Integrals of periodic motion and periodic solutions for classical equations of relativistic string with masses at ends. I. Integrals of periodic motion

摘要

Boundary equations for the relativistic string with masses at ends areformulated in terms of geometrical invariants of world trajectories of massesat the string ends. In the three--dimensional Minkowski space $E^1_2$, thereare two invariants of that sort, the curvature $K$ and torsion $\kappa$.Curvatures of trajectories of the string ends with masses are always constant,$K_i = \gamma/m_i (i =1,2,)$, whereas torsions $\kappa_i(\tau)$ obey a systemof differential equations with deviating arguments. For these equations withperiodic $\kappa_i(\tau+n l)=\kappa(\tau)$, constants of motion are obtained(part I) and exact solutions are presented (part II) for periods $l$ and $2l$where $l$ is the string length in the plane of parameters $\tau$ and $\sigma \(\sigma_1 = 0, \sigma_2 =l)$.