Three-body problem

The differential equations that govern the motions of three gravitating bodies are not integrable and cannot be solved to give explicit formulas for the positions of the bodies as a function of time. For most initial conditions, the dynamical system for three orbiting bodies is chaotic, and the only way to predict their motions is to estimate them using numerical methods. The three-body problem is a special case of the n-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. The mathematical statement of the three-body problem can be given in terms of the Newtonian equations of motion for vector positions r i = ( x i , y i , z i ) {\displaystyle \ \mathbf {r} _{i}=(x_{i},y_{i},z_{i})\ } of three gravitationally interacting bodies with masses m i {\displaystyle m_{i}} : r ¨ 1 = − G m 2 ( r 1 − r 2 ) | r 1 − r 2 | 3 − G m 3 ( r 1 − r 3 ) | r 1 − r 3 | 3 , r ¨ 2 = − G m 3 ( r 2 − r 3 ) | r 2 − r 3 | 3 − G m 1 ( r 2 − r 1 ) | r 2 − r 1 | 3 , r ¨ 3 = − G m 1 ( r 3 − r 1 ) | r 3 − r 1 | 3 − G m 2 ( r 3 − r 2 ) | r 3 − r 2 | 3 . {\displaystyle {\begin{aligned}{\ddot {\mathbf {r} }}_{1}&=-Gm_{2}{\frac {\left(\mathbf {r} _{1}-\mathbf {r} _{2}\right)}{\ \left|\mathbf {r} _{1}-\mathbf {r} _{2}\right|^{3}}}-Gm_{3}{\frac {\left(\mathbf {r} _{1}-\mathbf {r} _{3}\right)}{\ \left|\mathbf {r} _{1}-\mathbf {r} _{3}\right|^{3}}}\ ,\\{\ddot {\mathbf {r} }}_{2}&=-Gm_{3}{\frac {\left(\mathbf {r} _{2}-\mathbf {r} _{3}\right)}{\ \left|\mathbf {r} _{2}-\mathbf {r} _{3}\right|^{3}}}-Gm_{1}{\frac {\left(\mathbf {r} _{2}-\mathbf {r} _{1}\right)}{\ \left|\mathbf {r} _{2}-\mathbf {r} _{1}\right|^{3}}}\ ,\\{\ddot {\mathbf {r} }}_{3}&=-Gm_{1}{\frac {\left(\mathbf {r} _{3}-\mathbf {r} _{1}\right)}{\ \left|\mathbf {r} _{3}-\mathbf {r} _{1}\right|^{3}}}-Gm_{2}{\frac {\left(\mathbf {r} _{3}-\mathbf {r} _{2}\right)}{\ \left|\mathbf {r} _{3}-\mathbf {r} _{2}\right|^{3}}}~.\end{aligned}}} where G {\displaystyle \ G\ } 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 P i {\displaystyle P_{i}} represent three particles with masses m i {\displaystyle m_{i}} , distances P i P j = r i j , {\displaystyle \ P_{i}P_{j}=r_{ij}\ ,} and coordinates q i j {\displaystyle \ q_{ij}\ } ( i , j = 1 , 2 , 3 ) {\displaystyle \ (i,j=1,2,3)\ } in an inertial coordinate system ... the problem is described by nine second-order differential equations.