My book mentions the following equation:
$$p = \nabla S\tag{1.2}$$ where $S$ is the action integral, nabla operator is gradient, $p$ is momentum.
After discussing it with @hft, on here, it turns out book must have meant impulse is only such at initial and final points.
$$ \frac{\partial S_{cl}}{\partial x_2} = \left.\frac{\partial L}{\partial \dot x}\right|_{x_{cl}}(t_2) = p_2 $$ and also $$ \frac{\partial S_{cl}}{\partial x_1} = -\left.\frac{\partial L}{\partial \dot x}\right|_{x_{cl}}(t_1) = p_1. $$
While this is definitely clear, I still feel doubtful as I asked my professor very abruptly(couldn't have chance to have conversation) and he said that $p = \nabla S$ not only applies to initial and final points, but everywhere.
I think, this must be correct. If we imagine that potential energy of the system is not dependent on time, $\frac{\partial L}{\partial \dot x}$ would give the non-time dependent function which means impulse or speed is always the same. Hence we can conclude $\frac{\partial S_{cl}}{\partial x} = p_x$ (note, i wrote with respect to x, not x2).
Q1: Is the assumption correct?
Q2: If the potential energy is still non-time dependent and kinetic energy is given by the $\frac{1}{2}mv^2$, i don't think $\frac{\partial S_{cl}}{\partial x} = p_x$ wouldn't hold true. I'm trying to see the case when to be careful about this. Let's assume potential energies are never time dependent. Would I still need to be careful? Would be good if you could summarize the thoughts.