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Using distributions is a trick that people use whenever dealing with systems that do not have the required smoothness and integrability conditions and is in general only a mathematical technique to nevertheless solve those problems. Maxwell equations, to start with, require both sides to be differentiable (at least a few times) and integrable and when you ...

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$\partial \vec{B}/\partial t \neq 0$ on the surface, but you don't need it to be zero. By your logic, you have $$(E_{1t} - E_{2t}) l = \iint_S [\nabla \times \vec{E}] . \vec{n}\, d S = - \iint_S \left[ \frac{\partial \vec{B}}{\partial t} \right] . \vec{n}\, d S = - \frac{d}{dt} \left[ \iint_S \vec{B} . \vec{n}\, d S \right] = - \frac{d\Phi}{dt}.$$ ...

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The tangential electric field is just the electric field projected onto the surface. So the requirement of continuity means that the projections of $\mathbf{E_{1}}$ and $\mathbf{E_{2}}$ onto the surface must be equal, which is what the equation describes. If the surface in question is the $xy$-plane, and $\mathbf{\hat n}$ is oriented in the $z$-direction, ...

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It only is a symmetry when $F$ is independent of $\dot q$ (disregarding more complicated cases like Qmechanic mentions), exactly for the reason that you state: only then the equations of motion remain the same. Your possible solutions in order: Do we have to introduce boundary conditions so that q˙ is not varied at the endpoints as well? This seems ...

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OP is asking about (1) Euler-Lagrange (EL) equations and (2) Noether's (first) theorem. 1) Let us start with EL eqs. OP is pondering what happens to EL eqs. if we change the Lagrangian $L$ with a total time derivative $$\tag{1} \tilde{L}~:=~L+\frac{dF}{dt}$$ in various settings. Often we assume that Lagrangians do not depend on \$\ddot{q},\dddot{q}, ...

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