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When doing research in theoretical physics, a frequent task one encounters is trying to express some physical quantity as a function of other quantities. A lot of times this can't be done analytically, but even when it can - it sometimes results in very long and complicated symbolic expressions.

Although technically such an expression is a "solution", it is not of much use for a researcher that wants to gain physical insight from the solution, and maybe rely on it to do more research.

What are some general strategies a researcher can use to gain insight when confronted with such expressions?

For starters - here are some ideas I use:

  • Check certain limits of the expression, i.e. when one of the variables is very low or very high - these are often simpler and can shed light on the behavior in the general case.
  • Look for a recurring pattern in the expression and give it a name. The newly defined variable usually has some physical significance in itself, and also when all the occurrences of the pattern are replaced with the new variable the whole expression becomes simpler.
  • Substitute some of the variables with reasonable numerical estimates, and plot the expression as a function of the rest of the variables.
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1 Answer 1

up vote 9 down vote accepted

In addition to what you have listed:

  1. Use the Buckingham Pi theorem to create as many non-dimensional numbers as possible from combinations of dimensional numbers. This simplifies the expression but also allows you to reduce the number of variations on variables that need studied.

  2. Non-dimensionalize all of your variables using suitable reference measures for the problem you are studying. Use this to do an order-of-magnitude assessment of terms to decide if some terms are insignificant under certain conditions.

  3. Perform a perturbation analysis. For example, if you have an equation for a wave, $\psi$, substitute in $\psi = \overline{\psi}+\psi'$ where $\overline{\psi}$ is the average wave value (in time, space or any of your other independent variables) and $\psi'$ is a disturbance on that mean. Then expand all terms and collect the means together and the perturbations together. Do a lot of manipulation and you'll get an expression for the mean behavior and the response to disturbances. This isn't always "simpler" but gives tremendous insight. You could then substitute in simple functions, like trigonometric functions, for that disturbance and study how it grows or shrinks and under what conditions.

  4. Learn what types of terms do what. Which terms transport or convect your variable? Which terms produce or dissipate your variable? These types of terms usually have typical forms in terms of derivatives. You can group terms together to come up with a simple conservation equation (time change = transport + production - dissipation) where all those terms can be lumped together. You can then attempt models for each individual group of terms.

Of course, all of these can be combined with one another to do all sorts of complex study.

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