Consider the classical problem of extremizing a functional of the form $$S[x] = \int_a^b L\left(t,x,\dot{x}\right)\ dt.$$ In almost all cases of consideration, the integrand $L$ is considered to be a $C^2$ function of its variables and in that case the condition for an extremal is given by the Euler-Lagrange equations: $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) - \frac{\partial L}{\partial x} = 0.$$ Is there anything we can say in the case that $L$ is discontinuous (say piecewise $C^2$)? Note that I am not asking about broken extremals, i.e. solutions $x(t)$ which are piecewise smooth. I am asking whether we can generalize the Euler-Lagrange equations to the case where $L$ is discontinuous.
Motivation: The reason I am asking this question is in the context of Lagrangian optics. In that case, the equations of motion are given by extremizing the optical path length given by $$S[x] = \int_a^b n(x(t))\ dt,$$ where $n(x)$ is the refractive index as a function of position. In many cases of interest (such as a lens) the refractive index is a discontinuous function of position.