Final positions and velocities of two bodies, given initial position and velocities, masses, and a force law $f(r)$

Question

How would I find the final positions and velocities of two bodies given the initial positions and velocities of both objects, the masses of both objects, and some force law $f(r)$, which is some function of distance $r$? I understand one way to approximate the final positions and velocities is using a simulation that uses time step increments, but how would I calculate the final positions and velocities of two bodies without using time step increments? How would I calculate the final positions and velocities by hand?

Answer 1

You solve Newton's, Lagrange's or Hamilton equations which leads to a system of ordinary differential equations with the given initial positions and velocities as initial conditions. For the two-body problem, the solution can often be found analytically by using conservation laws (energy, momentum, angular momentum, ...).