developed in the galactic disk. In Section \[sec:SpiralDynamics\], we focus on a typical spiral arm and describe its dynamical evolution, i.e., the amplification (Section \[sec:AmplifySpiralArm\]) and destruction (Section \[sec:DampingSpiralArm\]) processes. Section \[sec:StarAroundSpiralArm\] examines the evolution of stars around the growing spiral arm from the viewpoint of the angular momentum-energy space. Finally, in Section \[sec:Discussion\], we compare our simulation results with observations, and we provide a perspective view on grand-design spirals. The grand-design spirals (i.e., $m=2$ spirals) [^2] will be the subject of our future studies.

Models and Methods {#sec:Method}
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The simulations of the formation and evolution of spiral arms were carried out using a three-dimensional $N$-body simulation. The initial point of our simulation was a disk galaxy in a nearly equilibrium state. The density profile of the stellar disk in this case is given by $$\rho_{\rm sd}(R,z) = \frac{M_{\rm sd}}{4\pi {R_{\rm sd}}^2 z_{\rm sd}}
\exp(-R/R_{\rm sd}) {\rm sech}^2(z/z_{\rm sd}),\label{eq:stellardisk}$$ where $M_{\rm sd}$ is the mass of the stellar disk, $R_{\rm sd}$ is the radial scale-length, and $z_{\rm sd}$ is the vertical scale-length. Since the distribution function of a stellar disk is unknown, we generated the equilibrium state of the stellar disk by using an empirical method: We first generated 