Three Body Problem Example

[Jenette Bregantini]

In physics, specifically classical mechanics, the three-body problem involves taking the initial positions and velocities (or momenta) of three point masses that orbit each other in space and calculating their subsequent trajectories using Newton's laws of motion and Newton's law of universal gravitation.[1]

Unlike the two-body problem, the three-body problem has no general closed-form solution.[1] When three bodies orbit each other, the resulting dynamical system is chaotic for most initial conditions, and the only way to predict the motions of the bodies is to calculate them using numerical methods.

The three-body problem is a special case of the n-body problem. Historically, the first specific three-body problem to receive extended study was the one involving the Moon, Earth, and the Sun.[2] In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles.

In the restricted three-body problem, a body of negligible mass (the "planetoid") moves under the influence of two massive bodies. Having negligible mass, the planetoid exerts force on the two massive bodies that may be neglected; therefore the resulting system can be analyzed and described as a two-body motion problem.[3][5][failed verification] With respect to a rotating reference frame, the two co-orbiting bodies are stationary, and the third can be stationary as well at the Lagrangian points, or move around them, for instance on a horseshoe orbit. It can be useful to consider the effective potential. Usually this two-body motion is taken to consist of circular orbits around the center of mass, and the planetoid is assumed to move in the plane defined by the circular orbits.

There is no general closed-form solution to the three-body problem.[1] In other words, it does not have a general solution that can be expressed in terms of a finite number of standard mathematical operations. Moreover, the motion of three bodies is generally non-repeating, except in special cases.[8]

However, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists an analytic solution to the three-body problem in the form of a Puiseux series, specifically a power series in terms of powers of t1/3.[9] This series converges for all real t, except for initial conditions corresponding to zero angular momentum. In practice, the latter restriction is insignificant since initial conditions with zero angular momentum are rare, having Lebesgue measure zero.

An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Therefore, it is necessary to study the possible singularities of the three-body problems. As is briefly discussed below, the only singularities in the three-body problem are binary collisions (collisions between two particles at an instant) and triple collisions (collisions between three particles at an instant).

Collisions of any number are somewhat improbable, since it has been shown that they correspond to a set of initial conditions of measure zero. But there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So Sundman's strategy consisted of the following steps:

The corresponding series converges extremely slowly. That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 108000000 terms.[10]

In 1772, Lagrange found a family of solutions in which the three masses form an equilateral triangle at each instant. Together with Euler's collinear solutions, these solutions form the central configurations for the three-body problem. These solutions are valid for any mass ratios, and the masses move on Keplerian ellipses. These four families are the only known solutions for which there are explicit analytic formulae. In the special case of the circular restricted three-body problem, these solutions, viewed in a frame rotating with the primaries, become points called Lagrangian points and labeled L1, L2, L3, L4, and L5, with L4 and L5 being symmetric instances of Lagrange's solution.

In 1893, Meissel stated what is now called the Pythagorean three-body problem: three masses in the ratio 3:4:5 are placed at rest at the vertices of a 3:4:5 right triangle, with the heaviest body at the right angle and the lightest at the smaller acute angle. Burrau[11] further investigated this problem in 1913. In 1967 Victor Szebehely and C. Frederick Peters established eventual escape of the lightest body for this problem using numerical integration, while at the same time finding a nearby periodic solution.[12]

In 2013, physicists Milovan Šuvakov and Veljko Dmitrašinović at the Institute of Physics in Belgrade discovered 13 new families of solutions for the equal-mass zero-angular-momentum three-body problem.[8][14]

In 2017, researchers Xiaoming Li and Shijun Liao found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem.[20] This was followed in 2018 by an additional 1,223 new solutions for a zero-angular-momentum system of unequal masses.[21]

In 2018, Li and Liao reported 234 solutions to the unequal-mass "free-fall" three-body problem.[22] The free-fall formulation starts with all three bodies at rest. Because of this, the masses in a free-fall configuration do not orbit in a closed "loop", but travel forward and backward along an open "track".

In 2023, Ivan Hristov, Radoslava Hristova, Dmitrašinović and Kiyotaka Tanikawa published a search for "periodic free-fall orbits" three-body problem, limited to the equal-mass case, and found 12,409 distinct solutions.[23]

Using a computer, the problem may be solved to arbitrarily high precision using numerical integration although high precision requires a large amount of CPU time. There have been attempts of creating computer programs that numerically solve the three-body problem (and by extension, the n-body problem) involving both electromagnetic and gravitational interactions, and incorporating modern theories of physics such as special relativity.[24] In addition, using the theory of random walks, an approximate probability of different outcomes may be computed.[25][26]

The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Philosophi Naturalis Principia Mathematica, when Newton was trying to figure out if any long term stability is possible, especially the system of our Earth, the Moon, and the Sun. He was guided under the major Renaissance astronomers Nicolaus Copernicus, Tycho Brahe and Johannes Kepler to the beginning of the gravitational three-body problem.[27] In Proposition 66 of Book 1 of the Principia, and its 22 Corollaries, Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions. In Propositions 25 to 35 of Book 3, Newton also took the first steps in applying his results of Proposition 66 to the lunar theory, the motion of the Moon under the gravitational influence of Earth and the Sun.[28] Later, this problem was also applied to other planets' interactions with the Earth and the Sun.[27]

The physical problem was first addressed by Amerigo Vespucci and subsequently by Galileo Galilei, as well as Simon Stevin, but they did not realize what they contributed. Though Galileo determined that the speed of fall of all bodies changes uniformly and in the same way, he did not apply it to planetary motions.[27] Whereas in 1499, Vespucci used knowledge of the position of the Moon to determine his position in Brazil.[29] It became of technical importance in the 1720s, as an accurate solution would be applicable to navigation, specifically for the determination of longitude at sea, solved in practice by John Harrison's invention of the marine chronometer. However the accuracy of the lunar theory was low, due to the perturbing effect of the Sun and planets on the motion of the Moon around Earth.

Jean le Rond d'Alembert and Alexis Clairaut, who developed a longstanding rivalry, both attempted to analyze the problem in some degree of generality; they submitted their competing first analyses to the Acadmie Royale des Sciences in 1747.[30] It was in connection with their research, in Paris during the 1740s, that the name "three-body problem" (French: Problme des trois Corps) began to be commonly used. An account published in 1761 by Jean le Rond d'Alembert indicates that the name was first used in 1747.[31]

From the end of the 19th century to early 20th century, the approach to solve the three-body problem with the usage of short-range attractive two-body forces was developed by scientists, which offered P.F. Bedaque, H.-W. Hammer and U. van Kolck an idea to renormalize the short-range three-body problem, providing scientists a rare example of a renormalization group limit cycle at the beginning of the 21st century.[32] George William Hill worked on the restricted problem in the late 19th century with an application of motion of Venus and Mercury.[33]

At the beginning of the 20th century, Karl Sundman approached the problem mathematically and systematically by providing a functional theoretical proof to the problem valid for all values of time. It was the first time scientists theoretically solved the three-body problem. However, because there was not a qualitative enough solution of this system, and it was too slow for scientists to practically apply it, this solution still left some issues unresolved.[34] In the 1970s, implication to three-body from two-body forces had been discovered by V. Efimov, which was named the Efimov effect.[35]

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