Let $Q$ have local coordinates $(q_1,\dots,q_n)$ and let $D$ be described by independent one-forms $\omega_a=\sum_i A_{a i}(q)dq_i$, $a=1,\dots,k$, through the equations $(\omega_a)_q(\dot q)=0$. Then a Lagrange-d'Alembert trajectory satisfies \begin{align*} \frac{d}{dt}\frac{\partial L}{\partial \dot q_i}-\frac{\partial L}{\partial q_i} =\sum_{a=1}^k \mu_a(t)A_{a i}(q), \qquad \sum_i A_{a i}(q)\dot q_i=0. \end{align*}