For a functional of the form:
$$S(q)=\int_{t_{1}}^{t_{2}}L(q,\dot{q})dt,\tag{1}$$ where $\dot{q}=\frac{\partial q}{\partial t}$, one finds that extrema are reached (to first order) for the condition:
$$0=S(q+\delta q)-S(q)=\int_{t_{1}}^{t_{2}}dt(\frac{dL(q,\dot{q})}{dq}-\frac{\partial}{\partial t}\frac{\partial L(q,\dot{q})}{\partial\dot{q}})\delta q(t).\tag{2}$$
Then it's simply stated that:
$$0=\frac{dL(q,\dot{q})}{dq}-\frac{\partial}{\partial t}\frac{\partial L(q,\dot{q})}{\partial\dot{q}}.\tag{3}$$
I could be wrong, but it seems like the integral form yields a more general set of solutions than the lattermost equation? In the particular problem I'm working on, the domain of integration is a compact space such that $t_{1}=t_{2}$, lets just say it's over a circle of some radius $R$. Then especially here, won't these two equations diverge in their solutions? Maybe I'm missing something.