Oscillations and stability of polytropic filaments
  We study the oscillations and stability of self-gravitating cylindrically
symmetric fluid systems and collisionless systems. This is done by studying
small perturbations to the equilibrium system and finding the normal modes,
using methods similar to those used in astroseismology. We find that there is a
single sequence of purely radial modes that become unstable if the adiabatic
exponent is less than 1. Nonradial modes can be divided into p modes, which are
stable and pressure-driven, and g modes, which are are gravity driven. The g
modes become unstable if the adiabatic exponent is greater than the polytrope
index. These modes are analogous to the modes of a spherical star, but their
behavior is somewhat different because a cylindrical geometry has less symmetry
than a spherical geometry. This implies that perturbations are classified by a
radial quantum number, an azimuthal quantum number, and wavelength in the z
direction, which can become arbitrarily large. We find that decreasing this
wavelength increases the frequency of stable modes and increases the growth
rate of unstable modes. We use use variational arguments to demonstrate that
filaments of collisionless matter with ergodic distribution functions are
stable to purely radial perturbations, and that filaments with ergodic
power-law distribution functions are stable to all perturbations.
