I did not realize that simply apply the law of physics on particles would not maintain the conservation of energy. I thought about for a while and realize it was because that, say when a particle rotates around an object because of gravity, the velocity and acceleration changes in every infinitesimal time point. Yet the computer cannot simulate that process.

I was also thinking. if we don’t enforce “real time simulation”, maybe we could calculate the energy of the whole system in the beginning state (E0). Then for each step of simulation we could try minimizing (E – E0)^2 using gradient descent with some other constraints.

Not sure if that’s going to work. It’s just a thought.