Ubiquity of the Sturm-Liouville problem in multilayer systems
1 Centro de Investigación en Ciencias, Universidad Autónoma del Estado de Morelos - Ave. Universidad 1001, C.P. 62209, Cuernavaca, Morelos, México
2 Instituto de Física, Benemérita Universidad Autónoma de Puebla - Calle 18 Sur y San Claudio, Edif. 110-A, Ciudad Universitaria, C.P. 72570, Puebla, México
Received: 11 May 2016
Accepted: 22 July 2016
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.
PACS: 02.30.Xx – Calculus of variations / 02.30.Hq – Ordinary differential equations / 73.21.-b – Electron states and collective excitations in multilayers, quantum wells, mesoscopic, and nanoscale systems
© EPLA, 2016