Negative Energy and Angular Momentum Modes of Thin Accretion Disks
  This work derives the linearized equations of motion, the Lagrangian density, the Hamiltonian density, and the canonical angular momentum density for general perturbations [$\propto \exp(im\phi)$ with $m=0,\pm 1,..$] of a geometrically thin self-gravitating, homentropic fluid disk including the pressure. The theory is applied to ``eccentric,'' $m=\pm 1$ perturbations of a geometrically thin Keplerian disk. We find $m=1$ modes at low frequencies relative to the Keplerian frequency. Further, it shown that these modes can have negative energy and negative angular momentum. The radial propagation of these low frequency $m=1$ modes can transport angular momentum away from the inner region of a disk and thus increase the rate of mass accretion. Depending on the radial boundary conditions there can be discrete low-frequency, negative-energy, $m=1$ modes. 
L. Zhang, R. V. E. Lovelace