On November 3rd of 2018, I uploaded a video to youtube regarding the Geodesic Equation, and it contained what I believed to be the most intuitive explanation to date of the mathematics behind it:
https://www.youtube.com/watch?v=_Y6vDTKN-Go
Its current title is: World's Easiest Geodesic Equation Derivation - NO CHRISTOFFEL SYMBOLS (The Michael Scott Method)
Like the description says, the video is divided into three main parts.
Part 1 - Construct a coordinate grid, and stretch the lines of the grid into a non-flat surface. The symbol ex represents this stretch.
Part 2 - Show that the typical second-derivative equals zero equation for a line that doesn't bend must be modified from d/ds (dx/ds) = 0 to d/ds (dx/ds * ex) = 0 to incorporate the varying lengths of the grid lines between points.
Part 3 - Show how d/ds (dx/ds * ex) = 0 can be modified into a form more like the Wikipedia equation (the one with Christoffel Symbols) by using the product rule of differential calculus and a some high-school level algebra.
Part 1 is kind of interesting because rather than express a metric as the hypotenuse of a triangle from a x grid line and a y grid line, it's just expressed as two functions, one for the x grid line and another for the y grid line.
The ideas presented in this video are a much simpler (in my opinion) formation of the ideas found in the book General Relativity - Introduction for Physicists by Hobson, Efstathiou, and Lasenby, particularly the ideas on Page 72, section 3.14, Intrinsic Derivative of a Vector Along a Curve.
And of course I named the derivation The Michael Scott Method because I think it would be interesting to see something in mathematics named after Michael Scott, the boss from The Office. I put Meredith and Stanley into the beginning of the video as well, because I like them too.
Some additional remarks not featured in the video:
I don't like mathematics to be so formal that you can't understand any of it. I think it should be taught on an intuitive level FIRST and a formal level SECOND. Formalism firms it up, but intuition makes it accessible. I'm not the kind of guy who thinks, "Oh, well, Gauss and Riemann and Einstein thought of all that stuff and we can trust their expertise. So we can keep their work difficult so it's only accessible to the top mathematicians if we want." I think a huge part of progress in education is to make difficult topics more accessible as time goes on. I'm sure a long time ago they did not teach calculus in high school like they do now. So why not put something in place for geodesics to be potentially taught at the high school level some day?
I think advances in learning should be like advances in territorial exploration. In the days when the frontier was largely unexplored, first one tough guy found a suitable area in the rough wilderness for settlement, and after it was discovered, it had to be made accessible for people who weren't quite as tough as the first guy. If it's a place truly worthy of human presence, by all means, clear a path so people can reach it and see its beauty. If it's best left alone, or sparsely settled, then I guess leave it that way.
Perhaps certain knowledge should be kept limited so that only few have access to it, but I do not think geodesics should be that way. I made an effort to try to clear a path to them, even if it turns out to be ultimately fruitless.
