We study the general aspects of the dynamics of an ion trapped in an ideal multipolar radio-frequency trap while interacting with a dense cold atomic gas. In particular, we explore the dynamical stability, the energy relaxation, and the characteristic harmonic motion exhibited by a trapped ${\mathrm{Yb}}^{+}$ ion in different multipolar potentials and immersed in various cold atomic samples (Li, Na, Rb, Yb). For this purpose, we use two different molecular-dynamics simulations, one based on a time-event drive algorithm and the other based on the stochastic Langevin equation. Relevant values for experimental realizations, such as the associated ion's lifetimes and observable distributions, are presented along with some analytical expressions that relate the ion's dynamical properties with the trap parameters.