Numerical Solutions

N-Body Quotes

On p.89 of Atmospheric and Space Flight Dynamics: Modeling and Simulation with MATLAB by Professor Ashish Tewari (bio) 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 a Master's thesis 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. ”

A Princeton University programming assignment says:

  “ In 1687, Isaac Newton formulated the principles governing the motion of two particles under the influence of their mutual gravitational attraction in his famous Principia. However, Newton was unable to solve the problem for three particles. Indeed, in general, solutions to systems of three or more particles must be approximated via numerical simulations. ”

The paper Global Error Measures for Large N-Body Simulations describes:

  “ N-body systems are chaotic, which implies that small perturbations to a solution, such as numerical errors, are exponentially magnified with the passage of time. Although this is widely recognized, its impact on qualitative properties of numerical N-body simulations is not well understood. Animated movies of large N-body simulations, like spiral galaxies or cosmological systems, are very exciting to watch and often look quite reasonable, but it is little more than an “article of faith” that the results are qualitatively correct (Heggie, 1988). ”

General Quotes

The book Nuclear Astrophysics: A Course of Lectures tells us on p.259:

  “ Solutions generated by numerical methods are generally only approximations to the exact solution of the underlying equations. However, much more complex systems of equations can be solved numerically than can be solved analytically. Thus, approximate solutions to the exact equations found by numerical methods often provide far more insight than exact solutions to approximate equations that can be solved analytically. ”

The abstract of a medical research paper Simulation and air-conditioning in the nose says:

  “ In general, numerical simulations only calculate predictions in a computational model, e. g. realistic nose model, depending on the setting of the boundary conditions. Therefore, numerical simulations achieve only approximations of a possible real situation. ”