Symmetries and Conservation Laws of a System of Timoshenko Beam Type with Smooth Coefficients

Abstract

Recently Yoon, Ru and Mioduchowski have introduced a model based on the classical Timoshenko beam theory, describing the propagation of transverse waves in double-wall carbon nanotubes regarded as a system of two separate nested tubes interacting via van der Waals forces. In the present work, we study the group properties of a system of equations generalizing the governing equations of the aforementioned model in which some of the coefficients are assumed to depend on the spatial variable. The full group consisting of all local point one parameter smooth automorphisms admitted by the regarded system is established. Next, the subgroup of those of them that leave invariant the functional whose Fréchet gradient (Euler-Lagrange equations) is exactly the regarded system of equations is obtained.

Finally, all conservation laws bijective to the set of the divergence symmetries of the foregoing functional are determined.