How to derive things:
1. Start with a good cartoon
2. Consider the simplest approach to the physics
3. Be clever in choosing your reference frame
4. Add numbers to your cartoon
5. Do the math and/or arithmetic
6. Iterate as necessary until doable and obvious after-the-fact
At some point in my academic career (college? grad school?) I decided to get serious with my studies, and build a habit of going through derivations, step by step, with pencil and paper, in order to understand.
It is an only an occasional habit, part because of laziness, but also because if I spent my time walking in detail through every derivation, I didn’t feel as if I would have enough time left for my other activities.
One learns a lot of researcher shenanigans going through derivations. Be wary of the line that says something like: “We can obtain equation 3-9 from equations 3-5 through 3-18 and application of algebra. If you want to do the algebra, budget yourself many hours-days-possibly even weeks.
And then there are the assumptions. What if one of them is wrong? How does that change the derivation? You can almost guarantee that the assumptions were required to make the problem doable for the scientist.
The first thing anyone should know about derivations is that they often seem obvious in retrospect, obscuring the combination of inspiration and perspiration (and frustration) that was invested into the derivation. While a nice derivation is neat and tidy and perhaps a little bit exciting, the process the scientist (more likely: scientists) took to do the derivation was most likely totally messy, filled with wrong turns, and dead ends, and endless frustrations. The derivations are the stories that are told after the real life has been lived.
In my scientific life, I have derived one new equation (Kavner et al., 2005, eqn. 12). It definitely ranks up there as one of my top life experiences. It took about six weeks of work (most days, from 8 in the morning until lunchtime—at my desk with pencils and a stack of paper) to cover enough wrong paths and make enough simplifying assumptions so that I could complete the derivation in one sitting.
It was so much fun that I’m working on another, more complicated one that generalizes the first one and fixes some of its problems. This one I’ve been working on for far longer—on and off for a few years. I promise I’ll share it when I’m able to.
In the meantime, here is an annotated derivation of the Hugoniot-Rankine shock wave equations. The derivation is from a combination of Poirier’s textbook: Introduction to the Physics of the Earth’s Interior and my notes from my PhD advisor, R. Jeanloz. Annotations are mine.