How to approximately (or exactly, if it's easier) find a function which describes the 'density' of the path of a 2D double pendulum?
Image from the program at http://dllu.net/dp/
Here are some initial step(s) I would take: I would use the solutions to the usual 2D double pendulum to calculate the path for arbitrary m,M,l and L, and then 'compress' the path into a circular probability distribution (the circumference) and then make an equation describing the vertical probability.
- To simplify, 'compress' all the points into a circle. The thickness of each infinitesimal arc on the circumference of the circle represents the probability that the path of the pendulum would be on the radial line (the line that intersects the center of the circle and the infinitesimal arc).