The action functional for a linear elastic medium with dislocations is given. The equations of motion following from this action reproduce the Peach-K{o}hler and Lorentzian forces experienced by dislocations. The explicit expressions for singular and finite parts of the self-force acting on a curved dislocation are derived in the framework of linear theory of elasticity of an isotropic medium. The velocity of dislocation is assumed to be arbitrary but less than the shear wave velocity. The nonrelativistic and ultrarelativistic limits are investigated. In the ultrarelativistic limit, the explicit expression for the leading contribution to the self-force is obtained. In the case of slowly moving dislocations, the effective equations of motion derived in the present paper reproduce the known results.
The explicit expressions for the average number of twisted photons radiated by a charged particle in an elliptical undulator in the classical approximation as well as in the approach accounting for the quantum recoil are obtained. It is shown that radiation emitted by a particle moving along an elliptical helix which evolves around the axis specifying the angular momentum of twisted photons obeys the selection rule: $m+n$ is an even number, where $m$ is a projection of the total angular momentum of a twisted photon and $n$ is the harmonic number of the undulator radiation. This selection rule is a generalization of the previously known selection rules for radiation of twisted photons by circular and planar undulators and it holds for both classical and quantum approaches. The class of trajectories of charged particles that produce the twisted photon radiation obeying the aforementioned selection rule is described.
