If two of the bodies, say m1 and m2 in the three-body problem move in circular, coplanar orbits about their common center of mass and the mass say m3 of the third body is too small to affect the motion of the other bodies, the problem of the motion of the third body is called the circular, restricted, three body problem. The two revolving bodies are called the primaries; their masses are arbitrary but have such internal mass distributions that they may be considered point masses.
The equations of motion of the third body in a dimensionless sidereal (inertial) coordinate 
system with the mean motion
, are [
where 
is given as
denotes the mass of the smaller primary when the total mass of the primaries has been normalized to unity.
and 
are the distances of the third body from the primaries which are located at
;
, these coordinates are functions of the time t and are given as
, the coordinate system related to the system 
by certain transformation, is also called sidereal coordinate system. In this respect the system 
of Equation (7) is called sidereal spherical coordinate system.
and 
are given in terms of 
and 
from the previous equations.
.
are given as
, 
and
; 
can be computed from Equation (7), while
, 
and 
can be computed from Equations (1)-(3), so we get
where
,
,
,
.
(say) both:
, and 2) The Cartesian sidereal coordinates and velocities
.
. While the Cartesian sidereal coordinates and velocities 
are obtained by the substitutions in the direct transformation formulae (Equations (7) and (8)), rather than solving the six order system of Equations (1), (2) and (3). By this way, great time can be saved. 
at
2) the final time 
at 
and the inverse transformations to compute the initial values
.
(functions of
) to construct the analytical forms of equations of motion as first order system.
from step 1 to solve numerically the above differential system of step 2 for 
, (note that
).
from step 3 and the direct transformations of Equations (7) and (8) to compute numerically 
and 
.
(left) and 
(right).