Ubiquity of the Sturm-Liouville problem in multilayer systems

We postulate a general Lagrangian density which leads to the equations of motion of the well-known cases of the electromagnetic field, the theory of elasticity, and some other particular problems in classical physics. When the mentioned cases are studied in multilayer systems, i.e. , when the constitutive parameters depend on one of the Cartesian coordinates, all of them lead us to a matrix Sturm-Liouville problem whose equation of motion, in its most general form, can be derived from the postulated Lagrangian density too. These results demonstrate the ubiquitous character of the Sturm-Liouville matrix problem and consequently the relevance of its study from the mathematical, physical and numerical point of view. The consequences of this curious result are briefly analyzed.