Three-body problem
Historically, the first specific three-body problem to receive extended study was the one involving the Earth, the Moon, and the Sun.
The mathematical statement of the three-body problem can be given in terms of the Newtonian equations of motion for vector positions
[4]: 8 The problem can also be stated equivalently in the Hamiltonian formalism, in which case it is described by a set of 18 first-order differential equations, one for each component of the positions
;[citation needed] however, this time dependence can be removed through a transformation to a rotating reference frame, which simplifies any subsequent analysis.
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.
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.
[14] In 1993, physicist Cris Moore at the Santa Fe Institute found a zero angular momentum solution with three equal masses moving around a figure-eight shape.
For instance, the probability of a binary–binary scattering event[clarification needed] resulting in a figure-8 orbit has been estimated to be a small fraction of a percent.
[18] 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 2015, physicist Ana Hudomal discovered 14 new families of solutions for the equal-mass zero-angular-momentum three-body problem.
[19] In 2017, researchers Xiaoming Li and Shijun Liao found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem.
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.
[25][26] The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Philosophiæ Naturalis Principia Mathematica, in which Newton attempted to figure out if any long term stability is possible especially for such a system like that of the Earth, the Moon, and the Sun, after having solved the two-body problem.
[27] Guided by major Renaissance astronomers Nicolaus Copernicus, Tycho Brahe and Johannes Kepler, Newton introduced later generations to the beginning of the gravitational three-body problem.
[28] 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.
[28] 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.
[30] 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 Académie Royale des Sciences in 1747.
[31] It was in connection with their research, in Paris during the 1740s, that the name "three-body problem" (French: Problème des trois Corps) began to be commonly used.
[33] George William Hill worked on the restricted problem in the late 19th century with an application of motion of Venus and Mercury.
[36] In 2017, Shijun Liao and Xiaoming Li applied a new strategy of numerical simulation for chaotic systems called the clean numerical simulation (CNS), with the use of a national supercomputer, to successfully gain 695 families of periodic solutions of the three-body system with equal mass.
[37] In 2019, Breen et al. announced a fast neural network solver for the three-body problem, trained using a numerical integrator.
[41] In both classical and quantum mechanics, however, there exist nontrivial interaction laws besides the inverse-square force that do lead to exact analytic three-body solutions.
In these two respects it is analogous to (insoluble) models having Coulomb interactions, and as a result has been suggested as a tool for intuitively understanding physical systems like the helium atom.
[45] One can draw parallels between the motion of a passive tracer particle in the velocity field of three vortices and the restricted three-body problem of Newtonian mechanics.
Physically, a relativistic treatment becomes necessary in systems with very strong gravitational fields, such as near the event horizon of a black hole.
However, the relativistic problem is considerably more difficult than in Newtonian mechanics, and sophisticated numerical techniques are required.
Even the full two-body problem (i.e. for arbitrary ratio of masses) does not have a rigorous analytic solution in general relativity.
