The Sitnikov problem is a restricted version of the three-body problem named after Russian mathematician Kirill Alexandrovitch Sitnikov that attempts to describe the movement of three celestial bodies due to their mutual gravitational attraction. A special case of the Sitnikov problem was first discovered by the American scientist William Duncan MacMillan in 1911, but the problem as it currently stands wasn't discovered until 1961 by Sitnikov.
\left(m1=m2=\tfrac{m}{2}\right)
(m3=0)
m=1
2\pi
a=1
E
| E= | 1 | \left( |
| 2 |
| dz | |
| dt |
\right)2-
| 1 | |
| r |
After differentiating with respect to time, the equation becomes:
| d2z | =- | |
| dt2 |
| z | |
| r3 |
This, according to Figure 1, is also true:
r2=a2+z2=1+z2
Thus, the equation of motion is as follows:
| d2z | |
| dt2 |
=-
| z | |
| \left(\sqrt{1+z2 |
\right)3}
which describes an integrable system since it has one degree of freedom.
If on the other hand the primary bodies move in elliptical orbits then the equations of motion are
| d2z | |
| dt2 |
=-
| z | |
| \left(\sqrt{\rho(t)2+z2 |
\right)3}
where
\rho(t)=\rho(t+2\pi)
Although it is nearly impossible in the real world to find or arrange three celestial bodies exactly as in the Sitnikov problem, the problem is still widely and intensively studied for decades: although it is a simple case of the more general three-body problem, all the characteristics of a chaotic system can nevertheless be found within the problem, making the Sitnikov problem ideal for general studies on effects in chaotic dynamical systems.