I read recently about speculation that spots (geographic features) on the earth’s surface move toward or away from the spin axis.
Here is a qualitative argument that ensures that this is the case.
It suggests some numerical models whose nature might become clear to me as I write these notes.
Warning: There may be no useful ideas below.
The equations for spinning rigid bodies have been known for a long time.
It is complicated.
I think that the easiest useful statment for this discussion is as follows:
From on-board the spinning rigid object, both the angular momentum vector and the angular velocity vector undergo periodic movements relative to the body proper.
By “angular momentum” I mean that measured in a static frame and then reported and transformed to the axes that folllow the body.
(I am pretty sure that the motion measured from the static frame are almost mever periodic.)
This means that stresses on the body due to the accelerations due to the spin, are periodic.
A spinning elastic body is yet more complex for the shape of the body will respond to the periodic forces periodically.
Indeed the tensor of inertia will change periodically as well.
There is no disipative mechanism available and so the total energy of the system will be constant and there is presumably a solution of the equations with one period like the spinning rigid body.
When a plastic body changes its shape it disipates some of its energy as heat.
A spinning plastic body cannot change its angular momentum.
The “ground state” of a spinning plastic body must therefore be one of those few configurations where the angular momentum and angular velocity align with one of the principle axes of the inertia tensor, i.e. the angular velocity is an eigen vector of the inertia tensor.
In this state the stresses on each part of the body are constant and thus the strains and shape of the body are constant and there is no more disipation.
Alas a perfectly plastic body changes its shape upon any stress.
To study the earth we will need to include gravity to keep it from coming apart.
I have heard the claim that a spinning body is “not stable” unless it spins about either the least or the greatest eigen vector of its inertia tensor.
I don’t know what this means.
It might be true that when a spinning body spins about the intermediate eigen vector, it will eventually spin about one of its other axes if perturbed, but not without some form of disipation.
If you add centrifugal and Corioliss forces to the ordinary F=ma forces, you can do newtowian physics in a rotating frame.
What would it mean to do the calulation from on-board the body?