Earth’s rotation: observations and theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 295

The time derivative of a vector in the inertial frame is related to that in the rotat-
ing frame, by expression (3.72). Thus, the equation for rotational motion becomes

L

= + ω × L ,

Γ (4.23)

with L denoting the time derivative of the total angular momentum in the rotat-
ing frame. On substituting expression (4.22) for the total angular momentum into
(4.23), we obtain the equation for rotational motion attributed to Liouville in 1858,

d
dt ( I

Γ = · ω + ) + ω × ( I · ω + ). (4.24)

Adopting the geocentric Cartesian co-ordinate system of Section 4.2, the angular
velocity takes the form (4.1)

ω =Ω ( m + e 3 ). (4.25)

For most problems, large-scale polar wandering being an obvious exception, the
rotation axis does not travel far from the reference axis and the axis of figure does
not depart appreciably from the reference axis. Under these conditions,

⎝ ⎠

A + r 11 + c 11 r 12 + c 12 r 13 + c 13

I · ω = r 12 + c 12 A + r 22 + c 22 r 23 + c 23

r 13 + c 13 r 23 + c 23 C + r 33 + c 33

⎝ ⎠

Ω m 1

× Ω m 2 , (4.26)

Ω (1 + m 3 )

where A is the equatorial moment of inertia in the absence of disturbance and C
is the axial moment of inertia under the same condition. The r ij are contributions
to the inertia tensor caused by deformation under the changed centrifugal force,
while the c ij represent all other contributions to changes in the inertia tensor. The
quantities i , m i , r ij and c ij are then all regarded as small and their squares and
products are neglected. The components of the total angular momentum, to this
level of approximation, become

L 1 = A Ω m 1 + r 13 Ω+ c 13 Ω+ 1 , (4.27)

L 2 = A Ω m 2 + r 23 Ω+ c 23 Ω+ 2 , (4.28)

L 3 = C Ω+ r 33 Ω+ c 33 Ω+ C Ω m 3 + 3 . (4.29)