Dynamical Ages of the Young Local Associations with $\textit{Gaia}$
  The Young Local Associations constitute an excellent sample for the study of
a variety of astrophysical topics, especially the star formation process in
low-density environments. Data from the Gaia mission allows us to undertake
studies of the YLAs with unprecedented accuracy. We determine the dynamical age
and place of birth of a set of associations in a uniform and dynamically
consistent manner. There are nine YLAs in our sample $\epsilon$ Chamaeleontis,
TW Hydrae, $\beta$ Pictoris, Octans, Tucana-Horologium, Columba, Carina, Argus
and AB Doradus. We designed a method for deriving the dynamical age of the YLAs
based on the orbital integration. It involves a strategy to account for the
effect of observational errors and we tested it using mock YLAs. Finally, we
applied it to our set of nine YLAs with astrometry from the first Gaia data
release and complementary on-ground radial velocities from the literature. Our
orbital analysis yields a first estimate of the dynamical age of 3$^{+9}_{-0}$
Myr, 13$^{+7}_{-0}$ Myr and 5$^{+23}_{-0}$ Myr for $\epsilon$ Cha, $\beta$ Pict
and Tuc-Hor, respectively. For four other associations (Oct, Col, Car and Arg),
we provide a lower limit for the dynamical age. Our rigorous error treatment
indicates that TW Hya and AB Dor deserve further study. Our dynamical ages are
compatible spectroscopic and isochrone fitting ages obtained elsewhere. From
the orbital analysis, we suggest a scenario with two episodes of star
formation: one ~40 Myr ago in the first quadrant that gave birth to $\epsilon$
Cha, TW Hya and $\beta$ Pic, and another 5-15 Myr ago close to the Sun that
formed Tuc-Hor, Col, and Car. Future Gaia data will provide the necessary
accuracy to improve the present results, especially for the controversial age
determinations, and additional evidence for the proposed scenario, once a
complete census of YLAs and better membership can be obtained.
