Two-body problem

A problem dealing with the motion of two material points $P _ {1}$ and $P _ {2}$ with masses $m _ {1}$ and $m _ {2}$, respectively, moving in three-dimensional Euclidean space $E ^ {3}$ when acted upon by the mutual Newton attracting forces. The problem is a special case of the $n$-body problem, which may be described by a system of ordinary differential equations of order $6n$, and has 10 independent integrals: 6 of motion of the centre of inertia, 3 of law of areas (equivalently, conservation of angular momentum) and 1 of energy conservation [1]. The two-body problem also has three Laplace integrals (one of which is independent of the preceding ones) and is completely integrable [2].

The integration of the two-body problem is more conveniently effected in a special system of coordinates, in which these integrals are employed. If the origin of the Cartesian coordinates $x , y , z$ is placed at the centre of mass $( m _ {1} \vec{r} {} _ {1} + m _ {2} \vec{r} {} _ {2} ) / ( m _ {1} + m _ {2} )$ and the axis $z$ is directed along the relative angular momentum vector, then the motion of the relative position vector $\vec{r} {} _ {1} - \vec{r} {} _ {2} = ( x, y, z)$ takes place in the plane $z = 0$ and satisfies the system

$$\tag{1 } \mu \ddot{x} = - f x r ^ {- 3} ,\ \mu \ddot{y} = - f y r ^ {- 3} ,$$

where $r = \sqrt {x ^ {2} + y ^ {2} }$, $\mu = m _ {1} m _ {2} /( m _ {1} + m _ {2} )$ is the reduced mass and $f$ is the gravitational constant. The system (1) has four integrals:

$$x \dot{y} - y \dot{x} = c \ \textrm{ (law of areas), }$$

$$\frac{1}{2} \mu ( {\dot{x} } {} ^ {2} + {\dot{y} } {} ^ {2} ) - f r ^ {- 1} = h \textrm{ (energy) } ,$$

$$\mu ^ {2} c \dot{y} - \mu f x r ^ {- 1} = \lambda _ {1} \ \textrm{ and } \ \ \mu ^ {2} c \dot{x} + \mu f y r ^ {- 1} = - \lambda _ {2} \ \textrm{ (Laplace) },$$

which are interconnected by the relation

$$\lambda _ {1} ^ {2} + \lambda _ {2} ^ {2} = \ 2 \mu ^ {3} h c ^ {2} + \mu ^ {2} f ^ { 2 } .$$

Here

$$\tag{2 } c ^ {2} = \lambda _ {1} x + \lambda _ {2} y + \mu r ,$$

i.e. the orbits of the relative position vector are conical sections with parameter $p = c ^ {2} / \mu$, major semi-axis $a = - \mu / ( 2h)$, eccentricity $e = \mu ^ {- 1} \sqrt {1 + 2hc ^ {2} }$, longitude of pericentre $\omega$ ( $\lambda _ {1} = \mu e \cos \omega$, $\lambda _ {2} = \mu e \sin \omega$), and with focus at the coordinate origin. The location of the relative positive vector on the orbit is determined by the true anomaly $v$, counted from the direction towards the pericentre; (2) then implies that $r = p / ( 1 + e \cos v )$. If $c \neq 0$, three types of orbits are possible:

I) If $h < 0$, they are ellipses.

II) If $h > 0$, they are hyperbolas.

III) If $h = 0$, they are parabolas.

If $c = 0$, the motion is rectilinear. The two-body problem describes an unperturbed Kepler motion of a planet with respect to the Sun or of a satellite with respect to a planet, etc.

References

[1]	C.L. Siegel, "Vorlesungen über Himmelmechanik" , Springer (1956)
[2]	V.K. Abalakin, et al., "Handbook of celestial mechanics and astrodynamics" , Moscow (1971) (In Russian)
[a1]	H. Poincaré, "Les méthodes nouvelles de la mécanique céleste" , 1–3 , Gauthier-Villars (1899)
[a2]	C.L. Siegel, J. Moser, "Lectures on celestial mechanics" , Springer (1971)
[a3]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[a4]	R. Abraham, J.E. Marsden, "Foundations of mechanics" , Benjamin (1978)