If you restrict yourself to certain types of objects and environments, you can use totally different techniques. For a game dev class in uni, I wrote air hocke

If you restrict yourself to certain types of objects and environments, you can use totally different techniques. For a game dev class in uni, I wrote air hockey. This is straightforward because you only have circles and lines, and there's no gravity so objects can't rest on each other. I wrote an event-based physics engine, which exploits the fact that the equations to find the time t of a future collision are closed-form and you can just solve for t. If t is greater than the time to the next frame, just step everything forward; if it's less, step to that collision, then find the next one. Collision response is easy too: just choose a reference frame where one of the pucks is stationary, then solve conservation of momentum and energy. This excludes some kinds of friction models, but if you choose one where deceleration due to friction is independent of speed (perfectly valid for lowish speeds), you're fine. Another cool thing about an event-based system is that it makes it trivial to write an AI: just have the system try N moves toward the nearest puck, and run the simulation forward a few collisions to see if any of those moves results in a puck going in the opponent's goal.