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?
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1A typical upper-division or graduate text on mechanics spends at least a chapter on this subject. And not a short or easy chapter, either. I think the topic is simply too big for a Stack Exchange question. – dmckee --- ex-moderator kitten Mar 12 '18 at 16:18
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1Possible duplicate of Kepler problem in time: how do two gravitationally attracted particles move? – sammy gerbil Mar 13 '18 at 18:54
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It's not a duplicate because the alleged duplicate is on the inverse square law, not in the case of a generic force law $f(r)$. – Anders Gustafson Mar 13 '18 at 20:19
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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, ...).
freecharly
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+1 This very vague answer is the best one I can think of for this very vague question. – Mike Mar 15 '18 at 16:27