On a matrix method for the study of small perturbations in galaxies

A matrix method is formulated in a Lagrangian representation for the solution of the characteristic value problem governing modes of oscillation and instability in a collisionless stellar system. The underlying perturbation equations govern the Lagrangian displacement of a star in the six-dimensional phase space. This matrix method has a basis in a variational principle. The method is formulated here for modes with a discrete spectrum of frequencies, and it is developed in detail for radial oscillations of a spherical system with a finite radius. A set of basis vectors suitable for the representation of the Lagrangian displacement in this case is derived from solutions of the Lagrangian perturbation equations for radial perturbations in a homogeneous sphere. The basis vectors are made divergence free in the six-dimensional phase space in accordance with the requirement of Liouville's theorem that the flow of the system in the phase space must be incompressible. The basis vectors are made orthogonal with respect to a properly constructed set of adjoint vectors with the aid of a Gram-Schmidt procedure. Some basis vectors are null vectors in the sense that their inner products with their own adjoint vectors vanish. The characteristic frequencies of the lowest radial modes are calculated in several approximations for the stellar-dynamical counterparts of gaseous polytropes, which span a considerable range of central concentrations. The present formulation of the matrix method can be generalized, for example, for non-radial modes in spherical systems and for modes in axisymmetric systems, with the aid of other, suitably constructed sets of basis vectors.