On stationary oscillations of galaxies
This paper describes a general investigation of stationary oscillations of galaxies. It begins with a linear analysis of modes of oscillation with continuous spectra of real frequencies. Such modes are gravitational analogues of the van Kampen modes of oscillation in plasmas. The characteristic value problem governing modes of the van Kampen type in a galaxy is solved with the aid of a modified version of the matrix method of Kalnajs in which the perturbation of the distribution function is expressed in terms of generalized functions. In general, there is no characteristic equation governing the frequencies in the continuous spectrum. However, isolated frequencies in the continuous spectrum do satisfy a characteristic equation which, for stellar systems, is a counterpart of the dispersion relation proposed by Vlasov for plasma oscillations. The linear analysis also provides a characteristic equation for modes with a discrete spectrum of real and/or complex frequencies. The second part of the paper describes a perturbation theory for a stationary oscillation of a galaxy with a small but finite amplitude. Integrals of the stellar motion are constructed with the aid of canonical perturbation theory and used in conjunction with the theorem of Jeans in order to specify the density of stars in the six-dimensional phase space. These oscillations are slightly nonlinear counterparts of the modes of the van Kampen type, and they are stellar-dynamical counterparts of the nonlinear plasma waves described by Bernstein, Greene, and Kruskal. Fully nonlinear models of stationary oscillations of galaxies can be constructed with the aid of Schwarzschild's numerical method for the solution of the fundamental integral equation describing the self-consistency of a stellar system.