The Three-Body Problem, Classical Version: given three objects/particles interacting according to a 1/r2 law, which means gravity or electricity, can we write and solve the equation of motion in closed form?

Answer: we can write down the equation of motion in closed form, but we can’t solve it to reach a closed-form solution. That means to know the positions and momenta as a function of time, we need to either a) run a numerical simulation with discrete time steps or b) use approximations and/or perturbation theory to reach an approximate closed-form solution.

The Three-Body Problem, Quantum Relativistic Version: given three wave functions in relativistic space interacting according to a 1/r2 law, which means gravity or electricity, can we write and solve the equation of motion in closed form?

Answer: Also can’t do it.

The Three-Body Problem, How the Universe Does It: I suspect the universe runs a numerical simulation with infinitely-small time steps. The universe doesn’t rely on equations of motion: the universe is a nearly-infinity-body problem all the time, so it’s impossible to get an equation, and anyway the future doesn’t exist yet so even for two bodies the universe can’t rely on an equation to know what’s next.

Examples

Classical: Sun-Earth-Moon

Quantum: Hydrogen molecule ion (H2+), Helium atom (He)