Dynamics of flat galaxies. IV. The integral equation for normal modes in matrix form.

If the Poisson equation for the disk geometry is solved by means of biorthonormal expansions, the integral equation that determines the normal modes becomes an infinite matrix equation for the expansion coefficients. The matrix can be reduced to a sum of a real and an imaginary symmetric matrix. Purely oscillatory modes can be derived from a variational principle. All derivations are Lagrangian in nature and based on orbits described in action-angle variables. They include both direct and retrograde stars. The various components of the matrix equation have simple physical meanings that allow them to be computed in any coordinate system.