Now, the equations of motion of mass in the co-rotating frame are
specified in Equations (1056)-(1058). Note that the motion
in the - plane is complicated by presence of the Coriolis acceleration. However, the motion parallel to the -axis simply
corresponds to motion in the potential . Hence, the condition for
the stability of the Lagrange points (which all lie at ) to small displacements parallel
to the -axis is simply (see Section 3.2)

(1096) |

Suppose that a Lagrange point is situated in the - plane at coordinates
.
Let us consider small amplitude - motion in the vicinity of this point by writing

where and are infinitesimal. Expanding about the Lagrange point as a Taylor series, and retaining terms up to second-order in small quantities, we obtain

(1100) |

Finally, substitution of Equations (1097)-(1099), and (1101) into the equations of - motion, (1056) and (1057), yields

(1102) | |||

(1103) |

since .

Let us search for a solution of the above pair of equations
of the form
and
. We obtain

(1104) |

Now, it is convenient to define

(1106) | |||

(1107) | |||

(1108) | |||

(1109) |

where all terms are evaluated at the point . It thus follows that

(1110) | |||

(1111) | |||

(1112) |

Consider the co-linear Lagrange points, , , and . These all
lie on the -axis, and are thus characterized by
,
, and
. It follows,
from the above equations, that and . Hence,
,
, and . Equation (1105) thus yields

(1113) |

(1114) |

(1115) |

Let us now consider the triangular Lagrange points, and . These points
are characterized by
. It follows that , ,
,
and . Hence, , , and
,
where the upper/lower signs corresponds to and , respectively.
Equation (1105) thus yields

(1116) |

(1117) |

(1118) |

We thus conclude that the and Lagrange points are *stable* equilibrium
points, in the co-rotating frame, provided that mass is less than about of
mass . If this is the case then mass can orbit around these points
indefinitely. In the inertial frame, the mass will share
the orbit of mass about mass , but will stay approximately *ahead of*
mass , if it is orbiting the point, or *behind*, if it is orbiting the point--see Figure 55. This type of behavior has been observed in the Solar System. For instance,
there is a sub-class of asteroids, known as the *Trojan asteroids*, which are trapped
in the vicinity of the and points of the Sun-Jupiter system (which easily
satisfies the above stability criterion), and consequently share Jupiter's orbit around the Sun,
staying approximately *ahead of*, and *behind*, Jupiter, respectively.
Furthermore, the and points of the Sun-Earth system are occupied by clouds of dust.