Semiholonomic Constraints in Rolling Without Slipping I was reading Goldstein's Classical Mechanics and it is mentioned there, on page 50 (third edition), that rolling without slipping is a semiholonomic constraint, which is defined on page 49 as a constraint of the form
$$f_\alpha = \sum_{k= 1}^n a_{ak} \dot{q}_k + a_0 = 0. $$
I didn't understand how the author reached that conclusion and couldn't find any explanation in the book. Can anyone explain it with equations? Why is rolling without slipping a case of semiholonomic constraints?
 A: Well, rolling without slipping is a condition sometimes defined as the implication that the total velocity of the point of contact between the rolling body and the surface must be $0$.
As an example imagine a homogeneous sphere rolling to the positive side of the x-axis, the center of mass of the sphere is moving with velocity vector $\dot{x}$, and because of the rolling motion, each particle composing also has another component of the velocity given by $v_{rot}=r\dot{\theta}$, where $r$ is the distance from the axis of rotation, in this case, the center of mass of the sphere, and $\dot{\theta}$ is the angular velocity of the body.
If you study the velocity in the point of contact of the rolling object that isn't slipping, both velocities described before hand have opposite directions, and for the condition to be true should also have the same magnitude.
$$\dot{x}=R\dot{\theta}\rightarrow\dot{x}-R\dot{\theta}=0=f_0$$
Comparing this condition with the definition of semiholonomic constrain  it is in fact of that form, using the same notation as the definition one gets that $q_1=x$, $q_2=\theta$, $a_{01}=1$, $a_{02}=-R$ and $a_0=0$.
Sometimes the condition of rolling without slipping is defined such that the total distance traveled by the point of contact is the same as the perimeter swept by the angle, using the same example as before we get that $x=R\theta$. This clearly is a geometric constrain and not a semiholonomic, but if you derive it by the time you get exactly the first condition stated.
In either case for this example, you can see how the condition is of the semiholonomic form.
I hope it helps!
