Three-body problem

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https://en.wikipedia.org/wiki/Three-body_problem

Approximate trajectories of three identical bodies located at the vertices of a scalene triangle and having zero initial velocities. It is seen that the center of mass, in accordance with the law of conservation of momentum, remains in place.

In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation. The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists, as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required. The Lagrangian and dynamic equations of the three-body problem were outlined in relative coordinates and relative accelerations, which allows to solve the dynamical equations of the Sun-Earth-Moon system in an analytic way.

Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth, 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 ${\displaystyle \mathbf {r_{i}} =(x_{i},y_{i},z_{i})}$ of three gravitationally interacting bodies with masses ${\displaystyle m_{i}}$:

${\displaystyle {\begin{aligned}{\ddot {\mathbf {r} }}_{\mathbf {1} }&=-Gm_{2}{\frac {\mathbf {r_{1}} -\mathbf {r_{2}} }{|\mathbf {r_{1}} -\mathbf {r_{2}} |^{3}}}-Gm_{3}{\frac {\mathbf {r_{1}} -\mathbf {r_{3}} }{|\mathbf {r_{1}} -\mathbf {r_{3}} |^{3}}},\\{\ddot {\mathbf {r} }}_{\mathbf {2} }&=-Gm_{3}{\frac {\mathbf {r_{2}} -\mathbf {r_{3}} }{|\mathbf {r_{2}} -\mathbf {r_{3}} |^{3}}}-Gm_{1}{\frac {\mathbf {r_{2}} -\mathbf {r_{1}} }{|\mathbf {r_{2}} -\mathbf {r_{1}} |^{3}}},\\{\ddot {\mathbf {r} }}_{\mathbf {3} }&=-Gm_{1}{\frac {\mathbf {r_{3}} -\mathbf {r_{1}} }{|\mathbf {r_{3}} -\mathbf {r_{1}} |^{3}}}-Gm_{2}{\frac {\mathbf {r_{3}} -\mathbf {r_{2}} }{|\mathbf {r_{3}} -\mathbf {r_{2}} |^{3}}}.\end{aligned}}}$

where ${\displaystyle G}$ is the gravitational constant. This is a set of 9 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 ${\displaystyle \mathbf {r_{i}} }$ and momenta ${\displaystyle \mathbf {p_{i}} }$:

${\displaystyle {\frac {d\mathbf {r_{i}} }{dt}}={\frac {\partial {\mathcal {H}}}{\partial \mathbf {p_{i}} }},\qquad {\frac {d\mathbf {p_{i}} }{dt}}=-{\frac {\partial {\mathcal {H}}}{\partial \mathbf {r_{i}} }},}$

where ${\displaystyle {\mathcal {H}}}$ is the Hamiltonian:

${\displaystyle {\mathcal {H}}=-{\frac {Gm_{1}m_{2}}{|\mathbf {r_{1}} -\mathbf {r_{2}} |}}-{\frac {Gm_{2}m_{3}}{|\mathbf {r_{3}} -\mathbf {r_{2}} |}}-{\frac {Gm_{3}m_{1}}{|\mathbf {r_{3}} -\mathbf {r_{1}} |}}+{\frac {\mathbf {p_{1}} ^{2}}{2m_{1}}}+{\frac {\mathbf {p_{2}} ^{2}}{2m_{2}}}+{\frac {\mathbf {p_{3}} ^{2}}{2m_{3}}}.}$

In this case ${\displaystyle {\mathcal {H}}}$ is simply the total energy of the system, gravitational plus kinetic.

Dynamic equations in relative coordinates

If the following relative coordinates are defined

${\displaystyle {\boldsymbol {\rho _{1}}}=\mathbf {r_{3}} -\mathbf {r_{2}} ,\qquad {\boldsymbol {\rho _{2}}}=\mathbf {r_{1}} -\mathbf {r_{3}} ,\qquad {\boldsymbol {\rho _{3}}}=\mathbf {r_{2}} -\mathbf {r_{1}} }$

then the equations of the motion are written in a remarkably simple and elegant form 

${\displaystyle {\ddot {\rho _{1}}}=-G\mu {\frac {\boldsymbol {\rho _{1}}}{\rho _{1}^{3}}}+Gm_{1}{\Big (}{\frac {\boldsymbol {\rho _{1}}}{\rho _{1}^{3}}}+{\frac {\boldsymbol {\rho _{2}}}{\rho _{2}^{3}}}+{\frac {\boldsymbol {\rho _{3}}}{\rho _{3}^{3}}}{\Big )}}$

${\displaystyle {\ddot {\rho _{2}}}=-G\mu {\frac {\boldsymbol {\rho _{2}}}{\rho _{2}^{3}}}+Gm_{2}{\Big (}{\frac {\boldsymbol {\rho _{1}}}{\rho _{1}^{3}}}+{\frac {\boldsymbol {\rho _{2}}}{\rho _{2}^{3}}}+{\frac {\boldsymbol {\rho _{3}}}{\rho _{3}^{3}}}{\Big )}}$

${\displaystyle {\ddot {\rho _{3}}}=-G\mu {\frac {\boldsymbol {\rho _{3}}}{\rho _{1}^{3}}}+Gm_{3}{\Big (}{\frac {\boldsymbol {\rho _{1}}}{\rho _{1}^{3}}}+{\frac {\boldsymbol {\rho _{2}}}{\rho _{2}^{3}}}+{\frac {\boldsymbol {\rho _{3}}}{\rho _{3}^{3}}}{\Big )}}$

where ${\displaystyle \mu =m_{1}+m_{2}+m_{3}}$. It was proven that every linear combination of squares of the absolute coordinates of three bodies is equal to a linear combination of the squares of their relative coordinates and of the square of the center-of-mass coordinate, which is a generalization of a geometrical theorem of Leibniz 

${\displaystyle m_{1}\mathbf {r_{1}} ^{2}+m_{2}\mathbf {r_{2}} ^{2}+m_{3}\mathbf {r_{3}} ^{2}={\frac {m_{2}m_{3}}{\mu }}{\boldsymbol {\rho _{1}}}^{2}+{\frac {m_{1}m_{3}}{\mu }}{\boldsymbol {\rho _{2}}}^{2}+{\frac {m_{1}m_{2}}{\mu }}{\boldsymbol {\rho _{3}}}^{2}+\mu {\Big (}{\frac {m_{1}\mathbf {r_{1}} +m_{2}\mathbf {r_{2}} +m_{3}\mathbf {r_{3}} }{\mu }}{\Big )}^{2}}$

Since this equality holds for coordinates as well as for velocities, the Lagrangian of the three-body problem in the center-of-mass frame is 

${\displaystyle {\mathcal {L}}={\frac {m_{2}m_{3}}{2\mu }}{\boldsymbol {{\dot {\rho }}_{1}}}^{2}+{\frac {m_{1}m_{3}}{2\mu }}{\boldsymbol {{\dot {\rho }}_{2}}}^{2}+{\frac {m_{1}m_{2}}{2\mu }}{\boldsymbol {{\dot {\rho }}_{3}}}^{2}+{\frac {Gm_{2}m_{3}}{\rho _{1}}}+{\frac {Gm_{1}m_{3}}{\rho _{2}}}+{\frac {Gm_{1}m_{2}}{\rho _{1}}}+{\boldsymbol {\lambda }}\cdot ({\boldsymbol {\rho _{1}}}+{\boldsymbol {\rho _{2}}}+{\boldsymbol {\rho _{3}}})}$

The relative coordinates are not independent, because ${\displaystyle {\boldsymbol {\rho _{1}}}+{\boldsymbol {\rho _{2}}}+{\boldsymbol {\rho _{3}}}=0}$. This holonomic constraint is introduced into the Lagrangian through the scalar product by the Lagrange multiplier vector ${\displaystyle {\boldsymbol {\lambda }}}$. On applying the Lagrange equations of the motion, the relative accelerations are obtained

${\displaystyle {\boldsymbol {{\ddot {\rho }}_{1}}}=-G\mu {\frac {\boldsymbol {\rho _{1}}}{\rho _{1}^{3}}}+{\frac {\mu }{m_{2}m_{3}}}{\boldsymbol {\lambda }},\qquad {\boldsymbol {{\ddot {\rho }}_{2}}}=-G\mu {\frac {\boldsymbol {\rho _{2}}}{\rho _{2}^{3}}}+{\frac {\mu }{m_{1}m_{3}}}{\boldsymbol {\lambda }},\qquad {\boldsymbol {{\ddot {\rho }}_{3}}}=-G\mu {\frac {\boldsymbol {\rho _{3}}}{\rho _{3}^{3}}}+{\frac {\mu }{m_{1}m_{2}}}{\boldsymbol {\lambda }}}$

The addition of the three equations allows to know ${\displaystyle {\boldsymbol {\lambda }}}$ because ${\displaystyle {\boldsymbol {{\ddot {\rho }}_{1}}}+{\boldsymbol {{\ddot {\rho }}_{2}}}+{\boldsymbol {{\ddot {\rho }}_{3}}}=0}$

${\displaystyle {\boldsymbol {\lambda }}={\frac {Gm_{1}m_{2}m_{3}}{\mu }}{\Big (}{\frac {\boldsymbol {\rho _{1}}}{\rho _{1}^{3}}}+{\frac {\boldsymbol {\rho _{2}}}{\rho _{2}^{3}}}+{\frac {\boldsymbol {\rho _{3}}}{\rho _{3}^{3}}}{\Big )}}$

and the initial expressions for the relative accelerations are retrieved.

Restricted three-body problem

The circular restricted three-body problem is a valid approximation of elliptical orbits found in the Solar System, and this can be visualized as a combination of the potentials due to the gravity of the two primary bodies along with the centrifugal effect from their rotation (Coriolis effects are dynamic and not shown). The Lagrange points can then be seen as the five places where the gradient on the resultant surface is zero (shown as blue lines), indicating that the forces are in balance there.

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 force that the planetoid exerts on the two massive bodies may be neglected, and the system can be analysed and can therefore be described in terms of a two-body motion. 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.

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.

Mathematically, the problem is stated as follows. Let ${\displaystyle m_{1,2}}$ be the masses of the two massive bodies, with (planar) coordinates ${\displaystyle (x_{1},y_{1})}$ and ${\displaystyle (x_{2},y_{2})}$, and let ${\displaystyle (x,y)}$ be the coordinates of the planetoid. For simplicity, choose units such that the distance between the two massive bodies, as well as the gravitational constant, are both equal to ${\displaystyle 1}$. Then, the motion of the planetoid is given by

${\displaystyle {\begin{aligned}{\frac {d^{2}x}{dt^{2}}}=-m_{1}{\frac {x-x_{1}}{r_{1}^{3}}}-m_{2}{\frac {x-x_{2}}{r_{2}^{3}}}\\{\frac {d^{2}y}{dt^{2}}}=-m_{1}{\frac {y-y_{1}}{r_{1}^{3}}}-m_{2}{\frac {y-y_{2}}{r_{2}^{3}}},\end{aligned}}}$

where ${\displaystyle r_{i}={\sqrt {(x-x_{i})^{2}+(y-y_{i})^{2}}}}$. In this form the equations of motion carry an explicit time dependence through the coordinates ${\displaystyle x_{i}(t),y_{i}(t)}$. However, this time-dependence can be removed through a transformation to a rotating reference frame, which simplifies any subsequent analysis.

Solutions

General solution

There is no general closed-form solution to the three-body problem, meaning there is no 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 power series in terms of powers of t^1/3. 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 will be 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, whether binary or triple (in fact, any number), are somewhat improbable, since it has been shown that they correspond to a set of initial conditions of measure zero. However, 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:

1. Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as regularization.
2. Proving that triple collisions only occur when the angular momentum L vanishes. By restricting the initial data to L ≠ 0, he removed all real singularities from the transformed equations for the three-body problem.
3. Showing that if L ≠ 0, then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by using Cauchy's existence theorem for differential equations, that there are no complex singularities in a strip (depending on the value of L) in the complex plane centered around the real axis (shades of Kovalevskaya).
4. Find a conformal transformation that maps this strip into the unit disc. For example, if s = t^1/3 (the new variable after the regularization) and if |ln s| ≤ β, then this map is given by

${\displaystyle \sigma ={\frac {e^{\frac {\pi s}{2\beta }}-1}{e^{\frac {\pi s}{2\beta }}+1}}.}$

This finishes the proof of Sundman's theorem.

Unfortunately, the corresponding series converges very 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 10^8000000 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 formulae. In the special case of the circular restricted three-body problem, these solutions, viewed in a frame rotating with the primaries, become points which are referred to as L₁, L₂, L₃, L₄, and L₅, and called Lagrangian points, 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. Burrau further investigated this problem in 1913. In 1967 Victor Szebehely and C. Frederick Peters established eventual escape 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–Henon–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.

An animation of the figure-8 solution to the three-body problem over a single period T ≃ 6.3259.

In 1993, a zero angular momentum solution with three equal masses moving around a figure-eight shape was discovered numerically by physicist Cris Moore at the Santa Fe Institute. Its formal existence was later proved in 2000 by mathematicians Alain Chenciner and Richard Montgomery. The solution has been shown numerically to be stable for small perturbations of the mass and orbital parameters, which raises the intriguing possibility that such orbits could be observed in the physical universe. However, it has been argued that this occurrence 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 1%.

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 1223 new solutions for a zero-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 of the three body problem 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 forwards and backwards along an open "track".

Numerical approaches

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.

History

The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Principia (Philosophiæ Naturalis Principia Mathematica). 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 the Earth and the Sun.

The physical problem was addressed by Amerigo Vespucci and subsequently by Galileo Galilei; in 1499, Vespucci used knowledge of the position of the Moon to determine his position in Brazil. 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 the 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. 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. An account published in 1761 by Jean le Rond d'Alembert indicates that the name was first used in 1747.

In 2019, Breen et al. announced a fast neural network solver, trained using a numerical integrator.

Other problems involving three bodies

The term 'three-body problem' is sometimes used in the more general sense to refer to any physical problem involving the interaction of three bodies.

A quantum mechanical analogue of the gravitational three-body problem in classical mechanics is the helium atom, in which a helium nucleus and two electrons interact according to the inverse-square Coulomb interaction. Like the gravitational three-body problem, the helium atom cannot be solved exactly.

In both classical and quantum mechanics, however, there exist nontrivial interaction laws besides the inverse-square force which do lead to exact analytic three-body solutions. One such model consists of a combination of harmonic attraction and a repulsive inverse-cube force. This model is considered nontrivial since it is associated with a set of nonlinear differential equations containing singularities (compared with, e.g., harmonic interactions alone, which lead to an easily solved system of linear differential equations). 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.

The gravitational three-body problem has also been studied using general relativity. 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.

n-body problem

The three-body problem is a special case of the n-body problem, which describes how n objects will move under one of the physical forces, such as gravity. These problems have a global analytical solution in the form of a convergent power series, as was proven by Karl F. Sundman for n = 3 and by Qiudong Wang for n > 3 (see n-body problem for details). However, the Sundman and Wang series converge so slowly that they are useless for practical purposes; therefore, it is currently necessary to approximate solutions by numerical analysis in the form of numerical integration or, for some cases, classical trigonometric series approximations. Atomic systems, e.g. atoms, ions, and molecules, can be treated in terms of the quantum n-body problem. Among classical physical systems, the n-body problem usually refers to a galaxy or to a cluster of galaxies; planetary systems, such as stars, planets, and their satellites, can also be treated as n-body systems. Some applications are conveniently treated by perturbation theory, in which the system is considered as a two-body problem plus additional forces causing deviations from a hypothetical unperturbed two-body trajectory.