Action at a distance and equations of motion of a system of two massive points connected by a relativistic string

Dynamical equations in the theory of a relativistic string with point masses at the ends are formulated solely in terms of geometrical invariants of the worldlines of the massive ends of the string. In three-dimensional Minkowski space $\mathbf E_2^1$ , these invariants – the curvature $k$ and torsion $\varkappa$ – make it possible to completely recover the world surface of the string up to its position as a whole. It is shown that the curvatures $k_i$, $i=1,2$, of the trajectories are constants that depend on the string tension and the masses at its ends, while the torsions $\varkappa_i(\tau)$, $i=1,2$, satisfy a system of second-order differential equations with shifted arguments. A new exact solution of these equations in the class of elliptic functions is obtained.