($T_{\rm rot} = 3.6-4.0$), and further, these stars are dissociated from the original arm ($T_{\rm rot} = 4.2-4.6$). This behavior is similar to that of the “density wave” from a certain viewpoint, but the spiral arms are short-lived and co-rotating (left panel of Figure \[fig:PatternSpeed\]). Thus, even if the arms appeared in the early linear phase, they are not completely explained by the picture of stationary density waves. The middle column of images in Figure \[fig: orbit\_linear\] shows the plots of the azimuth angle ($\phi$) versus the angular momentum $L_{z}$ curve instead of the $\phi-R$ plot. It is clear that all the stars along the $\phi-L_{z}$ plane oscillate horizontally when the angular momentum of each star is conserved. The images in the right column of Figure \[fig: orbit\_linear\] show the so-called Lindblad diagram, where the angular momentum $L_{z}$ of each star is plotted against its total energy $E$. Most of the stars show no significant movement from their original position, thereby suggesting that their energy is also conserved.

On the other hand, the behavior of the stars is very different in the non-linear phase, where the spiral arms are well-developed and non-steady (refer to previous sections). Figure \[fig:EvolutionSpiral\] shows images 