Application of Lie Transformation Group Methods to Classical Theories of Plates and Rods

Abstract

In the present paper, a class of partial differential equations related to various plate and rod problems is studied by Lie transformation group methods. A system of equations determining the generators of the admitted point Lie groups (symmetries) is derived. A general statement of the associated group-classification problem is given. A simple intrinsic relation is deduced allowing to recognize easily the variational symmetries among the ''ordinary'' symmetries of a self-adjoint equation of the class examined. Explicit formulae for the conserved currents of the corresponding (via Noether's theorem) conservation laws are suggested. Solutions of group-classification problems are presented for subclasses of equations of the foregoing type governing stability and vibration of plates, rods and fluid conveying pipes resting on variable elastic foundations and compressed by axial forces. The obtained group-classification results are used to derive conservation laws and group-invariant solutions readily applicable in plate statics or rod dynamics.