

Mathematics Modeling Physics
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 newcomer 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
modelmaking 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 "semiempirical" 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 xy 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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