On canonical methods for the study of small perturbations in galaxies

Abstract

This paper describes a ‘canonical representation’ for the study of small perturbations in galaxies. The canonical representation is formulated in terms of the generating function for the canonical transformation that relates the perturbed coordinates and momenta of a star to the unperturbed coordinates and momenta. The fundamental equation of the canonical representation is a linear, partial differential equation governing the first-order part of the generating function. The Eulerian and Lagrangian representations of small perturbations are derived from the canonical representation. The characteristic value problem governing modes of oscillation of a galaxy is formulated in both the canonical and the Eulerian representations, and the two formulations are shown to be mutually adjoint. Variational principles are constructed for the solution of the characteristic value problem governing modes in the canonical and Eulerian representations, and the variational principles in the two representations are shown to be equivalent. The variational principles are made bases for the formulation of matrix methods of the Rayleigh‐Ritz type for the solution of the characteristic value problem. An earlier formulation of the characteristic value problem governing modes of oscillation of a galaxy in the Lagrangian representation is reviewed, and it is shown that the canonical, Lagrangian and Eulerian representations of the characteristic value problem are equivalent. The canonical representation of perturbations also arises in formulations of the characteristic value problem governing the modes of oscillation of a collisionless plasma. In particular, the canonical representation is implicit in Case’s formulation of the characteristic value problem governing the van Kampen modes in a homogeneous plasma. Ke yw ords: instabilities ‐ stellar dynamics ‐ methods: analytical ‐ galaxies: kinematics and dynamics ‐ galaxies: structure.