Spinors in Classical Mechanics and Geometry?

I'm trying to deepen my understanding of spinors by looking at applications in simple problems, preferably unrelated to quantum mechanics. For this purpose I'd like to refrain from discussing definitions, generalizations, and all manner of group theory jargon as much as possible, unless you can demonstrate it with a good example.

To motivate discussion and give a flavor of what I'm looking for, I offer two ways of looking a familiar problems in "spinor" coordinates.

Consider $$\mathbb{R}^2$$ with $$x$$ and $$y$$ as Cartesian coordinates. We can define a complex number $$\xi$$ as:

$$\xi=(x^2+y^2)^{1/4}e^{i\phi/2}$$

Where:

$$\phi=\tan^{-1}\left(\frac{y}{x}\right)$$

And see that it's essentially just a change of coordinates. We can go back to $$x$$ and $$y$$ by using:

$$x=\frac{1}{2}\left(\xi^2+{\xi^*}^2\right)$$

$$y=\frac{1}{2i}\left(\xi^2-{\xi^*}^2\right)$$

Which has the funny property that both $$\xi$$ and $$-\xi$$ map to the same point of $$x$$ and $$y$$. For the arclength of a line, we know that:

$$s=\int\text{d}t\sqrt{\dot{x}^2+\dot{y}^2}$$

In "spinor" coordinates this becomes:

$$s=2\int\text{d}t\sqrt{\xi\dot{\xi}\xi^*\dot{\xi^*}}$$

It's unclear to me whether writing this offers any advantages or interesting properties. Is there any? What if I want to make local transformations to the curve that keep it's total arclength invariant?

We also know a particle moving in free space is given by:

$$\ddot{x}=0$$ $$\implies x(t)=x(0)+\dot{x}(0)t$$

And likewise for $$y$$.

However, converting to spinor coordinates in the Lagrangian and solving for equations of motion yield:

$$\ddot{\xi}+\frac{\dot{\xi}^2}{\xi}=0$$

Which is solved by:

$$\xi(t)=\sqrt{\xi(0)^2+2\xi(0)\dot{\xi}(0)t}$$

And likewise for the conjugate.

Is there any insight to be gained from this? In polar coordinates certain problems become much simpler. Is there any advantages of applications of spinor coordinates in classical mechanics?

• In two dimensions there shouldnt be any difference in the chosen coordinates, since $SO(2) \cong Spin(2)$. So I do not quite understand what you mean by ''spin-coordinates'' (you for sure have taken different coordinates, but I do not understand how they are relatet to ''spin''). I might be able to answer your question, but I would need a clear definition of ''spin-coordinates'', if possible coupled with your understanding/a definition of ''spin-structure''. There are other difficutlies with your approach, assuming we both use the same definitions-which is why I'd need yours. – Creo Oct 16 '18 at 9:00