Differentiating a Physics Simulator

Have you seen something happen that seems almost impossible to replicate? Like dropping something and have it land in just the right way, or shoot an arrow and have it land on its tip?

How would you shoot an arrow that lands like THAT?

From a simulation standpoint, you might ask what are the initial conditions required to produce that outcome. To solve this, you could attempt to search through the whole search space. For instance, you could try all possible velocity and position of the arrow and select the ones that land on it's tip. However this is plain impractical as despite only having 7 degrees of freedom (3 for rotation, 3 for position and one for initial velocity), the search space is already huge and chances are, the solutions are but a small portion of the unwieldy search space.

Ok first off, why should we care?

The arrow problem is arguable classified under 'cool' but definitely not 'practical'. However, these kinds of problems where there is a huge search space and we are searching for only a small subset of said space is ubiqu