should one use fine grids in the action space. Polyachenko’s method is somehow costly because it results in a large linear system of equations to assure point-wise convergence in the action space and mean convergence of Fourier expansions in the space of the radial angle. Extension of his method to nonlinear regime is another challenging problem yet to be investigated. @T05 has also followed an approach similar to @P05 and studied the instability of stellar disks surrounding massive objects. His eigenvalue equations involve action variables, and practically, need to be solved over a discretized grid in the space of actions.

Recent developments in fluid mechanics [@DG95; @MS99] inspired me to formulate the dynamics of stellar systems in a new framework, which is capable of solving the CBE not only in the linear regime, but also in its full nonlinear form. The method systematically searches for smooth solutions of the CBE by expanding the perturbed DF using Fourier series of angle variables and an appropriate set of trial functions in the space of actions. Coefficients of expansion are unknown time-dependent amplitude functions whose evolution equations are obtained by the Petrov-Galerkin projection [@F72] of the CBE. That is indeed taking the weighted 