Phase-space distortion as a key to unraveling galactic bar buckling

For the first time, we investigate the resonant structure of $N$-body galactic bar at the stage of buckling using action-angle variables. We studied the evolution of vertical actions ($J_z$) and angles associated with vertical resonance ($\theta_\mathrm{res}=\theta_z - \theta_R$) for all orbits in the bar. For this purpose, we divide the orbits into types according to the behavior (libration or circulation) of their resonant angle with respect to fixed points $\theta_\mathrm{res}=0$ and $\pi$ (vertical resonance). We show that during buckling, flat bar orbits circulating with increasing $\theta_\mathrm{res}$ transformed into banana-shaped librating orbits (resonant capture) or circulating orbits with decreasing $\theta_\mathrm{res}$ (resonant heating). The orbital transformation is accompanied by an increase in $J_z$ and the formation of a boxy/peanut-shaped (B/PS) bulge. During buckling, the phase space $J_z - \theta_\mathrm{res}$ undergoes a distortion creating an asymmetry in the position of the fixed points $\theta_\mathrm{res}=0$ and $\pi$ and in banana-shaped orbits near these points. The fixed point $\theta_\mathrm{res}=0$ may disappear completely. This also breaks the symmetry between the orbits, which are captured into resonance or go into circulation with decreasing $\theta_\mathrm{res}$ near $\theta_\mathrm{res}=0$ and $\pi$. At the same time, near $\theta_\mathrm{res}=0$, banana-shaped orbits with low vertical action $J_z$ appear. This reopens the path of orbital transformation through the fixed point $\theta_\mathrm{res}=0$. The phase space transformation and orbit transformation occur in a coordinated manner and lead to smoothing of phase space perturbations and restoration of symmetry between orbits.