Dear Gary,
Thanks for your kind words on my essay.
I have looked at your paper "The Wave Equation and Rotation" to which you refer in your post. I am familiar with the quaternions, and also with the Euler equation. It is an interesting idea to use Euler-type equations for quaternions; however, it leads to contradiction. I will demonstrate this below.
From the pages 2 and 3 in your paper it is obvious that you want to add the following three equations to the quaternions:
[math](1) \ \ e^{ix} = cosx i \cdot sinx[/math]
[math](2) \ \ e^{jx} = cosx j \cdot sinx[/math]
[math](3) \ \ e^{kx} = cosx k \cdot sinx[/math]
Using (1) and (2), we get
[math](4) \ \ i \cdot j = e^{i \pi /2} \cdot e^{j \pi /2} = e^{i \pi /2 j \pi /2}[/math]
[math](5) \ \ j \cdot i = e^{j \pi /2} \cdot e^{i \pi /2} = e^{j \pi /2 i \pi /2}[/math]
However, since addition is commutative for quaternions, we get
[math](6) \ \ e^{i \pi /2 j \pi /2} = e^{j \pi /2 i \pi /2}[/math]
Thus, from (4) - (6) we get
[math](7) \ \ i \cdot j = j \cdot i[/math]
For the quaternions, however, we have
[math](8) \ \ i \cdot j = - j \cdot i[/math]
That proves the contradiction. The equations (1) - (3) can thus not be added to the (theory of the) quaternions. I would advise you to reconsider this aspect of your theory.
Best regards, Marcoen
