Geodesic
Initial-boundary problem for the relativistic string with massive ends
Teoretičeskaâ i matematičeskaâ fizika, Tome 101 (1994) no. 2, pp. 253-271

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It is shown that the initial-boundary-value problem for a relativistic string with masses at the ends can be solved for the most general form of specification of the initial position and initial velocities of the points of the string. An investigation is made of the connection between the freedom in the parametrization of the initial curve and the reparametrization invariance preserving linearity of the equations of motion of the string. The posed problem is solved by extending the solution determined by the initial conditions from the restricted initial region to the entire world surface of the string by means of boundary conditions of various types: the mass at a given end is equal to zero, is infinitely large, or is finite. In the last case it is shown that the problem of the extension reduces to the solution of a normal system of ordinary differential equations. Specific examples are considered. 
@article{TMF_1994_101_2_a7,
     author = {B. M. Barbashov and G. S. Sharov},
     title = {Initial-boundary problem for the relativistic string with massive ends},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {253--271},
     publisher = {mathdoc},
     volume = {101},
     number = {2},
     year = {1994},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1994_101_2_a7/}
}
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B. M. Barbashov; G. S. Sharov. Initial-boundary problem for the relativistic string with massive ends. Teoretičeskaâ i matematičeskaâ fizika, Tome 101 (1994) no. 2, pp. 253-271. http://geodesic.mathdoc.fr/item/TMF_1994_101_2_a7/