So I am a student and decided (for some bizarre reason) to attempt to tackle general relativity for my final astrophysics and computational physics project this term. I have been doing a lot of reading and in general things make sense conceptually. The tensor math is a bit over my head but once I sit down with a pencil and paper to actually work through them I am hoping things will begin to make more sense in that department. I am however still faced with one question that I can't seem to settle:
How do you move from the geodesic equations derived from, say, the Schwarzschild metric to equations of motion in real 3D Cartesian space that could be used to calculate the orbit of a point mass around a spherically symmetric body?
I'm not sure if I'm missing the point and you don't actually use the geodesic equations to obtain said equations of motion, or I just haven't searched hard enough in my studies to find the connection. Any help would be greatly appreciated, as I need some tangible equations of motion to calculate orbits numerically for my computational physics project.