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

TL;DR

This paper derives integrals of motion for a relativistic string with masses at ends, using geometric invariants, and presents exact solutions for periodic torsion functions in three-dimensional Minkowski space.

Contribution

It formulates boundary equations in terms of geometric invariants and finds constants of motion and exact solutions for periodic torsion functions of the string ends.

Findings

Curvatures of string end trajectories are constant, $K_i = rac{ ext{constant}}{m_i}$.

Torsions satisfy differential equations with periodic solutions.

Constants of motion are identified for periodic torsion functions.

Abstract

Boundary equations for the relativistic string with masses at ends are formulated in terms of geometrical invariants of world trajectories of masses at the string ends. In the three--dimensional Minkowski space $E^1_2$, there are 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 system of differential equations with deviating arguments. For these equations with periodic $\kappa_i(\tau+n l)=\kappa(\tau)$, 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 $\sigma \ (\sigma_1 = 0, \sigma_2 =l)$.