Euler-Lagrange confusion

Question

Consider the action $S = \int dt \sqrt{G_{ab}(q)\dot{q}^a\dot{q}^b}.$

Now for computing the Euler-Lagrange equations, we need the time derivative of $\frac{\partial L}{\partial \dot{q}^c} = \frac{1}{\sqrt{G_{ab}(q)\dot{q}^a\dot{q}^b}}G_{dc}(q)\dot{q}^d$.

Do we also need to take the time derivative of the denominator? If you do then the equations of motions become ugly, and my intuition says that they should be pretty normal.

The only explanation I can think of, is that the root is a scalar since it is fully contracted, so there is no need to consider the time derivative.

Answer 1

In general you do have to take the derivative of the inverse square-root factor. However if you choose $$dt^2 = G_{ab}dq^adq^b$$ then the factor becomes $1$.