View source for Two-body problem

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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 [[#References|[1]]]. The two-body problem also has three Laplace integrals (one of which is independent of the preceding ones) and is completely integrable [[#References|[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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  C.L. Siegel,   "Vorlesungen über Himmelmechanik" , Springer  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.K. Abalakin,   et al.,   "Handbook of celestial mechanics and astrodynamics" , Moscow  (1971)  (In Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Poincaré,   "Les méthodes nouvelles de la mécanique céleste" , '''1–3''' , Gauthier-Villars  (1899)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  C.L. Siegel,   J. Moser,   "Lectures on celestial mechanics" , Springer  (1971)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  V.I. Arnol'd,   "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Abraham,   J.E. Marsden,   "Foundations of mechanics" , Benjamin  (1978)</TD></TR>
</table>