Take a closed system of particles. Each has it's own mass, but that's really about it for the sakes of this argument. Now suppose there is some standard force law that applies between particles, whatever it may be. (In general it will depend only on the separation between particles and thus be translation invariant but not always). Now, the force on some given particle is therefore dependant on the position of ALL the particles. Let's work in one dimension, x for simplicity... So what we have here is an acceleration field which depends on x (all of them). Now suppose that we know the initial configuration of the system (positions and velocities of all the particles), how does one go about finding the configuration of the system at some later time, t? I'm looking for equations here, not explanations. I understand the mechanics of the problem but I'm having trouble visualizing the mathematics. Mainly because my instinct is to integrate the acceleration function with respect to time to find the overall change in velocty... but the acceleration field depends on position, not time... This is throwing me off. To summarise, I'm looking for the mathematical procedure one would follow to calculate the configuration of a system of this kind after a time, t has elapsed - given the initial configuration. (I want positions and velocities in case that isn't clear). Purely Newtonian mechanics also, I'm not worried about SR just yet and let's not complicate things. (I know my maths... I will understand whatever equations you give me, I just need a hand seeing the process here) Thanks :) EXTRA: If anyone feels like it, my next question would be: What if the acceleration depends on both positions and velocities? Such as an electrically charged particle moving in a magnetic field. How would this affect the equations?