Why do Euler poles for plate motions not stay fixed in time?

Question

Surely if a plate motion is a straight line across a sphere, this can be described by a fixed Euler pole, however I've been reading about a problem which I confess I don't fully understand which is titled the "Three Plate Problem" and talks about if three plates are moving relative to eachother, one of these relative motions will be described by an Euler pole that wanders. Why is this?

Answer 1

I think I can do an OK job explaining, but I could not find a proper diagram to illustrate this. Perhaps, if you could locate a professor at your university to draw this out and explain it would be best, as doing an illustration on computer is a bit time consuming. Lets start out with a two plate system, plate $A$ and $B$.

${}_{A}E_B$ is the relative plate motion of $B$ with respect to $A$, meaning that $A's$ motion is fixed while $B$ is free to rotate.

Since this is only a two plate mode, $A$ and $B's$ relative plate motions are always the same, no matter where they are in time, you can always go back and forawrd in time, as nothing changes the relationship between $A$ and $B$. IE, there is nothing to interfere with their relative plate motions.

Only 1 Euler pole is needed to fully describe the motions of two plates, in general. Lets now add a 3rd plate, plate C:

${}_AE_B$ is the plate motion of $B$ with respect to $A$

${}_BE_C$ is the plate motion of $C$ with respect to $B$

${}_CE_A$ is the plate motion of $A$ with respect to $C$

These three motions describe the 3 plate system. So now lets imagine we are standing on $A$ looking at $B$ and $C$, what happens? Since we are on $A$, we move relative to $C$, and $B$ moves relative to us. But wait, a problem occurs, because $C's$ motion is dependent on $B$ being relatively fixed, and $A's$ motion is dependent on $C$ being relatively fixed, meaning if two poles are fixed, the third and final pole must wander to accommodate the reference frame. IE, because All of the plates are moving relative to each other (depending on the reference frame), the Euler poles are changing over time. This gets more complex as we keep on adding plates into the system.