Date sön 06 januari 2019

There is a claim in Randell Mills GUTCP that because the precession of the loops, the total angular momentum is twice the spinn momentum for a single loop. Let's try to analyze this.

First of all if each loop precesses and we add up all paths we would still get a constant loop distribution. So the charge is still a constant value on the path.

Now over to the question of what we should expect when it comes to total angular momentum. Note that from wikipedia (Precession)[https://en.wikipedia.org/wiki/Precession] we would expect the movement to be torque free and hence if $$\alpha$$ is the angel between the total angular momentum and the symmetry line we have

$$L_p = \frac{L_s}{cos(\alpha)}$$

Now take the angel $$\beta$$ between the precession axis and the symmetry axis then we know from the fact that the sinus parts onto the total angular mumentum take each other out we must have

$$sin(\alpha) = \frac{sin(\beta - \alpha)}{cos(\alpha)},$$

or

$$sin(\alpha)cos(\alpha) = sin(\beta - \alpha).$$

The magnitude of the total angular momentum is

$$Total(\alpha) = L_s f(\alpha),$$

with

$$f(\alpha) = cos(\alpha) + \frac{cos(\beta-\alpha)}{cos(\alpha)}$$

using the constraint between $$\alpha,\beta$$ this can be rewritten as

$$f(\alpha)) = cos(\alpha) + \sqrt{\frac{1}{cos(\alpha)^2}+cos(\alpha)^2 - 1}$$

or with $$f(\alpha) = g (cos(\alpha))$$

$$g(x) = x + \sqrt{1/x^2+x^2-1}.$$

Lets find the extreme value of this, taking the derivative lead to

$$g'(x) = 1 + \frac{-1/x^3 + x}{\sqrt{\cdots}}$$

and putting $$g'(x) = 0$$ and simplify and squered we get

$$1 + 1/x^6-3/x^2 = 0$$

This can be solved via a third order equations, the interesting solution, $$x_0$$ is

$$x_0 = cos(\alpha_0) = 0.80790$$

with

$$f(\alpha_0) = 1.8964$$

These numbers are not close to the ones claimed by Mills. Not sure where the misstake is here but it is close and I find the calculation interesting. The extreme point is a minimal e.g. you will get minimal total angular momentum for a specific spinn angular momentum. Perhaps we should minimize energy in stead.