The configuration space of a rolling ball.

Question

This Wikipedia article mentions that the configuration space of a rolling ball is $\Bbb{C}^5$. I don't understand why that is.

The position of the center of mass, that's a point in $\Bbb{R}^3$. The axis and velocity of rotation, that's another point in $\Bbb{R}^3$ (this would give the orientation of the axis, the direction of rotation and also the velocity). Now we might also have a rate of change of axis. We may include as many derivatives with respect to time as we like. So we're essentially looking at $\Bbb{R}^{6+3d}$, where $d$ is the number of derivatives with respect to time of the change of axis that we take.

How can we be sure that the configuration space is exactly $\Bbb{C}^5$?

Answer 1

The configuration space is not $\mathbb{C}^5$. (And I don't see where in the Wikipedia article this is claimed.)

First: we are talking about a ball that is rolling on another ball. This means,

• The center of the ball that is moving is going to be a point on some sphere, and so can be parametrized by $\mathbb{S}^2$.
• The orientation of the ball that is moving is described as a relative rotation from some initial fixed orientation, and can be parametrized by $SO(3)$.

Thus the configuration space is $\mathbb{S}^2 \times SO(3)$ which has real dimension 5.

---

In terms of the group $G_2$, this is also not quite right. The correct description is found in this expository article: basically

• the ball that is rolling is not a real ball, but a spinorial ball; so its internal degrees of freedom is not described by $SO(3)$, but by its double cover $SU(2)$.
• the ball that is being rolled on is not a real ball, but one whose antipodes are identified, so instead of $\mathbb{S}^2$ the correct space is $\mathbb{R}\mathbb{P}^2$.