this is a property similar to that of the “groove” mode hypothesized by @SellwoodLin1989. This point is beyond the scope of this paper. We will investigate this elsewhere.

The time evolutions of the angular momentum of stars associated with a spiral arm at $T_{\rm rot} \simeq 4.0$ and $T_{\rm rot} \simeq 12.2$ are plotted in the left top panels of Figures \[fig:L-t04\] and \[fig:L-t12\], respectively. In the early phase ($T_{\rm rot} \simeq 4.0$), both the angular momentum and random energy do not change by more than 10% during one rotational period (top right panel of Figure \[fig:L-t04\]). On the other hand, in the non-linear phase ($T_{\rm rot} \simeq 12.2$), the fraction of the angular momentum changes by $\sim$ 50% (top right panel of Figure \[fig:L-t12\]). It is clear that the angular momentum of the stars changes significantly due to their scattering by a well-developed spiral at $T_{\rm rot} = 12.2$. This corresponds to the hypothesis that the guiding center of the epicycle motion of each star undergoes radial motion.

The change in the normalized random energy shown in the left bottom panels of Figures \[fig:L-t04\] and \[fig:L-t12\] indicates that certain stars with relatively small initial random energies experience a large 