DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

L
lagoon A shallow, sheltered bay that lies between
a reef and an island, or between a barrier
island and the mainland.
Lagrange points Five locations within a
three-body system where a small object will always
maintain a fixed orientation with respect to
the two larger masses though the entire system
rotates about the center of mass. If the largest
mass in the system is indicated by M1 and the
second largest mass is M2, the five Lagrange
points are as follows: in a straight line with M1
and M2 and just outside the orbit of M2 (usually
called the L1 point); in a straight line with M1
and M2 and just inside the orbit of M2 (L2); in
a straight line with M1 and M2 and located in
M2’s orbit 180 ◦ away from M2 (i.e., on the other
side of M1) (L3); and 60 ◦ ahead and behind M2
within M2’s orbit (L4 and L5). In the sun-Jupiter
system, the Trojan asteroids are found at the L4
and L5 locations. In the Earth-sun system, the
Lagrangian points L1 and L2 are both on the sun-
Earth line, about 236 RE (≈ 0.01 AU) sunward
and anti-sunward of Earth, respectively. The
other points are far from Earth and therefore too
much affected by other planets to be of much
use, e.g., L3 on the Earth-sun line but on the far
side of the sun. However, L1 (or its vicinity) is a
prime choice for observing the solar wind before
it reaches Earth, and L2 is similarly useful for
studying the distant tail of the magnetosphere.
Spacecraft have visited both regions — ISEE-3,
WIND, SOHO and ACE that of L1, ISEE-3 and
GEOTAIL that of L2. Neither equilibrium is
stable, and for this and other reasons spacecraft
using those locations require on-board propulsion.
Of the Lagrangian points of the Earth-moon
system, the two points L4 and L5, on the moon’s
orbit but 60 ◦ on either side of the moon, have
received some attention as possible sites of space
colonies in the far future. Their equilibria are
stable.
© 2001 by CRC Press LLC
Lagrangian
Lagrangian In particle mechanics, a function
L = L(x i ,dx j /dt, t) of the coordinate(s)
of the particle x i , the associated velocity(ies)
dx j /dt, and the parameter t, typically time,
such that the equations of motion can be written:
d
dt
∂
∂L
∂x i
∂t
− ∂L
= 0 .
∂xi In this equation (Lagrange’s equation) the partial
derivatives are taken as if the coordinates
xi and the velocities dxj /dt were independent.
The explicit d/dt acting on ∂L
∂( ∂xi differentiates
∂t )
xi and dxi /dt. For a simple Lagrangian with
conservative potential V ,
L = T − V = 1
2 m
dx i 2
j
/dt − V x ,
one obtains the usual Newtonian equation
d
dt mdxi
dt =−∂V .
∂xi Importantly, if the kinetic energy term T and
the potential V in the Lagrangian are rewritten
in terms of new coordinates (e.g., spherical), the
equations applied to this new form are again the
correct equations, expressed in the new coordinate
system.
A Lagrangian of the form L = L(xi ,dxj /dt,
t) will produce a resulting equation that is second
order in time, as in Newton’s equations. If
the Lagrangian contains higher derivatives of the
coordinates, then Lagrange’s equation must be
modified. For instance, if L contains the acceleration,
a j = d2x j
,
dt2 so that
L = L
x i ,dx j /dt, d 2 x k /dt 2
,t ,
the equation of motion becomes
− d2
dt 2
∂
∂L
d 2 x i
dt 2
+ d
dt
∂
∂L
∂x i
∂t
− ∂L
= 0 ,
∂xi which will in general produce an equation of motion
containing third time derivatives. The presence
of higher derivatives in the Lagrangian produces
higher order derivatives in the equation of