# General force between two point particles, one of which has “spin”

Consider two points in the empty (isotropic and homogeneous) space: since the only vector that "makes sense" (the only vector that we can define) is the vector given by the difference of the two positions $$\vec{r}$$, the force $$\vec{f}$$ between these two point particles must be "central", $$\vec{f} = F(r) \hat{r}$$.

How to build, using a purely geometric reasoning, the most general force between two points, one of which has a formal "spin" that defines a preferred direction $$\hat{s}$$?

Now we have 2 objects: $$\vec{s}$$ and $$\vec{r}$$, so one may guess that $$\vec{f} = ( \hat{r}-\hat{s} \hat{r}\cdot\hat{s}) Q(|\hat{r}-\hat{s} \hat{r}\cdot\hat{s}|, \vec{r}\cdot \vec{s}) + \hat{s} P(|\hat{r}-\hat{s} \hat{r}\cdot\hat{s}|, \vec{r}\cdot \vec{s}) +\hat{r} \times \hat{s} W(|\hat{r}-\hat{s} \hat{r}\cdot\hat{s}|, \vec{r}\cdot \vec{s})$$

Is this correct or other terms are possible? is there a clever way to write such a force? The point is that the preferred direction reduces the symmetry of the problem from $$O(3)$$ to $$O(2)$$, the rotations in the plane orthogonal to $$\hat{s}$$.