  We study the Hamiltonian formalism of the non-minimally coupled Weyl
connection gravity (NMCWCG) in order to check whether Ostrogradsky
instabilities are present. The Hamiltonian of the NMCWCG theories is obtained
by foliating space-time into a real line (representing time) and
\mbox{3-dimensional} space-like hypersurfaces, and by considering the spatial
metric and the extrinsic curvature of the hypersurfaces as the canonical
coordinates of the theory. Given the fact that the theory we study contains an
additional dynamical vector field compared to the usual NMC models, which do
not have Ostrogradsky instabilities, we are able to construct an effective
theory without these instabilities, by constraining this Weyl field.
