The Two-Body Problem, Classical Version: given two 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: yes – we can write down and solve the equation of motion in closed form. That means, given the initial positions and momenta, we can pick any time value with any precision, plug it in, and get the positions and momenta of the particles with corresponding precision.

The Two-Body Problem, Quantum Relativistic Version: given two 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: I don’t know.

The Two-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 the universe can’t rely on an equation to know what’s next.

Examples

Classical: star-planet, binary star

Quantum: Hydrogen Atom (H)