Nonholonomic constraint

Constraints that are not holonomic constraint.

There is, mainly, two reasons for being nonholonomic:

The constraint is of the form

f (t, r_{i}, {\dot{r}}_{N}) = 0

and this ODE cannot be integrated. In this case it is described as a Pfaffian equation that is non integrable. We can add this ODE to the equations of motion, and try to solve the system.
I think they are also called semi-holonomic.

The constraint is of the form

f (t, r_{1}, r_{2}, \dots, r_{N}) \leq 0.

I don't know how proceed in this case.