Classification of motions admitting linearization of boundary conditions for the relativistic string with massive ends

Abstract

For the relativistic string with masses at its ends, a classification of motions (world surfaces) admitting a parametrization under which the equations of motion and the boundary conditions are linear (due to the fact that the parameter for the trajectories of the string ends is proportional to the natural parameter) is carried out. These motions can be represented as Fourier series with respect to the eigenfunctions of a generalized Sturm-Liouville problem. The completeness of the family of these eigenfunctions in class C is proved. It is shown that in Minkowski spaces of dimensions2+1 and3+1, the motions in question reduce to a uniform rotation of one or several (spatially coincident) rectilinear strings (the world surface is a helicoid). In spaces of higher dimensions, some other nontrivial motions of this type are also possible.