For the Lagrangian $2L=\dot{q}^2-q\dot{q}+q^2$
(A) $q+\dot{q}$
(B) $q-\dot{q}$
(C) $q\dot{q}$
(D) $\dot{q}-q$
I have a doubt here. in fiew books $p$ is said to be generalized momentum. Does these generalized momentum and conjugate momentums are same? If yes the by definition
$p=\frac{\partial L}{\partial \dot{q} }= \dot{q}-\frac{q}{2}$. In that case all the four given options are wrong.