The double pendulum problem is kind of a historical/mathematical marvel. It is very easy to predict the motion of a common, everyday clock pendulum that swings back and forth; but add another pendulum at the end of that, and the complexity shoots through the roof in the most unexpected ways. It puzzled early mathematicians for years.
The system begins to exhibit chaotic motion, and the mathematics needed to solve the motion goes from simple polynomial math to Lagrange differential equations requiring a full college program of Calculus to solve. Here’s a full solution write-up.
Solving is one problem, but making something useful out of it is even more impressive. Here are two robots that can balance not only a double pendulum, but a triple! One application would be to have self-parking and skid-controllable multi-trailer semis. There are probably many more.
If the original double-pendulum mathematicians could see this, they would consider is sorcery. If you’ve ever tried to balance a broom on your hand, you should also marvel: