Error in the example "A rolling disk using the Lagrange method"

[AlbertoRamilRego]

The 3D case of the disk rolling on a plane is a non-holonomic problem (see https://en.wikipedia.org/wiki/Nonholonomic_system), to obtain the correct equations of motion within the Lagrangian formalism, the method of the indeterminate multipliers must be applied. For this, 5 coordinates must be used (for example, the XY coordinates of the contact point and the three Euler angles) and 2 multipliers corresponding to the two components of the speed of the disk at the contact point, which must be zero.
Although both kinetic and potential energy (in the documentation the potential has the wrong sign) can be calculated from the point of contact so that only the Euler angles appear, using these expressions for the Lagrangian does not obtain the equations of motion correct as these angles do not fully describe the position of the disc.
The correct solution of the equations of motion would be:
{q1 '': -2 * q2 '* q3' / cos (q2), q2 '': 4 * g * sin (q2) / (5 * r) + sin (2 * q2) * q1 '** 2 / 2 + 6 * cos (q2) * q1 '* q3' / 5, q3 '': - (10 * cos (q2) * q1 '- 12 * tan (q2) * q3') * q2 '/ 6}