  We consider a system of spinless fermions in a nearly one-dimensional
cylindrical trap, with the Fermi level such that only the lowest-lying states
with angular momenta l = +hbar, 0, or -hbar about the axis of the trap are
occupied. We treat the particles in these states as comprising three separate
Luttinger liquids, with the possibility for forward and backward scattering
between them. We determine the normal modes in the system in the presence of
this scattering, finding two sets of linear harmonic fluid modes and one set of
modes that are self-coupled through a sine-Gordon interaction.
