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Hi, I work on N-body orbit determination software for asteroid detection and planetary defense (for example, https://github.com/moeyensj/thor). Ask me anything.
You're right that orbit determination is unstable over long runs, and very sensitive to initial conditions. But the good news is that the solar system is big. Most trajectories do not intersect with the earth's location. So perturbing an orbit that we think has a small chance of hitting us tends to be safe.
Also, of course, after deflection, any object is going to be monitored really closely for a long time.
Mostly Python and Fortran. See for example https://github.com/oorb/oorb.
The hardest problems are always the social ones. How do you get uptake of a new method, how do you get funding, how do you politely tell a collaboration they are doing the wrong thing, etc.
But if you mean pure technical stuff - the hardest problem I had to solve was rethinking some of the inner loops of the THOR algorithm. The problem was essentially to speed up a Hough transform in 6D space. Lots of time spent profiling CPU cache timings to get that fast.