  Birkhoff normal forms are commonly used in order to ensure the so called
"effective stability" in the neighborhood of elliptic equilibrium points for
Hamiltonian systems. From a theoretical point of view, this means that the
eventual diffusion can be bounded for time intervals that are exponentially
large with respect to the inverse of the distance of the initial conditions
from such equilibrium points. Here, we focus on an approach that is suitable
for practical applications: we extend a rather classical scheme of estimates
for both the Birkhoff normal forms to any finite order and their remainders.
This is made for providing explicit lower bounds of the stability time (that
are valid for initial conditions in a fixed open ball), by using a fully
rigorous computer-assisted procedure. We apply our approach in two simple
contexts that are widely studied in Celestial Mechanics: the H\'enon-Heiles
model and the Circular Planar Restricted Three-Body Problem. In the latter
case, we adapt our scheme of estimates for covering also the case of resonant
Birkhoff normal forms and, in some concrete models about the motion of the
Trojan asteroids, we show that it can be more advantageous with respect to the
usual non-resonant ones.
