Numerical Solutions

The topic of Numerical Solutions typically refers to a response to the Three Body Problem, which claims that there are working celestial models with three or more bodies. It is suggested by some sources that numerical solutions exist which can simulate the n-body systems proposed by conventional astronomy. However, it is seen that 'numerical solutions' refers to methods of approximations. The numerical solutions for n-body problems where N > 2 do not fully simulate gravity and involve limited interaction and liberal assumptions.

N-Body Quotes
From p.89 of Atmospheric and Space Flight Dynamics: Modeling and Simulation with MATLAB (Archive) by Professor Ashish Tewari (bio) we read:



On p.2 of a Master's thesis Evaluation of mass loss in the simulation of stellar clusters using a new multiphysics software environment (Archive) by Guillermo Kardolus we see:





A Princeton University programming assignment (Archive) says:



The paper Global Error Measures for Large N-Body Simulations (Archive.is) describes:



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



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



From a question posted on researchgate.net (Archive):




 * Mohammad Firoz Khan, Ph.D. (bio) responds:



Jason Brownlee, Ph.D., tells us on machinelearningmastery.com (Archive):



http://www.math.pitt.edu/~sussmanm/2071Spring09/lab02/index.html (Archive)



Two Body Approximations
https://www.askamathematician.com/2011/10/q-what-is-the-three-body-problem/ (Archive)



https://academic.oup.com/mnras/article/440/1/719/1747624 (Archive)



https://hanspeterschaub.info/Papers/UnderGradStudents/ConicReport.pdf (Archive)



https://academic.oup.com/mnras/article/452/2/1934/1069988 (Archive)



On the subject of particle physics - https://academic.oup.com/mnras/article/440/1/719/1747624 (Archive)



Four Body Approximation
http://www.cds.caltech.edu/~marsden/volume/missiondesign/KoLoMaRo_DMissionBook_2011-04-25.pdf (Archive)

5.3 Bicircular Model 126 5. Trajectories in the Four-Body Problem



Galaxy Simulator
The following was provided to us as an example of the numerical solution of multiple bodies - The Numerical Solution of the N-Body Problem (Archive). Read the below quotes and decide whether the methods are describing a full simulation of gravity.

From the introduction of the paper:



Looking up 'Barnes Hut' we find: https://beltoforion.de/en/barnes-hut-galaxy-simulator/ (Archive)


 * The Barnes-Hut Galaxy Simulator



The above shows that there could be a numerical solution that doesn't use gravity fully, discrediting the "numerical solutions exist" idea. Like with the previous quotes and examples, liberal assumptions are made, rather than a true simulation of the laws involved.

Addendum
As stated above, the N-Body problem cannot be solved with analytical solutions for bodies greater than two. Professor Tewari says that "we cannot mathematically prove certain observed facts (such as the stability of the solar system) concerning N-body motion". Numerical solutions are described as "approximations of a possible real situation", which "give solution, not the understanding of the problem", and which are seen to employ multiple two body problems or fudges to simulate an N-Body system.

Would the planets, moons, or asteroids use gravitationally selective two body problems or mathematical fudges while traversing the solar system? If not, and if gravity would be universal between the bodies under a real scenario, then the miserable state of the model becomes apparent. Our fundamental cosmic outlook says that it should be possible for a star to have a planet which has a moon, yet the Three Body Problem shows us that this basic component of astronomy cannot be simulated, despite the best efforts of the greatest mathematicians of human history.