of the perpendicular velocity ($\Delta v_{\eta}$), since the Coriolis force works perpendicular to the direction of the velocity.

This indicates that the Coriolis force exerted on the stars in the damping phase exceeds the non-axisymmetric gravitational force due to the spiral perturbation. This causes stars to “escape” from the spiral perturbation, and eventually the spiral arm itself begins to thin out and fade.

The above argument suggests that the non-steady nature of stellar spirals is originated in the evolution of the orbits of the stars in the spirals. The phenomenon of swing amplification [@Toomre1981] is a part of this non-linear coupling between particles and waves, but it does not describe all observed phenomena. In the next section, we explore the orbital evolution of stars associated with the spiral arm in detail.

Orbital Evolution of Stars Around Spiral Arms {#sec:StarAroundSpiralArm}
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Figure \[fig: orbit\_linear\] shows the evolution of stars along the $\phi-R$ plane in the early phase of spiral development. At this stage of the simulation, we selected 15 particles associated with one of the three weak spiral arms that had evolved at $T_{\rm rot} = 4.0$. The figure shows that stars with epicycle motion are captured by the density enhancement 