Looking at phase diagrams is something that feels weird at first but turns out to be very nice. I think one reason it’s weird is that (when one follows the norm

Looking at phase diagrams is something that feels weird at first but turns out to be very nice. I think one reason it’s weird is that (when one follows the normal track of education through school and into a degree like physics or mathematics at university), this is the first time (at least in th uk school system) when one stands basically no chance of actually solving the equations but giving good useful qualitative information about the solutions is possible. I think part of why it can be hard to learn is that one must guess what is in the unknown areas whereas in past problems with limited information (eg in geometry), the puzzle is to derive just enough information on the boundary to solve the problem, so one may not be able to say something about the areas of some figures but might have something to say about their sum. I found the trick to phase diagrams was realising that they are mostly made of a few interesting features: at every fixed point the diagram is some combination of a circular motion and a scaling in/out motion, or it is some kind of saddle point going out in one direction and in at another (eg the fixed point at the top of the pendulum in the video). Then you just fill in the gaps. This feels hard at first because you can imagine there could be anything in the gaps but it is actually easy because the flow lines do not cross and one cannot have significant turbulence without fixed points (and those have all been found) so only boring things can be drawn in the gaps. The same is also true with drawing contour diagrams, just replace fixed point with stationary point, and flow line with isosurface.