Positive Lynden-Bell derivative as a ticket to the bar trap?

We have translated the results of N-body simulations of one barred model into the language of action variables and frequencies. Using this language, we analysed the behaviour of all orbits in the model on a large time scale at the stage of a mature bar. We show that the orbits join the bar while preserving their adiabatic invariant, which takes into account the 3D structure of the orbits. This allows us to apply the concept of the Lynden-Bell derivative for each of these orbits and trace how the sign of the derivative changes, i.e. how asynchronous changes in angular momentum L_z and orbital precession rate Ω_pr (normal orbital mode) change to synchronous (abnormal mode). The transition to the abnormal mode occurs when Ω_pr reaches the angular velocity of the pattern Ω_p, after which the orbit becomes stuck in the bar trap. All this happens against the background of secular changes in actions (L_z decreases, J_R and J_z increase). At the same time, corotation particles near two stable Lagrange points are also subject to secular changes in their actions. They increase L_z and drift to the periphery, shifting corotation outwards. We also show that a change in the orbital mode from normal to abnormal and the trapping of orbits in a bar is possible only when the bar speed decreases with time, regardless of what is causing the bar to slow down. Our findings clarify and expand the picture of bar formation and evolution in numerical models.