Editing Two-body problem (section)

===Displacement vector motion (2nd one-body problem)===
Dividing both force equations by the respective masses, subtracting the second equation from the first, and rearranging gives the equation
<math display="block">
\ddot {\mathbf{r}} = \ddot{\mathbf{x}}_{1} - \ddot{\mathbf{x}}_{2} = 
\left( \frac{\mathbf{F}_{12}}{m_{1}} - \frac{\mathbf{F}_{21}}{m_{2}} \right) =
\left(\frac{1}{m_{1}} + \frac{1}{m_{2}} \right)\mathbf{F}_{12}
</math>
where we have again used [[Newton's third law]] {{math|1='''F'''<sub>12</sub> = −'''F'''<sub>21</sub>}} and where {{math|'''r'''}} is the [[Displacement (vector)|displacement vector]] from mass 2 to mass 1, as defined above.

The force between the two objects, which originates in the two objects, should only be a function of their separation {{math|'''r'''}} and not of their absolute positions {{math|'''x'''<sub>1</sub>}} and {{math|'''x'''<sub>2</sub>}}; otherwise, there would not be [[translational symmetry]], and the laws of physics would have to change from place to place. The subtracted equation can therefore be written:
<math display="block">\mu \ddot{\mathbf{r}} = \mathbf{F}_{12}(\mathbf{x}_{1},\mathbf{x}_{2}) = \mathbf{F}(\mathbf{r})</math>
where <math>\mu</math> is the '''[[reduced mass]]'''
<math display="block">\mu = \frac{1}{\frac{1}{m_1} + \frac{1}{m_2}} = \frac{m_1 m_2}{m_1 + m_2}.</math>

Solving the equation for {{math|'''r'''(''t'')}} is the key to the two-body problem. The solution depends on the specific force between the bodies, which is defined by <math>\mathbf{F}(\mathbf{r})</math>. For the case where <math>\mathbf{F}(\mathbf{r})</math> follows an [[inverse-square law]], see the [[Kepler problem]].

Once {{math|'''R'''(''t'')}} and {{math|'''r'''(''t'')}} have been determined, the original trajectories may be obtained
<math display="block">\mathbf{x}_1(t) = \mathbf{R} (t) + \frac{m_2}{m_1 + m_2} \mathbf{r}(t)</math>
<math display="block">\mathbf{x}_2(t) = \mathbf{R} (t) - \frac{m_1}{m_1 + m_2} \mathbf{r}(t)</math>
as may be verified by substituting the definitions of '''R'''  and '''r''' into the right-hand sides of these two equations.