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I have been thinking about how to come up with a function of time for the n-body problem. Later I found that there still is no function of time known for the n-body problem, but I am still not sure where the brick wall for everyone is because I am quite sure my understanding has hit from very early on. I pretty much started from the ground up, where gravity is constant, and barely got through the basics. There are no explanations for what I am looking for to be found in any of my physics books, calculus book, or on the internet. I feel it is very basic yet I can't quite mathematically get anywhere, even though all of the concepts I understand clearly.

So from the start to where I am stuck:

-For a constant acceleration of gravity, such as near the earths surface where the acceleration of gravity = "g" [OPEN attachment A]

-Then integrate it to make the height as a function of time. [OPEN attachment B]

It is Easy so far, but when you make the height considerably far from earth and consider it over a long fall toward earth, the Gravitational Force on the object is constantly changing and the function of time would have to respond to a function of constantly changing acceleration:

where: F = ma = GmM(1/r[1/r^2])

divide by m : a = F/m = (GM)/r^2 !!!!!!

Now if we sum up the entire acceleration experienced for the object from the "initial r" to the "final r", we would simply integrate that function to get: [OPEN attachment C]

So the final velocity at any "final r" is:

V("final r") = initial V + GM[(1/ "final r") - (1/ "initial r")

This is as far as my understanding goes and now I would like to understand how to make this a function of time. Surely NASA can do something like this. They obviously know where all of the planets will be around the sun at any point in time and they obviously must have an understanding on how to send probes all over the solar system, orbiting various planets at various times and sling-shotting around this and that. Yet I can not find any literature on any of this.

ANY HELP WOULD BE GREAT. Once I got started on this not knowing how to do it has been driving me insane!

So from the start to where I am stuck:

-For a constant acceleration of gravity, such as near the earths surface where the acceleration of gravity = "g" [OPEN attachment A]

-Then integrate it to make the height as a function of time. [OPEN attachment B]

It is Easy so far, but when you make the height considerably far from earth and consider it over a long fall toward earth, the Gravitational Force on the object is constantly changing and the function of time would have to respond to a function of constantly changing acceleration:

where: F = ma = GmM(1/r[1/r^2])

divide by m : a = F/m = (GM)/r^2 !!!!!!

Now if we sum up the entire acceleration experienced for the object from the "initial r" to the "final r", we would simply integrate that function to get: [OPEN attachment C]

So the final velocity at any "final r" is:

V("final r") = initial V + GM[(1/ "final r") - (1/ "initial r")

This is as far as my understanding goes and now I would like to understand how to make this a function of time. Surely NASA can do something like this. They obviously know where all of the planets will be around the sun at any point in time and they obviously must have an understanding on how to send probes all over the solar system, orbiting various planets at various times and sling-shotting around this and that. Yet I can not find any literature on any of this.

ANY HELP WOULD BE GREAT. Once I got started on this not knowing how to do it has been driving me insane!