Numerical Solutions

On p.89 of Atmospheric and Space Flight Dynamics: Modeling and Simulation with MATLAB we read:

  “ In the next section, it will be shown that two additional integrals can be obtained when N = 2 from the considerations of relative motion of the two bodies. Hence, a two-body problem is analytically solvable. However, with N > 2, the number of unknown motion variables exceeds the total number of integrals; thus, no analytical solution exists for the N-body problem when N > 2. Due to this reason, we cannot mathematically prove certain observed facts (such as the stability of the solar system) concerning N-body motion. The best we can do is to approximate the solution to the N-body problem either by a set of two-body solutions or by numerical solutions. ”

On p.2 of Evaluation of mass loss in the simulation of stellar clusters using a new multiphysics software environment by Guillermo Kardolus we see:

  “ Although numerical solutions are only approximations, very complex problems can be solved numerically and with a high degree of accuracy. ”

  “ An interesting implementation of computational astrophysics is the numerical solution to the n-body problem: the problem of predicting position and velocity of a set of n gravitationally interacting particles. The differential equations describing the motion of the n particles can only be solved analytically for n = 2, and in certain special cases for n = 3 (e.g. the Lagrange points). Solving a higher order system seemed impossible and in 1885 a price was announced for a solution to the n-body problem. The problem could not be solved, it turned out that general solutions of order n ≥ 3 can only be approximated numerically. The current situation is that special purpose supercomputers can simulate in the order of 106 particles within a reasonable accuracy (Harfst et al., 2007). See also Chapter 2.2. ”