Numerical Solutions

N-Body Quotes

From 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. ”

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. ”

From On the Reliability of N-body Simulations:

  “ In time, the numerical solution diverges from the true solution and this error due to divergence will become more dominant. ”

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. ”

From a question posted on researchgate.net:

  “ 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:

  “ 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

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

Dr. Math at mathforum.org remarks:

  “ A numerical solution is a method of finding better and better approximations to the solution, good to more and more decimal places. Most numerical solutions start with a guess, and then use that guess to make another guess that's closer, and use the closer guess to get closer still. ”

Two Body Approximations

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

  “ 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

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

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

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.

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

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.