Earth’s rotation: observations and theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 294

4.3 The dynamics of polar motion and wobble
The basic dynamical equation for rotational motion equates the total external
applied torque, Γ , with the time rate of change of the total angular momentum,
L . Note that Γ and L are measured about the centre of mass in an inertial frame of
reference.
The total velocity is

v + ω × r , (4.16)

where v is the velocity relative to the rotating frame and ω is its angular velocity.
Then, the total angular momentum is

r

v ) ρ d L = × ( ω × r + V ,

(4.17)

V

with ρ the mass density, the integral extending over the whole Earth. Expanding
the vector triple product gives

r 2 ω r × ( ω × r ) = − ( r · ω ) r , (4.18)

and the i th component of the angular momentum becomes

= ω j

r 2

x i x j ρ d

i
j

L i δ − V+ i , (4.19)

V

where

r 2

x i x j ρ d

i
j

I ij = δ − V (4.20)

V

are the components of the inertia tensor of the Earth and

= ( r × v )ρ d V (4.21)

V

is the relative angular momentum about the centre of mass relative to the rotating
frame. Finally, the total angular momentum can be expressed as

L = I · ω + . (4.22)