Vibrational modes of a rotating string

Motivated by nonrelativistic models of a QCD (quantum chromodynamic) string, we examine the system of a nonrelativistic string in uniform rotational motion with one end fixed and with a mass (quark) attached to the other end. A QCD string has no purely longitudinal modes, so some constraint must be imposed upon the nonrelativistic system to exclude these modes. Accordingly, we examine the cases that the string is either inextensible or purely transverse. For each case we solve first the discretized string and then do the continuum case. We find the small amplitude oscillatory motions and frequencies of oscillation of the string and mass. We show that the assumption of a node at the end of the inextensible string produces frequencies that are very close to the actual frequencies. For the transverse string, we show that keeping centrifugal terms for the motion of the masses comprising the string leads to an unstable mode.