Older Efforts
The simulations can be done in either Cartesian or in spherical coordinates. The former are easier, but extremely calculation intensive, and only practical on an electronic computer. As such only the latter was used in former times. Strictly speaking not much less calculation intensive, but it was possible to start with some simple approximations and then to add perturbations, as much as needed to reach the wanted accuracy.
In essence this mathematical simulation of the Solar System is a form of the N-body problem. The symbol N represents the number of bodies, which can grow quite large if one includes 1 sun, 8 planets, dozens of moons and countless planetoids, comets and so forth. However the influence of the sun on any other body is so large, and the influence of all the other bodies on each other so small that the problem can be reduced to the analytically solvable 2-body problem. The result for each planet is an orbit, a simple description of its position as function of time. Once this is solved the influences moons and planets have on each other are added as small corrections. (Small compared to a full planetary orbit, some corrections might be still several degrees large, while measurements can be made to an accuracy of better than 1″).
Although this method is no longer used for simulations, it is still useful to find an approximate ephemeris as one can take the relatively simple main solution, perhaps add a few of the largest perturbations, and arrive without too much effort at the wanted planetary position. The disadvantage is that perturbation theory is very advanced mathematics.
Read more about this topic: Numerical Model Of The Solar System
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