Low frequency m=1 normal mode oscillations of a self-gravitating disc
  A continuous system such as a galactic disc is shown to be well approximated by an N-ring differentially rotating self-gravitating system. Lowest order (m=1) non-axisymmetric features such as lopsidedness and warps are global in nature and quite common in the discs of spiral galaxies. Apparently these two features of the galactic discs have been treated like two completely disjoint phenomena. The present analysis based on an eigenvalue approach brings out clearly that these two features are fundamentally similar in nature and they are shown to be very Low frequency Normal Mode (LNM) oscillations manifested in different symmetry planes of the galactic disc. Our analysis also show that these features are actually long-lived oscillating pattern of the N-ring self-gravitating system. 
Kanak Saha