  Dynamics and collapse of collisionless self-gravitating systems is described
by the coupled collisionless Boltzmann and Poisson equations derived from
$f(R)$-gravity in the weak field approximation. Specifically, we describe a
system at equilibrium by a time-independent distribution function $f_0(x,v)$
and two potentials $\Phi_0(x)$ and $\Psi_0(x)$ solutions of the modified
Poisson and collisionless Boltzmann equations. Considering a small perturbation
from the equilibrium and linearizing the field equations, it can be obtained a
dispersion relation. A dispersion equation is achieved for neutral
dust-particle systems where a generalized Jeans wave-number is obtained. This
analysis gives rise to unstable modes not present in the standard Jeans
analysis (derived assuming Newtonian gravity as weak filed limit of $f(R)=R$).
In this perspective, we discuss several self-gravitating astrophysical systems
whose dynamics could be fully addressed in the framework of $f(R)$-gravity.
