This is a physics question, but the motivation for it comes from game design.
I want to simulate the motion of an object in 2D space with several point sources of gravity (actually stars). The point sources will not move, so the gravitational field is static. (I'm also ignoring any relativistic effects.)
I know I can get a pretty accurate trajectory just by applying an appropriate acceleration per tick, but I'm wondering if there's some way to generate the equation of motion for an object given the formula for its gravitational potential. That way to get the coordinates at a given time, I could get it by plugging the time into that equation. I know that the evolution of a system follows the path of least action, but I don't know how to go further than that.
I haven't found much on the internet for solving this particular problem. I'm wondering if:
A) There is no exact solution for any significant number of point sources (I already know this is true if the sources are moving in each other's fields);
B) There is an exact solution but evaluating it for a given instant is significantly more computationally expensive than finding the acceleration using the simple approach;
C) It's possible to get an arbitrarily good approximation using an iterative process, but it takes so long to converge that it's not worth it.
I'm probably going to go with the simple approach in the end (maybe with pre-calculating the acceleration as a vector field), but I'm curious to know what an action-integral–based solution would look like and whether a solution like that is at all practical to do in real-time.