The earth-pole wobble
The rigid earth as a gyroscope
We can now apply the theory to the idealized earth.
may be, in the first
considered a huge rigid gyro spinning
around its instantaneous axis of rotation, going through its center o
Evidently, both equations are equations of simple harmonic
Solving the first equation, we get
w are some integration constants.
where K and
Substituting this result
back into the first equation of first order above, one obtains
be determined from the phase-lags of the two constituents X,Y.
to predictions, the hodograph is not completely circular, or put in other
words, the circular
Chandlerian motion (as the free nutation is sometimes
known as) does not account fu
In spite of this theoretical prediction, the actually observed
amount of wobble (amplitude) does not seem to decrease significantly over
an extended period of time.
The most sensible explanation for this
disagreement is that besides damping, there exis
Ellipsoid of inertia and principal axes of inertia
It is known from mechanics that the moments of inertia of a
rigid body B with respect to all possible axes going through a point Q,
when we interpret the reciprocals of their square roots as lengt
It is not difficult to see that the equation of the central ellipsoid of
inertia in 511 will acquire the following form:
This can be interpreted as follows-the eigenvectors of the
tensor of in
It is obviously symmetrical and is very often denoted thus
Its diagonal components (elements) are known as moments of inertia
with respect to the individual aKes
are usually called products of inertia or
as the vector of rotation of B.
It is easily seen that its meaning does
not change even for the general case of translating, non-rigid body.
third velocity in the gernal formula can be then understood as the
rotational velocity of B.
= JB cfw_(r-
rQ) x r 0 dB.
Its absolute time derivative reads:
= dt (J B -+
x r 0 dB - Br Q
r 0 da
= J 8cfw_r
dt = 0
x r + r x r 0dB - J -rQ - +
B +x r
due to the rigidity of B.
Euler's equations of rotation
Definition of coordinate systems
Let us consider a physical body B in a three dimensional space.
In order to be able to describe its motion in the space let us define the
following two Cartesian coordinate systems:
is the antisymmetric tensor
We can then write f ina 11 y for -+
+ w x a.
Note that applying this formula, valid for any free vector in
to w we get
since, as we have said earlier,
= 1 ,2,3).
usually called absolute velocity (velocity with respect to the
"absolute" system of coordinates S) and
known as relative velocity
(taken with respect to the relative syst