Three-body problem

The problem on the motion of three bodies, regarded as material points, mutually attracting one another according to Newton's law of gravitation (cf. Newton laws of mechanics). The classical example of the three-body problem is that of the motion of the Sun–Earth–Moon system. The three-body problem consists in finding the general solution of the system of differential equations

$$ m _ {i} \frac{d ^ {2} x _ {i} }{\partial t ^ {2} } = \ \frac{\partial U }{\partial x _ {i} } ,\ \ m _ {i} \frac{d ^ {2} y _ {i} }{\partial t ^ {2} } = \ \frac{\partial U }{\partial y _ {i} } ,\ \ m _ {i} \frac{d ^ {2} z }{dt ^ {2} } = \ \frac{\partial U }{\partial z _ {i} } , $$

$$ i = 1, 2, 3, $$

where $ x _ {i} , y _ {i} , z _ {i} $ are the rectangular coordinates of the body $ M _ {i} $ in some absolute coordinate frame with fixed axes, $ t $ is the time, $ m _ {i} $ is the mass of $ M _ {i} $, and $ U $ is the potential, which depends only on the mutual distances between the points. The function $ U $ is defined by the relation

$$ U = f \left ( \frac{m _ {1} m _ {2} }{\Delta _ {12} } + \frac{m _ {2} m _ {3} }{\Delta _ {23} } + \frac{m _ {3} m _ {1} }{\Delta _ {13} } \right ) ,\ f > 0 , $$

where the mutual distances $ \Delta _ {ij} $, $ i, j = 1, 2, 3 $, are given by the formula

$$ \Delta _ {ij} = \ \Delta _ {ji} = \ \sqrt {( x _ {i} - x _ {j} ) ^ {2} + ( y _ {i} - y _ {j} ) ^ {2} + ( z _ {i} - z _ {j} ) ^ {2} } . $$

From the properties of the potential one can derive ten first integrals of the equations of motion in the absolute system of coordinates. Six of them, called the integrals of motion of the centre of mass, determine the uniform rectangular motion of the centre of mass of the three bodies. The three integrals of the angular momentum fix the value and the direction of the angular momentum of the three-body system. The energy integral defines the constant magnitude of total energy of the system. H. Bruns (1887) proved that the equations of motion of the three-body problem have no other first integrals expressible in terms of algebraic functions of the coordinates and their derivatives. H. Poincaré (1889) further proved that the equations of motion of the three-body problem do not have transcendental integrals expressible in terms of single-valued analytic functions. C. Sundman (1912) found the general solution of the problem in the form of power series in a certain regularizing variable, converging at each instant. However, the Sundman series proved to be completely useless for qualitative investigations as well as for practical computations due to its extremely slow convergence.

The equations of the three-body problem admit five particular solutions, in which all three material points are in some fixed plane. Here, the configuration of the three bodies remains fixed and they describe Kepler trajectories with a common focus at the centre of mass of the system. Two of the particular solutions correspond to the case when the three bodies form an equilateral triangle at all times. This is the so-called triangular solution of the three-body problem, or the Lagrange solution. The three particular solutions corresponding to three bodies on one straight line are called the rectilinear particular solutions, or the Euler solutions.

For the general solution of the three-body problem, final motions have been studied in detail, that is, the limiting properties of the motion as $ t \rightarrow + \infty $ and $ t \rightarrow - \infty $.

A particular case of the three-body problem is the so-called restricted three-body problem, which is obtained from the general three-body problem in case the mass of one of the three bodies is so small that its influence on the motion of the other two bodies can be neglected. In this case, the bodies $ M _ {1} $ and $ M _ {2} $ with finite masses $ m _ {1} $ and $ m _ {2} $ move under the action of their mutual attraction along Kepler orbits. In the right-handed rectangular coordinate system $ G \xi \eta \zeta $ with origin $ G $ at the centre of mass of $ M _ {1} $ and $ M _ {2} $, with axis $ \xi $ directed along the line joining $ M _ {1} $ and $ M _ {2} $ and axis $ \zeta $ perpendicular to the plane of their motion, the motion of the third body $ M _ {3} $ of small mass is described by the following differential equations:

$$ \dot \xi dot - 2 \dot \nu \dot \eta - \dot \nu ^ {2} \xi - \dot \nu dot \eta = \ \frac{\partial W }{\partial \xi } , $$

$$ \dot \eta dot + 2 \dot \nu \dot \xi - \dot \nu ^ {2} \eta + \dot \nu dot \xi = \frac{\partial W }{\partial \eta } , $$

$$ \dot \zeta dot = \frac{\partial W }{\partial \zeta } , $$

where

$$ W = f \left ( \frac{m _ {1} }{r _ {1} } + \frac{m _ {2} }{r _ {2} } \right ) , $$

$ \nu $ is the true anomaly of the Kepler motion of $ M _ {1} $ and $ M _ {2} $, and $ r _ {1} $ and $ r _ {2} $ are the distances of $ M _ {3} $ from $ M _ {1} $ and $ M _ {2} $, respectively. In the case of the circular restricted three-body problem,

$$ \dot \nu = n = \textrm{ const } ,\ \ \dot \nu dot = 0, $$

the equations of motion of $ M _ {3} $ have also a first integral, called the Jacobi integral, of the form

$$ \dot \xi ^ {2} + \dot \eta ^ {2} + \dot \xi ^ {2} = \ n ^ {2} ( \xi ^ {2} + \eta ^ {2} ) + 2f \left ( \frac{m _ {1} }{r _ {1} } + \frac{m _ {2} }{r _ {2} } \right ) + C. $$

where $ C $ is an arbitrary constant. The surface defined by the equation

$$ n ^ {2} ( \xi ^ {2} + \eta ^ {2} ) + 2f \left ( \frac{m _ {1} }{r _ {1} } + \frac{m _ {2} }{r _ {2} } \right ) + C = 0 $$

is called the surface of zero velocity and is remarkable in that it determines the regions of possible motions of $ M _ {3} $ relative to $ M _ {1} $ and $ M _ {2} $. The restricted three-body problem has particular solutions similar to those of the general three-body problem. The positions of the body with small mass in these particular solutions are called the points of libration.

For the restricted three-body problem, various classes of periodic motion have been investigated.

Comments

KAM theory, cf. Quasi-periodic motion, applies to the restricted three-body problem.

References

[1]	G.N. Duboshin, "Celestial mechanics. Analytical and qualitative methods" , Moscow (1978) (In Russian)
[2]	M.F. Subbotin, "Introduction to theoretical astronomy" , Moscow (1968) (In Russian)
[a1]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[a2]	H. Poincaré, "Les méthodes nouvelles de la mécanique céleste" , 1–3 , Gauthier-Villars (1892–1899)
[a3]	V.I. Arnol'd, A. Avez, "Problèmes ergodiques de la mécanique classique" , Gauthier-Villars (1967) (Translated from Russian)
[a4]	C.L. Siegel, "Vorlesungen über Himmelsmechanik" , Springer (1956)
[a5]	C.L. Siegel, J. Moser, "Lectures on celestial mechanics" , Springer (1971)
[a6]	R. Abraham, J.E. Marsden, "Foundations of mechanics" , Benjamin/Cummings (1978)