![Sitnikov Problem Konfiguration](https://upload.wikimedia.org/wikipedia/commons/3/36/Sitnikov_Problem_Konfiguration.jpg)
The Sitnikov problem is a restricted version of the
three-body problem
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 ...
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
William Duncan MacMillan (July 24, 1871 – November 14, 1948) was an American mathematician and astronomer on the faculty of the University of Chicago. He published research on the applications of classical mechanics to astronomy, and is noted fo ...
in 1911, but the problem as it currently stands wasn't discovered until 1961 by Sitnikov.
Definition
The system consists of two primary bodies with the same
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
, which move in circular or elliptical Kepler orbits around their
center of mass. The third body, which is substantially smaller than the primary bodies and whose mass can be set to zero
, moves under the influence of the primary bodies in a plane that is perpendicular to the orbital plane of the primary bodies (see Figure 1). The origin of the system is at the focus of the primary bodies. A combined mass of the primary bodies
, an orbital period of the bodies
, and a radius of the orbit of the bodies
are used for this system. In addition, the
gravitational constant is 1. In such a system that the third body only moves in one dimension – it moves only along the z-axis.
Equation of motion
In order to derive the
equation of motion
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
in the case of circular orbits for the primary bodies, use that the total
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
is:
:
After
differentiating with respect to time, the equation becomes:
:
This, according to Figure 1, is also true:
:
Thus, the equation of motion is as follows:
:
which describes an
integrable system
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
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
:
where
is the distance of either primary from their common center of mass. Now the system has one-and-a-half degrees of freedom and is known to be chaotic.
Significance
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.
See also
*
Celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
*
Chaos theory
*
Two-body problem
In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting each ...
Literature
* K. A. Sitnikov: '' The existence of oscillatory motions in the three-body problems''. In: ''Doklady Akademii Nauk SSSR'', 133/1960, pp. 303–306, (English Translation in ''Soviet Physics. Doklady.'', 5/1960, S. 647–650)
* K. Wodnar: ''The original Sitnikov article – new insights''. In: ''Celestial Mechanics and Dynamical Astronomy'', 56/1993, pp. 99–101,
pdf* D. Hevia, F. Rañada: ''Chaos in the three-body problem: the Sitnikov case''. In: ''European Journal of Physics'', 17/1996, pp. 295–302,
pdf* Rudolf Dvorak, Florian Freistetter, J. Kurths, ''Chaos and Stability in Planetary Systems.'', Springer, 2005,
* J. Moser: "Stable and Random Motion", Princeton Univ. Press, 1973,
References
{{Reflist
External links
Sitnikov problem – Scholarpedia
Orbits
Classical mechanics