While there is an analytical solution for the two-body system, systems involving three or more masses do not have an analytical solution. Thus computer simulations of the time-evolution of multi-body systems are very common in astronomy.
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Fellhauer, M. (2008). Particle-Mesh Technique and Superbox. In: Aarseth, S.J., Tout, C.A., Mardling, R.A. (eds) The Cambridge N-Body Lectures. Lecture Notes in Physics, vol 760. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-8431-7_6
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