Lagrangian perturbations of Riemann ellipsoids

Abstract

Abstract The spectral method previously introduced for the solution of the Lagrangian perturbation equations of rotating, inviscid fluids is applied to the problem of the stability of a class of Riemann ellipsoids. The description of the basis previously outlined is completed and shown to have two particularly desirable features in the present context: it respects a further invariance of the linear operators unique to the present problem, and it diagonalizes the operator representing the perturbation of the self-gravitational potential. This diagonalization property holds both when the fluid is assumed incompressible and, as in the gasdynamic case considered here, when it is compressible. The infinite-dimensional perturbation equations reduce under the spectral method to a hierarchy of finite-dimensional systems, each having a canonical structure familiar from classical mechanics. This reduction and the nature of the finite-dimensional systems are outlined.