The chaotic behavior of a three-dimensional swinging Atwood’s machine system with additional Coulomb interactions is studied using geometric methods. The stability of dynamics, related to curvature properties of the configuration space manifold, is investigated through the Jacobi-Levi-Civita (JLC) equation for geodesic spread.The scalar curvature of the Atwood’s machine system under the given Jacobi metric is calculated. Meanwhile, the qualitative information provided by the Poincaré sections are compared with the results of the geometric investigation, and they are completely consistent.The results of numerical calculations indicate that in the swinging Atwood’s machine system with additional Coulomb interactions, the curvature along geodesics on the configuration space manifold fluctuates with positive values in chaotic regions.