Bm. Barbashov et Gs. Sharov, INITIAL-BOUNDARY-VALUE PROBLEM FOR THE RELATIVISTIC STRING WITH MASSIVE ENDS, Theoretical and mathematical physics, 101(2), 1994, pp. 1332-1345
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 for
m of specification of the initial position and initial velocities of t
he points of the string. An investigation is made of the connection be
tween 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 sol
ution 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 syst
em of ordinary differential equations. Specific examples are considere
d.