Geometrical method of solving the boundary-value problem in the theory of a relativistic string with masses at its ends

A differential-geometric formulation of the dynamics of a relativistic string with masses at its ends is considered in the Minkowski space $E_2^1$. The surface swept out by the string is described by differential forms and is bounded by two curves – the worldlines of its massive ends. These curves have a constant geodesic curvature, and their torsion is determined only up to an arbitrary function on the interval $[0,2\pi]$. Equations are obtained that determine the world surface of the string as a function of the curvature and torsion of the trajectories of its massive ends. For the choice of the constant torsions for which the mass points move along helices, the surface of the relativistic string is a helicoid.