  The present-day response of a Galactic disc stellar population to a
non-axisymmetric perturbation of the potential has previously been computed
through perturbation theory within the phase-space coordinates of the
unperturbed axisymmetric system. Such an Eulerian linearized treatment however
leads to singularities at resonances, which prevent quantitative comparisons
with data. Here, we manage to capture the behaviour of the distribution
function (DF) at a resonance in a Lagrangian approach, by averaging the
Hamiltonian over fast angle variables and re-expressing the DF in terms of a
new set of canonical actions and angles variables valid in the resonant region.
We then follow the prescription of Binney (2016), assigning to the resonant DF
the time average along the orbits of the axisymmetric DF expressed in the new
set of actions and angles. This boils down to phase-mixing the DF in terms of
the new angles, such that the DF for trapped orbits only depends on the new set
of actions. This opens the way to quantitatively fitting the effects of the bar
and spirals to Gaia data in terms of distribution functions in action space.
