Seeking a example of noholonomic constraint

[xephyrind]

I was reading up classical mechanics in Goldstein but needed some clarifications. I looked online and saw something that bothers me qutie a bit. In the online pdf below, on page 69 (or 74th screen scrolls), it states that Dot Cancellation does not work if the position vector is a function of BOTH generalized coordinates AND generalized velocities such that \vec_r = \vec_r(q_1, q_2, ...; \dot_q_1, \dot_q_2, \dot_q_3). On top, the PDF states that it happens for certain non-holonomic constraints. Since this dot cancellation is central to using D'Alembert's principle to arrive at the form of Euler-Lagrange Equations. I am wondering where this Dot Cancellation would fail.

Dot Cancellation: d(\vec_r)/dq = d(\vec_v)/d(\dot_q)
http://www.astro.caltech.edu/~golwala/ph106ab/ph106ab_notes.pdf

Question:
Would someone kindly give me an example such that the position vector \vec_r actually depends on generalized velocities as well as the constraints that causes it please? Thanks much.

 

[gtoguy97]

Answer: One example is a system with a non-holonomic constraint involving a rolling surface. In this case, the position of the system depends on both the generalized coordinates and velocities. The constraint can be expressed as:\vec_r \cdot \hat_n = 0,where \hat_n is a unit vector normal to the surface. This means that the position of the system can only lie on a certain surface. The velocity of the system must also be perpendicular to the surface, which implies that it is a function of both the generalized coordinates and velocities. As a result, dot cancellation does not work in this case.