Question about Classical mechanics John R. Taylor

[HomogenousCow]

In Taylor's advanced text on classical mechanics, he gives an example of a non-holonomic system, I find this part very strange.
He gives the example of a hard rubber ball being rolled in a triangle on a flat surface, the point is that if you take the ball out through the triangle and back to the initial position, the orientation of the ball will be different at the end.
I don't understand why that would make the system holonomic, if we treat the ball as a rigid body it would have 6 degrees of freedom, three numbers to specify its spatial position and three numbers to specify its orientation, it seems to me that only when we ignore the extended nature of the ball does the system become non-holonomic.

 

[Jano L.]

Here is one explanation:

Holonomic system is one in which the constraints on the coordinates are of the form

$$
f(q_1, q_2, ...) = 0.
$$

For example, mass point bound to a sphere of radius $r$ in space has constraint

$$
x^2 + y^2 + z^2 = r^2,
$$

and locally the condition

$$
xdx + ydy + zdz = 0
$$

has to be satisfied. The point has only two free degrees of freedom; for example, x and y. If we know displacements in these, the displacement in z can be calculated from the above equation.

The ball rolling without slipping is another kind of system, because the no-slipping condition cannot be written in the above way with coordinates only; the condition says that the velocity of the contact point on the ball is zero.

The constraint equation for such system will contain also the derivatives of the coordinates $x,y,\varphi,\vartheta,\alpha$, and the local version will contain time differential $dt$. There is no one definite relation between these 5 coordinates and no definite constraint on their changes; what will happen to them depends also on the angular velocity of the ball.

This means that although there is a constraint on the motion, one can make infinitesimal changes in four variables $x,y,\varphi,\vartheta$, but these still do not determine the change in $\alpha$ ; the value of the latter depends on the path chosen for the change of the former four coordinates.

 

[Jano L.]

Here is another, presumably simpler way to understand non-holonomic system:

if the system has constraint on motion which does not restrict accessible configurations, only the paths to it, it is non-holonomic. The ball can get into any state $x,y,\vartheta,\varphi,\alpha$, but the path cannot be arbitrary since it has to be such that no-slipping condition is satisfied.