  We investigate the coupling of matter to geometry in conformal quadratic Weyl
gravity, by assuming a coupling term of the form $L_m\tilde{R}^2$, where $L_m$
is the ordinary matter Lagrangian, and $\tilde{R}$ is the Weyl scalar. The
coupling explicitly satisfies the conformal invariance of the theory. By
expressing $\tilde{R}^2$ with the help of an auxiliary scalar field and of the
Weyl scalar, the gravitational action can be linearized, leading in the Riemann
space to a conformally invariant $f\left(R,L_m\right)$ type theory, with the
matter Lagrangian nonminimally coupled to the Ricci scalar. We obtain the
gravitational field equations of the theory, as well as the energy-momentum
balance equations. The divergence of the matter energy-momentum tensor does not
vanish, and an extra force, depending on the Weyl vector, and matter Lagrangian
is generated. The thermodynamic interpretation of the theory is also discussed.
The generalized Poisson equation is derived, and the Newtonian limit of the
equations of motion is considered in detail. The perihelion precession of a
planet in the presence of an extra force is also considered, and constraints on
the magnitude of the Weyl vector in the Solar System are obtained from the
observational data of Mercury. The cosmological implications of the theory are
also considered for the case of a flat, homogeneous and isotropic
Friedmann-Lemaitre-Robertson-Walker geometry, and it is shown that the model
can give a good description of the observational data for the Hubble function
up to a redshift of the order of $z\approx 3$.
