We restrict that $$\delta q\mid _{t_{1}}= \delta q\mid _{t_{2}}=0$$ while applying Hamilton Principle ($\delta\int_{t_{1}}^{t_{2}}Ldt=0$) to get Euler-Lagrange’s Equations. Hence adding a $$\frac{d}{dt}\Lambda(q,t)$$ doesn’t change the principle as $$\delta\int_{t_{1}}^{t_{2}}\frac{d}{dt}\Lambda(q,t)dt=\delta\Lambda\mid_{t_{1}}^{t_{2}}=0 \ .$$
My question is, why don’t we restrict $$\delta\dot{q}\mid_{t_{1}}=\delta\dot{q}\mid_{t_{2}}=0$$ as well, so that adding a $$\frac{d}{dt}\Lambda(q,\dot{q},t)$$ won’t change the principle? Why is it no longer feasible just because $$\dot{q}=\frac{dq}{dt} \ ?$$
