  It has been suggested that the gravitational potential can have a significant
role in suppressing the star formation in the nearby galaxies. To establish
observational constrains on this scenario, we investigate the connection
between the dynamics, through the circular velocity curves (CVCs) as a proxy of
the inner gravitational potential, and star formation quenching in 215
non-active galaxies across Hubble sequence from the Calar Alto Legacy Integral
Field Area (CALIFA) survey. Our results show that galaxies with similar CVCs
tend to have a certain star-formation quenching pattern. To explore these
findings in more details, we construct kpc-resolved relations of the equivalent
width of the H$\alpha$ ($W_{{\rm H}\alpha}$) versus the amplitude ($V_c$) and
shape ($\beta= d\ln V_c/ d\ln R$) of the circular velocity at given radius. We
find that the $W_{{\rm H}\alpha}-V_c$ is a declining relationship, where the
retired regions of the galaxies (the ones with $W_{{\rm H}\alpha}$ values below
3 \r{A}) tend to have higher $V_c$. Differently, $W_{{\rm H}\alpha}-\beta$ is a
bi-modal relationship, characterised by two peaks: concentration of the star
forming regions at a positive $\beta$ (rising CVC) and another one of the
retired regions with a negative $\beta$ (declining CVC). Our results show that
both the amplitude of the CVC, driven by the mass of the galaxies, and the
shape of the CVC, reflecting the internal structure of the galaxies, play an
important role in galaxy's quenching history.
