More complex constants and parameters

Prerequisites

In our article, Variables, constants, and parameters, we described how mathematical symbols may stand numbers (or, better, measurements) that we are looking to change (variables) or not (constants). This became a bit fuzzier when we realized that we could compare situations with different values of the constants (parameters). 

In real physical situations that we might be considering, there are more possibilities of variables and parameters. Under some situations we might want to consider a variable as a parameter (constant) or vice versa. 

For example, at some point in the course we might be consider a mass swinging from a pivot as an example of oscillations (a pendulum). Our swinging pendulum has a length of $L$, a mass of $m$ and experiences a gravitational field, $g$ (9.81 N/kg). If we pull it back to an angle $\theta_0$ and release it, it will swing back and forth between $\theta = \theta_0$ and $\theta = -\theta_0$ (assuming we can ignore friction and air resistance).

Those are kind of funny equations. How can both  $\theta = \theta_0$ and $\theta = -\theta_0$? Doesn't that imply $\theta =0$? Well... no. That's because the two $\theta$'s on the left of those equals signs are considered to be dependent variables and the equations mean that we are considering each of those equations to be happening at different times. So really what the equations should be is $\theta(t_1) = \theta_0$ and $\theta(t_2) = -\theta_0$ but we typically don't bother to be that explicit.

We are describing the motion in terms of the angle as some function of time: $\theta(t).$ We can also calculate a parameter, $f$, that tells us the number of cycles it oscillates in each second -- the frequency

It's clear that for this problem we want to take $t$ as our independent variable and the angle $\theta$ as our dependent variable. We'll find that the frequency $f$ and the function $\theta(t)$ both depend on our starting angle, $\theta_0$ (though not very much for small angles). We now have a lot of different kind of symbols:

  • an independent variable ($t$)
  • a dependent variable ($\theta$)
  • system parameters ($m$, $L$, $g$)
  • calculated parameters ($f$)
  • particular case parameters ($\theta_0$)

In fact, we'll find that $f$ depends on $L$, $g$, and $\theta_0$. If we're not paying attention to variables and parameters, this could look like we had a constant depending on a variable. 

You can imagine such an analysis could be of great importance in examples in biology and medicine. For example in our problem in which we analyze the growth of a worm, the size of certain biological parameters mean life or death to the worm. In medical situations, the presence of a certain level of hormone in a system might be considered a parameter controlling the rate of reactions in the system or another variable, depending on how we model the system. 

The interesting result is that in a mathematical model of a physical system, what we consider to be a variable or a constant depends on the system we are considering, and what questions we are asking.

In any physical situation you need to ask yourself: For the problem I am considering at this moment, which symbols should I treat as constants and which as variables? Which variables should I consider independent and which dependent?

It may seem easier in math where it always seemed clear: some things were variables and some things were constants. But realizing that in any particular situation we get to choose, gives us much more power to analyze and answer questions about physical systems.

Joe Redish 8/23/18

Article 518
Last Modified: August 23, 2018