  Using a variational approach based on a Lagrangian formulation and Gaussian
trial functions, we derive a simple dynamical system that captures the main
features of the time-dependent Schr\"odinger-Newton equations. With little
analytical or numerical effort, the model furnishes information on the ground
state density and energy eigenvalue, the linear frequencies, as well as the
nonlinear long-time behaviour. Our results are in good agreement with those
obtained through analytical estimates or numerical simulations of the full
Schr\"odinger-Newton equations.
