Editing Nonholonomic system (section)

===Rolling sphere===
This example is an extension of the 'rolling wheel' problem considered above.

Consider a three-dimensional orthogonal Cartesian coordinate frame, for example, a level table top with a point marked on it for the origin, and the ''x'' and ''y'' axes laid out with pencil lines.  Take a sphere of unit radius, for example, a ping-pong ball, and mark one point ''B'' in blue.  Corresponding to this point is a diameter of the sphere, and the plane orthogonal to this diameter positioned at the center ''C'' of the sphere defines a great circle called the equator associated with point ''B''.  On this equator, select another point ''R'' and mark it in red.  Position the sphere on the ''z''&nbsp;=&nbsp;0 plane such that the point ''B'' is coincident with the origin, ''C'' is located at ''x''&nbsp;=&nbsp;0, ''y''&nbsp;=&nbsp;0, ''z''&nbsp;=&nbsp;1, and ''R'' is located at ''x''&nbsp;=&nbsp;1, ''y''&nbsp;=&nbsp;0, and ''z''&nbsp;=&nbsp;1, i.e. ''R'' extends in the direction of the positive ''x'' axis.  This is the initial or reference orientation of the sphere.

The sphere may now be rolled along any continuous closed path in the ''z''&nbsp;=&nbsp;0 plane, not necessarily a simply connected path, in such a way that it neither slips nor twists, so that ''C'' returns to ''x''&nbsp;=&nbsp;0, ''y''&nbsp;=&nbsp;0, ''z''&nbsp;=&nbsp;1.  In general, point ''B'' is no longer coincident with the origin, and point ''R'' no longer extends along the positive ''x'' axis.  In fact, by selection of a suitable path, the sphere may be re-oriented from the initial orientation to any possible orientation of the sphere with ''C'' located at ''x''&nbsp;=&nbsp;0, ''y''&nbsp;=&nbsp;0, ''z''&nbsp;=&nbsp;1.<ref>The Nonholonomy of the Rolling Sphere, Brody Dylan Johnson, The American Mathematical Monthly, June–July 2007, vol. 114, pp.&nbsp;500–508.</ref>  The system is therefore nonholonomic.  The anholonomy may be represented by the doubly unique [[quaternion]] (''q'' and −''q'') which, when applied to the points that represent the sphere, carries points ''B'' and ''R'' to their new positions.