Context : The Young Local Associations ( YLAs ) 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 .

Aims : 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 .

Methods : We designed a method for deriving the dynamical age of the YLAs based on the orbital integration .
The method involves a strategy to account for the effect of observational errors .
We tested the method 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 .

Results : 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 Chamaeleontis , \beta Pictoris , and Tucana-Horologium , respectively .
For four other associations ( Octans , Columba , Carina , and Argus ) , we provide a lower limit for the dynamical age .
Our rigorous error treatment indicates that TW Hydrae and AB Doradus deserve further study .

Conclusions : The dynamical ages that we obtain are compatible spectroscopic and isochrone fitting ages obtained elsewhere .
From the orbital analysis , we suggest a scenario for these YLAs where there were two episodes of star formation : one \sim 40 Myr ago in the first quadrant that gave birth to \epsilon Chamaeleontis , TW Hydrae , and \beta Pictoris , and another 5 - 15 Myr ago close to the Sun that formed Tucana-Horologium , Columba , and Carina .
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 .