# Numerical Solutions

## Contents

## 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:

- “ 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* (Archive) 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.*”

A Princeton University programming assignment (Archive) 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* (Archive) 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. ”

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

- “ Q. What kind of problem solutions do you rate higher: analytical or numerical? More problems can be solved numerically, using computers. But some of the same problems can be solved analytically. What would your preference be? ”

- Mohammad Firoz Khan, Ph.D. responds:

- “ A researcher would like to solve it analytically so that it is clear what are premises, assumptions and mathematical rules behind the problem. As such problem is clearly understood. Numerical solution using computers give solution, not the understanding of the problem. It is quite blind. However, in emergency one may resort to this option. ”

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

- “ An analytical solution involves framing the problem in a well-understood form and calculating the exact solution. A numerical solution means making guesses at the solution and testing whether the problem is solved well enough to stop. ”

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

- “ With rare exceptions, a numerical solution is always wrong; the important question is, how wrong is it? ”

## Two Body Approximations

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

- “ Once a physicist gets a hold of all the appropriate equations and a big computer, they can start approximating things. With enough computing power and time, these approximations can be made amazingly good. Computer simulation and approximation is a whole science unto itself.

- But even with just mechanical pencil and paper there are cheats. For example, although there are more than three bodies in the solar system (the Sun, eight planets, dozens of moons, and millions of asteroids and comets), almost everything behaves, roughly, as though it were in a two body system. Basically, this is due to the pronounced size differences between things. As far as each planet is concerned, the only important body in the rest of the universe is the Sun. To get some idea of why; the Sun pulls on the Earth about 200 times harder than the Moon, and about 20,000 times harder than Jupiter. Nothing else even deserves a mention. So, if you want to calculate the orbits of all the planets, a “2-body approximation” will get you more than 99% of the way to the right answer. ”

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

- “ We developed a Keplerian-based Hamiltonian splitting for solving the gravitational N-body problem. This splitting allows us to approximate the solution of a general N-body problem by a composition of multiple, independently evolved two-body problems. While the Hamiltonian splitting is exact, we show that the composition of independent two-body problems results in a non-symplectic non-time-symmetric first-order map. A time-symmetric second-order map is then constructed by composing this basic first-order map with its self-adjoint. The resulting method is precise for each individual two-body solution and produces quick and accurate results for near-Keplerian N-body systems, like planetary systems or a cluster of stars that orbit a supermassive black hole. ”

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

- “ The patched-conic approximation has thus been developed as a more accurate solution to interplanetary transfer description. It involves partitioning the overall transfer into distinct conic solutions. For instance, as a spacecraft travels from Earth to Mars, its orbit is approximated as a hyperbolic departure, an elliptic transfer, and a hyperbolic arrival. The patched-conic approximation breaks the entire orbit down into several two-body problems. In other words, only one celestial body’s influence is considered to be acting upon the spacecraft at all times. ”

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

- “ In this paper, we present a new symplectic integrator for collisional gravitational N-body dynamics. The integrator is inspired by the non-symplectic and non-reversible integrator in Gonçalves Ferrari et al. (2014), SAKURA, and makes use of Kepler solvers. Like SAKURA we decompose the N-body problem into two-body problems. In contrast to SAKURA, our two-body problems are not independent. The integrator is reversible and symplectic and conserves nine integrals of motion of the N-body problem to machine precision. ”

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

- “ It seems that a truly rigorous and elegant solution will be achieved only by finding a mathematical transformation that reduces the many-body problem to a one-body problem. In such a formulation each atom, nucleus or electron can be treated alone with the contributions of all the others summed together. In Alder's opinion such a development will really allow working on the deep-lying problems of the quantum-mechanical structure of matter. Physics has a long history of reducing many-body problems to one-or two-body problems in order to find more powerful solutions, and Alder and his colleagues have high hopes of doing it for this one. ”

## 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

- “ As mentioned earlier, we use the equations of motion derived under the BCM assumptions as the underlying dynamical model. The bicircular problem is a simplified version of the restricted four-body problem. The objective is to describe the motion of a spacecraft of negligible mass under the gravitational attraction of the Earth, Moon, and Sun. “Negligible mass” means that the spacecraft does not influence the motion of the Earth, Moon, and Sun. This description follows that of Simo, Gomez, Jorba, and Masdemont [1995].

- In this model we suppose that the Earth and Moon are revolving in circular orbits around their center of mass (barycenter) and the Earth-Moon barycenter is moving in a circular orbit around the center of mass of the Sun-Earth-Moon system. The orbits of all four bodies are in the same plane. We remark that, with these assumptions, the motion of these three bodies is not coherent. That is, the assumed motions do not satisfy Newton’s equations. However, numerical simulation shows that, in some regions of phase space, this model gives the same qualitative behavior as the real system. Thus, the model is extremely useful for the study of some kinds of orbits, in particular the Hiten trajectory of Belbruno and Miller [1993] and more recently the “Shoot the Moon” trajectory of Koon, Lo, Marsden, and Ross [2001b]. ”

## 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:

- “ In the last few years, a group of algorithms has been developed in the astrophysics community which have come to be known as "tree codes" or "hierarchical codes." They are due to Appel, Barnes and Hut, and others. They are designed to work well in a variety of settings, including ones where there is a high degree of clustering. The basic idea is to replace groups of distant particles by their centers of mass, and to compute the interactions between groups via this approximation. ”

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

*The Barnes-Hut Galaxy Simulator*

- “ The Barnes-Hut Algorithm describes an effective method for solving n-body problems. It was originally published in 1986 by Josh Barnes and Piet Hut [1]. Instead of directly summing up all forces, it is using a tree based approximation scheme which reduces the computational complexity of the problem from O(N2) to O(N log N).

- It works by reducing the number of force calculations by grouping particles. The basic idea behind the algorithm is that the force which a particle group excerts on a single particle can be approximated by the force of a pseudo particle located at the groups center of mass. For instance, the force which the Andromeda galaxy excerts on the milky way can be approximated by a point mass located at the centre of the Andromeda galaxy. There is no need to integrate over all stars in the Andromeda galaxy provided the distance between the two galaxies is large enough. This approximation is valid as long as the distance from a point group to a particle is large and the radius of the group is small in relation to the distance between the group and the particle. ”

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.