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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 $\Bbb{C}^5$?

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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 $\mathbb{R}\mathbb{P}^2$.
Willie Wong
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  • I am quoting the relevant line from the Wiki article- "The space of configurations of the rolling ball is 5-dimensional, with a 2-dimensional distribution that describes motions of the ball where it rolls without slipping or twisting." –  Jul 31 '20 at 18:31
  • I see that I confused real dimension for complex dimension –  Jul 31 '20 at 18:38
  • I don't think the connection between the distribution that Cartan found in C^5 and the rolling motion of the ball is entirely trivial to see. But I think if you Tweet at John Baez about it he may tell you the answer. :-) – Willie Wong Jul 31 '20 at 18:52
  • Is it obvious that the configuration space is $\Bbb{S}^2\otimes SO(3)$? Let us fix an element of $\Bbb{S}^2$ (hence the position of the center of mass of the ball). Also, each relative orientation of the ball can be thought of as an element of $SO(3)$. Hence, from what I can gather, the configuration space must be a subset of $\Bbb{S}^2\oplus SO(3)$. How do we go from here to prove that it is $\Bbb{S}^2\otimes SO(3)$? –  Jul 31 '20 at 18:53
  • More or less, I'd say. SO(3) is the space of orientations, and the orientations are independent from the location. Note that this is without any restriction like "rolling without slipping", which would introduce additional constraints. – Willie Wong Jul 31 '20 at 18:54
  • Did you mean $\Bbb{S}^2\oplus SO(3)$, instead of $\otimes$, by any chance? –  Jul 31 '20 at 18:56
  • Actually, I meant $\times$. Typed an extra o. Thanks for checking. (fixed now) – Willie Wong Jul 31 '20 at 18:58
  • Thanks for the link to the expository article! –  Jul 31 '20 at 18:59