The production of light nuclei and hyper-nuclei in heavy-ion collisions, particularly at high baryon density, is crucial for understanding the dynamical evolution of the collision system and exploring the internal state of nuclear matter of compacted stellar object. Despite being a topic of ongoing debate, an improved theoretical understanding is necessary. In this work, production of light nuclei ($d$, $t$, $^{3}$He, $^{4}$He) and hyper-nuclei ($^{3}_{\Lambda}$H, $^{4}_{\Lambda}$H) was investigated using the JAM microscopic transport model combined with an afterburner coalescence process at $\sqrt{s_{\text{NN}}} =$ 3 GeV Au+Au collisions. The formation of a specific nucleus during the coalescence process is determined by its Wigner function. The comparison of the calculations for $\mathrm{p_T}$ spectra, average $\mathrm{p_T}$, and rapidity distributions to the measurements from the STAR experiment was performed. We investigated the dynamic information carried by light nuclei and determined the averaged spatial distance $\langle \Delta R \rangle$ and momentum difference $\langle \Delta P \rangle$ of constituent nucleons ($\Lambda$) for each nucleus species.
