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

Distribution functions for resonantly trapped orbits in the Galactic disc