In the process of discretely solving mathematical models, traditional numerical methods primarily focus on linear local numerical stability, making it difficult to accurately describe the system’s dynamic characteristics over long periods. This paper uses Lie groups to describe the configuration of rigid body motion and defines two variational formulas on Lie groups. By applying the discrete Hamiltonian variational principle and the discrete Legendre transformation, we derive the general forms of Lie group variational integrators and Hamel variational integrators for the Hamiltonian system of the rigid body. We simulate the 3D car pendulum model using both of these Lie group variational integrator algorithms and compare their performance in preserving the system’s group structure and energy. Simulation results show that the Hamel variational integrator achieves higher accuracy than the general form of the Lie group variational integrator and better preserves the system’s group structure and energy.