Three-body problem

• 1. Introduction

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  The three-body problem asks a question that sounds almost too simple for physics. Take three point masses floating in space, each tugging on the others through gravity. Give each one a starting position and a starting velocity, or momentum. Then use Newton's laws of motion and his law of universal gravitation to calculate where all three travel next.

  With two bodies, that calculation can be done exactly. Add a third body and the problem has defied mathematicians for centuries. Some of the best-known names in mathematics and astronomy have attacked it, and their partial victories turn out to be stranger than the failure itself.

  What goes wrong when a third body joins the system? Which special arrangements can be solved after all, and who found them? And what happens when the same puzzle appears inside an atom, in a swirling fluid, or near a black hole?

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• 2. Newton's Moon Problem

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  In 1687 Isaac Newton published his Philosophiæ Naturalis Principia Mathematica, and with it the first real foothold on the three-body problem. Having solved the two-body case, he wanted to know whether a system like the Earth, the Moon and the Sun could stay stable over the long term. He built on the work of Copernicus, Tycho Brahe and Kepler.

  In Proposition 66 of Book 1, along with its 22 Corollaries, Newton took the first steps toward defining how three massive bodies move under their mutual pulls. In Propositions 25 to 35 of Book 3, he applied those results to the lunar theory. That theory describes the Moon's motion under the combined gravitational influence of the Earth and the Sun.

  Earlier figures had brushed against the problem without recognizing it. In 1499 Amerigo Vespucci used the position of the Moon to work out where he was in Brazil. Galileo Galilei and Simon Stevin also contributed pieces of the physics, though they did not realize it. Galileo found that all falling bodies change speed uniformly and in the same way, but he never applied that idea to the motion of the planets.

  The question became a matter of technical importance in the 1720s, because an accurate solution could help sailors find their longitude at sea. In practice, John Harrison's invention of the marine chronometer solved that navigation problem. The lunar theory itself stayed inaccurate, since the Sun and the planets keep disturbing the Moon's path around the Earth.

  In 1747 the rivals Jean le Rond d'Alembert and Alexis Clairaut each submitted a first analysis of the problem to the Académie Royale des Sciences. Their research in Paris in the 1740s is where the name Problème des trois Corps, the three-body problem, came into common use. D'Alembert's own account, published in 1761, dates the first use of that name to 1747.

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• 3. Counting Equations

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  The astronomer Juhan Frank points out that the three second-order vector equations of motion are equivalent to 18 first-order scalar differential equations. The historian of mathematics June Barrow-Green describes another presentation, in which nine second-order equations do the same job. The problem can also be written in the Hamiltonian formalism, again as 18 first-order equations, one for each component of position and momentum. There the Hamiltonian is simply the total energy of the system, gravitational plus kinetic.

  Mathematicians often turn to the restricted three-body problem, which strips the puzzle down. In Barrow-Green's description, two bodies circle their centre of mass in circular orbits, forming a two-body system whose motion is already known. A third body, called a planetoid, is treated as massless. It moves in the plane of the other two, feels their pull and exerts none of its own. In Barrow-Green's words, "the problem is then to ascertain the motion of the third body."

  This version is easier to analyze, yet it still matches many real situations. That is why it has played such a large part in the history of the full problem.

  The restricted problem has a phase space with 4 dimensions but only one conserved quantity, called the Jacobi integral. Heinrich Bruns showed that there are no further algebraic conserved quantities. In 1889 Henri Poincaré showed that there are no further analytic ones either. When the phase space has more dimensions than the system has constants of motion, exact solution is impossible. The system is in fact chaotic.

  Which fate awaits a body depends on the value of the Jacobi integral. A body that starts out orbiting the larger mass may be captured by the secondary mass, or it may be ejected through the Lagrange points L2 or L3. If both large bodies exert radiation pressure, four more equilibrium points appear beside the five classical Lagrange points.

  In the late 19th century, George William Hill used the restricted problem to study the motion of Venus and Mercury.

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• 4. Numerical Answers

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  In 2019, Breen and colleagues announced a fast neural network solver for the three-body problem, trained using a numerical integrator. That tool sits at the end of a long reliance on computers. For most starting conditions, the only way to predict the motion of three orbiting bodies is to estimate it numerically.

  Using numerical integration, a computer can solve the problem to arbitrarily high precision. Programmers have also tried to build simulations of the n-body problem that include both electromagnetic and gravitational interactions. Some attempts incorporate modern physics such as special relativity.

  The theory of random walks offers a different route. With it, researchers can compute an approximate probability for each of the different possible outcomes. That probabilistic view returns later, when researchers ask how likely a particular strange orbit is to occur in nature.

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• 5. Sundman's Series

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  In 1912 the Finnish mathematician Karl Fritiof Sundman proved that the three-body problem does have an analytic solution, written as a Puiseux series, a kind of power series. The series converges for every real time, except when the starting conditions give zero angular momentum. That exception matters little in practice, because such starting conditions are rare. They have Lebesgue measure zero.

  The proof had to confront singularities, since the radius of convergence of a series is set by the distance to the nearest one. In the three-body problem, the only singularities are binary collisions, where two particles meet at an instant, and triple collisions, where all three do. Collisions of any number are improbable, because they correspond to a set of starting conditions of measure zero. Still, no criterion is known that would let someone exclude them from a given starting state.

  Sundman's strategy had four steps. First, he changed variables so the solution could be followed past a binary collision, a process called regularization. Second, he proved that triple collisions occur only when the angular momentum vanishes, so restricting the starting data removed every real singularity. Third, he showed that when angular momentum is not zero, the system stays strictly away from a triple collision. By Cauchy's existence theorem for differential equations, that means no complex singularities in a strip around the real axis. Fourth, he found a conformal transformation that maps this strip into the unit disc.

  The resulting series converges extremely slowly. In 1930 David Beloriszky calculated that using it for astronomical observations would require at least an astronomically large number of terms.

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• 6. Special Orbits

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  In 1767 Leonhard Euler found three families of periodic solutions in which the three masses sit on a straight line at every instant. In 1772 Joseph-Louis Lagrange found a family in which the masses form an equilateral triangle at every instant. Together, these are the central configurations of the problem. They hold for any ratio of masses, with the bodies moving on Keplerian ellipses, and they are the only four families with explicit analytic formulas.

  These exceptions stand out because three-body motion is generally non-repeating. In the circular restricted problem, viewed in a frame rotating with the two large bodies, the four families become the Lagrangian points L1 to L5. The points L4 and L5 are symmetric versions of Lagrange's triangle.

  Between 1892 and 1899, Henri Poincaré established that the restricted problem has an infinite number of periodic solutions. He also supplied techniques for continuing them into the general three-body problem.

  In 1893 Meissel posed what is now called the Pythagorean three-body problem. Three masses in the ratio 3:4:5 start at rest at the corners of a 3:4:5 right triangle. The heaviest sits at the right angle and the lightest at the smaller acute angle. Burrau studied it further in 1913. In 1967 Victor Szebehely and C. Frederick Peters used numerical integration to show that the lightest body eventually escapes, and they found a nearby periodic solution at the same time.

  In the 1970s, Michel Hénon and Roger A. Broucke each found solutions belonging to one family, now called the Broucke-Hénon-Hadjidemetriou family. In it the three objects have equal mass and can move in retrograde and direct forms. In some of Broucke's solutions, two of the bodies follow the same path.

  In 1993 the physicist Cris Moore at the Santa Fe Institute found a solution with zero angular momentum in which three equal masses chase one another around a figure-eight. In 2000 the mathematicians Alain Chenciner and Richard Montgomery proved that it formally exists. The orbit has been shown numerically to be stable under small changes to the masses and orbital parameters, which means it could in principle be seen in the physical universe. But the domain of stability is small, and the chance that a binary-binary scattering event produces a figure-8 orbit has been estimated at a small fraction of a percent.

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• 7. A Growing Catalogue

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  In 2013 the 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 problem. In 2015 the physicist Ana Hudomal discovered 14 more families for the same setup.

  In 2017 Xiaoming Li and Shijun Liao found 669 new periodic orbits of the equal-mass, zero-angular-momentum problem. Their method was a new strategy for simulating chaotic systems, called clean numerical simulation, run on a national supercomputer. In 2018 they followed with 1,223 new solutions for a zero-angular-momentum system in which the masses are unequal.

  Later in 2018, Li and Liao reported 234 solutions to the unequal-mass free-fall problem. In that formulation all three bodies start at rest. As a result, the masses never orbit in a closed loop. They travel forward and backward along an open track instead.

  In 2023 Ivan Hristov, Radoslava Hristova, Dmitrašinović and Kiyotaka Tanikawa published a search for periodic free-fall orbits, limited to the equal-mass case. They found 12,409 distinct solutions. In September 2023, according to reports, several possible solutions to the problem were found.

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• 8. Beyond Gravity

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  The helium atom is the quantum-mechanical cousin of the gravitational problem. It consists of a helium nucleus and two electrons that interact through the inverse-square Coulomb force. Like the gravitational version, it cannot be solved exactly.

  Not every force law is so stubborn. In both classical and quantum mechanics, some interaction laws other than the inverse-square force allow exact three-body solutions. One model combines a harmonic attraction with a repulsive inverse-cube force. It is nontrivial because its nonlinear differential equations contain singularities, whereas harmonic interactions alone give easily solved linear equations. Because it resembles the insoluble Coulomb models in these two respects, it has been suggested as a tool for building intuition about the helium atom.

  The point vortex model shows how a change in the rules changes everything. Vortices in a two-dimensional ideal fluid obey equations with only first-order time derivatives, so their relative positions fix their velocity rather than their acceleration. The three-vortex problem remains integrable, and at least four vortices are needed to produce chaos. A passive tracer particle in the flow of three vortices parallels the restricted three-body problem.

  General relativity brings the opposite difficulty. A relativistic treatment becomes necessary in systems with very strong gravitational fields, such as near the event horizon of a black hole. It is considerably harder than the Newtonian version and demands sophisticated numerical techniques. Even the full two-body problem, for an arbitrary ratio of masses, has no rigorous analytic solution in general relativity.

  In the 1970s, V. Efimov discovered that two-body forces have implications for three-body behavior, now called the Efimov effect. At the beginning of the 21st century, P. F. Bedaque, H.-W. Hammer and U. van Kolck built on work with short-range attractive two-body forces. They found a way to renormalize the short-range three-body problem, giving scientists a rare example of a renormalization group limit cycle.

  The three-body problem is a special case of the n-body problem, which describes how n objects move under a force such as gravity. Qiudong Wang proved a convergent power series for the general case, following Sundman's result for three bodies. Wang's series is also too slow to be of practical use, so solutions are approximated by numerical integration or, in some cases, classical trigonometric series. Atoms, ions and molecules can be treated as quantum n-body systems. Among classical systems, the term usually refers to a galaxy or a cluster of galaxies, though stars, planets and their satellites can also be treated this way.

  Some applications are handled with perturbation theory, which treats the system as a two-body problem plus extra forces. Those forces cause deviations from a hypothetical unperturbed two-body trajectory, the same simple pairing Newton had solved in the 17th century.

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