I have a question regarding the red term below. This is the integration by parts during the derivation of the Euler-Lagrange equation for continuous systems. Why is this not the time derivative $d/dt$.... why do we use the partial? I have seen the equation expressed in both ways and wondered what the physicality's are.
\begin{equation} \int^{t_2}_{t_1}dt\frac{\partial \mathcal L}{\partial (\partial _t\phi)}\frac{\partial }{\partial t}\delta \phi =\frac{\partial \mathcal L}{\partial (\partial _t\phi)}\delta \phi \bigg|^{t_2}_{t_1}-\int^{t_2}_{t_1}\color{red}{\frac{\partial }{\partial t}}\bigg(\frac{\partial\mathcal L}{\partial (\partial _t\phi)}\bigg)\delta \phi dt \end{equation}
This particular Lagrange density is a function of $\mathcal L=\mathcal L(\phi,\dot \phi ,\partial _x\phi)$
