Three-body problem

In physics, specifically classical mechanics, the three-body problem is to take the initial positions and velocities of three point masses orbiting each other in space and then to calculate their subsequent trajectories using Newton's laws of motion and Newton's law of universal gravitation.
Unlike the two-body problem, the three-body problem has no general closed-form solution, meaning there is no explicit formula for the positions of the bodies. When three bodies orbit each other, the resulting dynamical system is usually chaotic. For most initial conditions, the only way to predict the motions of the three bodies is to estimate them using numerical methods.
The three-body problem is a special case of the -body problem. Historically, the first specific three-body problem to receive extended study was the one involving the Earth, the Moon, and the Sun. 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.

Mathematical description

The mathematical statement of the three-body problem can be given in terms of the Newtonian equations of motion for vector positions of three gravitationally interacting bodies with masses :
where is the gravitational constant.
As astronomer Juhan Frank describes, "These three second-order vector differential equations are equivalent to 18 first order scalar differential equations." As June Barrow-Green notes with regard to an alternative presentation, if represent three particles with masses, distances and coordinates in an inertial coordinate system... the problem is described by nine second-order differential equations.
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 and momenta :
where is the Hamiltonian:
In this case, is simply the total energy of the system, gravitational plus kinetic.

Restricted three-body problem

In the restricted three-body problem formulation, in the description of Barrow-Green,

two... bodies revolve around their centre of mass in circular orbits under the influence of their mutual gravitational attraction, and... form a two body system... motion is known. A third body, assumed massless with respect to the other two, moves in the plane defined by the two revolving bodies and, while being gravitationally influenced by them, exerts no influence of its own.

Per Barrow-Green, "he problem is then to ascertain the motion of the third body."
The restricted three-body problem is easier to analyze theoretically than the full problem. It is of practical interest as well since it accurately describes many real-world problems, the most important example being the Earth–Moon–Sun system. For these reasons, it has occupied an important role in the historical development of the three-body problem.
The restricted 3-body problem has a 4-dimensional phase space, but only one conserved quantity, the Jacobi integral. It was shown by Heinrich Bruns that there are no more algebraic conserved quantities, and by Henri Poincaré in 1889 that there are no more analytic conserved quantities. Therefore, since the dimension of the phase space is larger than the number of constants of motion, the system is not exactly solvable; in fact, it is chaotic.
Depending on the value of the Jacobi integral, a body initially orbiting the larger mass may be able to be captured by the secondary mass or be ejected via Lagrange points L2 or L3.
A variant of this problem, where the two large bodies both exert radiation pressure, results in the addition of four additional equilibrium points in addition to the five classical Lagrange points.

Solutions

General solution

There is no general closed-form solution to the three-body problem. 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.
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. This series converges for all real, 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 and triple collisions.
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:

Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as regularization.
Proving that triple collisions only occur when the angular momentum vanishes. By restricting the initial data to, he removed all real singularities from the transformed equations for the three-body problem.
Showing that if, then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by Cauchy's existence theorem for differential equations, that there are no complex singularities in a strip in the complex plane centered around the real axis.
Find a conformal transformation that maps this strip into the unit disc. For example, if and if, then this map is given by

This finishes the proof of Sundman's theorem.
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 10 terms.

Special-case solutions

In 1767, Leonhard Euler found three families of periodic solutions in which the three masses are collinear at each instant. 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 formulas. 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 work summarized in 1892–1899, Henri Poincaré established the existence of an infinite number of periodic solutions to the restricted three-body problem, together with techniques for continuing these solutions into the general three-body problem.
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 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.
In the 1970s, Michel Hénon and Roger A. Broucke each found a set of solutions that form part of the same family of solutions: the Broucke–Hénon–Hadjidemetriou family. In this family, the three objects all have the same mass and can exhibit both retrograde and direct forms. In some of Broucke's solutions, two of the bodies follow the same path.
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. In 2000, mathematicians Alain Chenciner and Richard Montgomery proved its formal existence. The solution has been shown numerically to be stable for small perturbations of the mass and orbital parameters, which makes it possible for such orbits to be observed in the physical universe. But it has been argued that this is unlikely since the domain of stability is small. For instance, the probability of a binary–binary scattering event resulting in a figure-8 orbit has been estimated to be a small fraction of a percent.
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.
In 2015, physicist Ana Hudomal discovered 14 new families of solutions for the equal-mass zero-angular-momentum three-body problem.
In 2017, researchers Xiaoming Li and Shijun Liao found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem. This was followed in 2018 by an additional 1,223 new solutions for a zero-angular-momentum system of unequal masses.
In 2018, Li and Liao reported 234 solutions to the unequal-mass "free-fall" three-body problem. 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.