along the curve, thereby leading to the formation of structures referred to as “swarms of stars” along the $\phi-L_{z}$ and $R-\phi$ planes. The non-steady nature of the spiral arms originates in the dynamical interaction between these “swarming” stars with a [*non-linear*]{} epicycle motion and the high-density regions, i.e., the spiral arms moving with the galactic rotation. This is similar to the wave-particle interaction described previously; however, in this case, the high-density regions are not “waves”. Since the non-steady spirals move at the rate of the local galactic rotational speed (Section 3), in contrast to a spiral perturbation with a single pattern speed, co-rotating points are found everywhere on the spiral arms. Therefore, the motion of stars along the $E-L_{z}$ plane can be naturally understood as due to their scattering around co-rotating spirals [^4]. This behavior is similar to reported in previous studies [e.g., @SellwoodBinney2002; @Grand+2012; @Bird+2011; @Grand+2012b]. Furthermore, we observe that the $E-L_{z}$ curve of each star changes over a large range of radii. This is entirely different from what is expected in stationary density waves, where these changes are limited to the Lindblad resonances [@LyndenBellKalnajs1972]. Further, it is noteworthy that the structures formed self-induced in the angular momentum space; 