model for a problem, where no model exists. Here one usually starts with a "similar" for which the "answer" is known, and modifying that "simple" model to account for the difference between the problem at hand and the related known mode. For example, take the shape of an egg. Qualitatively, an egg shape is an ellipsoid of revolution with one end more pointed than the other. So a good starting point for the projection of an egg shadow onto the x-y plane, call it EGG, would be the equation for an ellipse, call that ELL. Now we have to make this symmetric shape asymmetric. We can do that by adding to it an asymmetric function, call it GAS. We also will need some parameter to set how large GAS is compared to EL. So we multiply GAS by a size parameter, call it L. Finally, we need to ensure that the value of EGG at the ends of the egg are zero, otherwise the egg is not closed at the ends. ELL already meets this boundary condition, but we must introduce another parameter to GAS to ensure the boundary condition of closure. Call that function, BND. Then the end result for our model for an egg shape is: EGG = ELL (+/-)(L*GAS*BND). So EGG is asymmetric, is closed at the ends, and can be made as skewed as we wish by adjusting the value of "L".

There are some books on mathematical modeling, for example see: "Guide to Mathematical Modeling" Dilwyn Edwards & Mike Hamson, published by CRC.

Your question is a very difficult one to answer as you can see, but your perception of, "What is going on here?" is healthy. Sorry for the long answer, but efficient methods of model building is a problem that I have given a lot of thought to.

Vince Calder


P. M.,

You ask an excellent question. Mathematics often works as a good model for anything that can be expressed with numbers. The numbers represent measurable quantities. The mathematics relates the numbers in a way that corresponds to how the measurable quantities relate. Mathematics is a nice tool to calculate expected values of future measurements, based on these models and equations.

Remember that expected values are not always exact. Mathematical models are approximations. They do not always yield what is actually measured. To express everything exactly would require more mathematics than you could write in a lifetime. Understand the concepts first. Use the mathematical models as a reminder and as a tool for estimations. A simple example is "F=mg". If you want to express the gravitational force on a falling ball exactly, you must consider the force between each possible pair of atoms and sum the vectors. F=mg yields a value that is close enough to use in most situations. Spending several years calculating and adding all individual vectors would not yield a noticeably different result. However, the effective gravitational force on a single electron flying though the air or on a distant comet requires a different model. F=mg only works for millions of molecules (like a baseball) close to the surface of the Earth.

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College



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