Stability of Lagrange points
We have seen that the five Lagrange points, to , are the equilibrium points
of mass in the corotating frame. Let us now determine whether or not
these equilibrium points are stable to small displacements.
The equations of motion of mass in the corotating frame are
specified in Equations (9.27)–(9.29). 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 2.7)

(9.67) 
This condition is satisfied everywhere in the  plane. Hence, the Lagrange points are all
stable to small displacements parallel to the axis. It thus remains to investigate
their stability to small displacements lying within the  plane.
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

(9.71) 
where
,
,
,
and so on. However, by definition, at a Lagrange point, so the expansion simplifies to

(9.72) 
Finally, substituting Equations (9.68)–(9.70), and (9.72)
into the equations of  motion, (9.27) and (9.28), and only retaining terms up to
first order in small quantities, we get
as
.
Let us search for a solution of the preceding pair of equations
of the form
and
. We obtain

(9.75) 
This equation only has a nontrivial solution if the determinant of the
matrix is zero. Hence, we get

(9.76) 
It is convenient to define


(9.77) 


(9.78) 


(9.79) 


(9.80) 
where all terms are evaluated at the point
. It thus follows that
Figure: 9.13
The solid, dashed, and dotted curves show as a function of at
, , and Lagrange points, respectively.

Consider the colinear Lagrange points, , , and . These all
lie on the axis, and are thus characterized by
,
, and
. It follows,
from the preceding equations, that and . Hence,
,
, and . Equation (9.76) thus yields

(9.84) 
where
. For a Lagrange point to be stable
to small displacements, all four of the roots, , of Equation (9.76) must
be purely imaginary. This, in turn, implies that the two roots of
the preceding equation,

(9.85) 
must both be real and negative.
Thus, the stability criterion is

(9.86) 
Figure 9.13 shows calculated at the three colinear Lagrange points as a function of , for all
allowed values of this parameter (i.e.,
). It can be seen that
is always greater than unity for all three points. Hence, we conclude that the colinear
Lagrange points, , , and , are intrinsically unstable equilibrium points in the corotating
frame.
Figure: 9.14
Positions of the Trojan asteroids (small circles) and Jupiter (large circle) projected
onto ecliptic plane (viewed from north) at MJD 55600. The axis is directed toward
vernal equinox. Raw data from JPL SmallBody Database.

Let us now consider the triangular Lagrange points, and . These points
are characterized by
. It follows that , ,
,
and . Hence,
,
, and
,
where the upper and lower signs corresponds to and , respectively.
Equation (9.76) thus yields

(9.87) 
for both points,
where
. As before, the stability criterion is that the two roots of the
preceding equation must both be
real and negative. This is the case provided that
, which
yields the stability criterion

(9.88) 
In unnormalized units, this criterion becomes

(9.89) 
Figure: 9.15
Positions of the Trojan asteroids (small circles) and Jupiter (large circle) at MJD 55600.
is normal distance from the ecliptic plane.
is the difference in
ecliptic longitude between the asteroids and Jupiter. Raw data from JPL SmallBody Database.

We thus conclude that the and Lagrange points are stable equilibrium
points, in the corotating frame, provided that mass is less than about percent 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 it will stay approximately ahead of
mass if it is orbiting the point, or behind if it is orbiting the point. See Figure 9.12. This type of behavior has been observed in the solar system. For instance,
there is a subclass of asteroids, known as the Trojan asteroids, that are trapped
in the vicinity of the and points of the SunJupiter system [which easily
satisfies the stability criterion in Equation (9.89)], and consequently share Jupiter's orbit around the Sun,
staying approximately ahead of and behind, Jupiter, respectively. These asteroids
are shown in Figures 9.14 and 9.15. The SunJupiter system is not the only dynamical system in the solar system that possess
Trojan asteroids trapped in the vicinity of its and points. In fact, the SunNeptune system has eight known Trojan asteroids,
the SunMars system has four, and the SunEarth system has one (designated 2010 TK7) trapped at the point.
The and points of the SunEarth system are also observed to trap clouds of interplanetary dust.