Suppose we have a non-holonomic mechanical system, say Lagrangian, for example the Chaplygin sleigh is a model of a knife in the plane. Its configuation space is $Q = S^1\times \mathbb{R}^2$ with local coordinates $q = (\theta,x,y)$. The non-holonomic constraint is "no admissible velocities are perpendicular to the blade" which is specified by a one-form
\begin{equation} \omega_q = \sin\theta dx + \cos\theta dy. \end{equation} When evaulating on a velocity $\dot{q}$ we specify the velocity constraint
$$ v = \omega_q\cdot \dot{q} = -\dot{x}\sin\theta + \dot{y}\cos\theta = 0. $$
What is the mathematical/physical meaning of the exterior derivative of the constraint one-form? Since $\omega_q$ is a one-form, in physics, we typically interpret it as a force. Mathematically, we integrate this form over a 'line' to get the work due to the force over the 'line'. The exterior derivative is a 2-form, (does this have any physical significance??)
$$ \boldsymbol{d}\omega = -\cos\theta d\theta \wedge dx - \sin\theta d\theta \wedge dy $$
If we evaluate this 2-form along a tangent (velocity) vector $\dot{q}$, we find the 'force'
$$ \alpha_q = \boldsymbol{d}\omega(\dot{q},.) = \left(\dot{x}\cos\theta + \dot{y}\sin\theta \right)d\theta - \cos\theta\dot{\theta}dx - \sin\theta \dot{\theta}dy. $$
Does this 'force', $\alpha_q \in T^*Q$, have any direct/obvious physical/mathematical physics/differential geometric interpretation?