The Theoretical Solution
This is the most elemental way of solving the n-body problem, the theoretical expression. It is often called "The n-body problem by Taylor series", which is an implementation of the Power series solution of differential equations.
We start by defining the differential equations system:
where (in ) the upper index indicates the second derivative with respect to time, represents the number of each body and the coordinate.
Because and are given as initial conditions, then every are known. Doing implicit derivation over every results in which at are known because each depends on known precalculated and given constants and then the Taylor series are constructed theoretically in such way, performing this process infinitely.
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