residual form of the CBE by integration over the action-angle space and deriving a system of nonlinear ordinary differential equations (ODEs) for the amplitude functions. The associated first order system of ODEs leads to a linear eigenvalue problem, which is solved using standard numerical methods.

In this paper I present my new method and apply it to explore the modal properties of a model galaxy. In a second paper, I will address the nonlinear evolution of modes and wave interactions. The paper is organized as follows. In sections \[sec::dynamical-theory\] and \[sec::linear-theory\], I use the Petrov-Galerkin method to project the CBE to a system of ODEs in the time domain and derive a system of linear eigenvalue equations. Section \[sec:modes-exp-disk\] presents the eigenfrequency spectra and their corresponding mode shapes of the cored exponential disk of JH. The stars of this model move in the field of a soft-centered logarithmic potential. I also investigate the effect of physical parameters of the equilibrium model on the modal content. In section \[sec:final-discussions\], I discuss on the nature of a self-gravitating mode and compare the performance of my method with other theories. Some fundamental achievements of this work are summarized in section \[sec:conclusions\].

NONLINEAR THEORY 