Symmetries and Conservation Laws of a System of Timoshenko Beam Type Equations

The paper is concerned with a system of equations of Timoshenko beam type describing free transverse vibration of a double-wall carbon nanotube regarded as a system of two separate nested nanotubes interacting via van der Waals forces. The problem is formulated in an exact variational statement. Then, the point Lie symmetry groups admitted by the governing equations and the associated functional are established. Finally, we derive the densities and fluxes of the conservation laws corresponding, by virtue of Bessel-Hagen’s extension of Noether’s theorem, to the admitted variational and divergence symmetries as well as the reciprocity relation.