I have a Lagrangian, that describes a Pendulum hanging on a spinning disk in 3D. It looks like this:
$$ L(p,\lambda, t) = \frac 12m\|\dot p\|^2-m g p\cdot\hat k+\lambda(\|p-q\|^2-l^2) $$ with $p$ and $q$ being: $$ \cases{ q=r(\hat i \cos(\omega t) + \hat j\sin(\omega t))+q_0 \hat k\\ p = (x(t),y(t),z(t)) }\ \ \ \ \ (1) $$
If I now want to add viscous damping which is defined with: $$ f = c \cdot \dot{p} $$ How do I do that, I have tried to just substract it like this: $$ L(p,\lambda, t) = \frac 12m\|\dot p\|^2-m g p\cdot\hat k+\lambda(\|p-q\|^2-l^2)- c \cdot \dot{p} $$
But when trying to calculate the Euler lagrangian, $c$ as the coefficient disapears through the derivative here: $$ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot p}\right)-\frac{\partial L}{\partial p}=0 $$ Does this mean it has no effect or is there something wrong with the damping term?
How would you add the damping to the Lagrangian?
