  We investigate the radiation from a charge rotating around conductors with
cylindrical symmetry. First the problem is considered with a charge rotating
around a conducting cylinder immersed in a homogeneous medium. The surface
charge and current densities induced on the cylinder surface are evaluated. A
formula is derived for the spectral-angular density of the radiation intensity.
In the second part, we study the radiation for a charge rotating around a
diffraction grating on a cylindrical surface with metallic strips parallel to
the cylinder axis. The effect of the grating on the radiation intensity is
approximated by the surface currents induced on the strips by the field of the
rotating charge. The expressions are derived for the electric and magnetic
fields and for the angular density of the radiation intensity on a given
harmonic. We show that the interference between the synchrotron and
Smith-Purcell radiations may lead to interesting features. In particular, the
behavior of the radiation intensity on large harmonics can be essentially
different from that for a charge rotating in the vacuum or around a solid
cylinder. Unlike to these limiting cases, for the geometry of diffraction
grating the radiation intensity on higher harmonics does not vanish for small
angles with respect to the cylinder axis. For given characteristics of the
charge, by the choice of the parameters of the diffraction grating, one can
have highly directional radiation near the normal to the plane of the charge
rotation. With decreasing energy, the relative contribution of the synchrotron
radiation decreases and the Smith-Purcell part is dominant.
