CLASSIFICATION OF MOTIONS ADMITTING LINEARIZATION OF BOUNDARY-CONDITIONS FOR THE RELATIVISTIC STRING WITH MASSIVE ENDS

For the relativistic string with masses at its ends, a classification of motions (world surfaces) admitting a parametrization under which th e equations of motion and the boundary conditions are linear (due to t he fact that the parameter for the trajectories of the string ends is proportional to the natural parameter) is carried out. These motions c an be represented as Fourier series with respect to the eigenfunctions of a generalized Sturm-Liouville problem. The completeness of the fam ily of these eigenfunctions in class C is proved. It is shown that in Minkowski spaces of dimensions 2 + 1 and 3 + 1, the motions in questio n 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.