With growing concerns over the impact threat of near-Earth small bodies and the utilization of space resources, the orbital evolution of binary asteroid systems has become an important subject in space science. When describing the full two-body dynamics of such systems, the gravitational potential of irregular bodies is determined by both their positions and attitudes, leading to coupling between translational and rotational motions, and the resulting model complexity significantly exceeds that of the classical two-body problem. Although existing numerical methods, such as explicit Runge–Kutta integrators, are widely used, they fail to preserve the symplectic structure and Lie group geometric properties of the system. In long-term simulations, key conserved quantities such as total energy and angular momentum are prone to numerical drift, severely undermining the long-term credibility of orbital predictions. While Lie group variational integrators can preserve geometric structure to a certain extent, their implicit equations are formulated on Lie group elements, which incurs substantial computational overhead. Therefore, how to construct a high-performance integration method that simultaneously achieves numerical accuracy, structure preservation, and computational efficiency has become a core challenge in the long-term evolutionary simulation of binary asteroid systems.
In a recent study published in Space: Science & Technology, the team led by Guo Yongxin from the College of Physics, Liaoning University, proposed a numerical method for the full two-body dynamics of a double-dumbbell system based on the Hamel variational integrator. The study employs the special Euclidean group SE(3) to uniformly describe the translational and rotational motions of the dumbbell-shaped rigid bodies, introduces a body-fixed coordinate frame attached to the second dumbbell, reduces the system motion to the evolution of relative position and attitude, and constructs discrete equations of motion on the Lie algebra based on the discrete Hamilton's principle, thereby achieving an efficient iterative scheme for the Hamel variational integrator. Simulation results demonstrate that in regular-shaped examples, the proposed method outperforms Lie group variational integrators in energy conservation and orthogonality preservation, and, since the implicit equations are formulated on the Lie algebra rather than on Lie group elements, it incurs lower computational cost. In irregular-shaped examples, the total energy error and rotation matrix orthogonality error of the proposed method are significantly superior to those of the Runge–Kutta method, which fails to maintain orthogonality structure and consequently leads to substantial accumulation of force and torque computation errors over time. Irregular shape also induces noticeable deviations in the barycentric trajectory of the double-dumbbell system along specific directions, indicating that the irregular geometry of celestial bodies has a non-negligible effect on full two-body motion. This study provides a numerical scheme that balances structure preservation and computational efficiency for high-precision long-term orbital evolution of binary asteroid systems, offering significant methodological support for planetary defense mission planning and long-term dynamical prediction of binary systems.
First, this paper focuses on the full two-body dynamics problem in the orbital evolution of binary asteroid systems. Binary asteroids constitute a significant proportion of near-Earth asteroids, with approximately 16% of near-Earth asteroids possibly being binary systems; the assessment of their impact threat and the planning of exploration missions both rely on high-precision predictions of the long-term dynamical behavior of such systems. In the full two-body problem, the gravitational potential of irregular bodies is determined by both their positions and attitudes, resulting in coupling between translational and rotational motions, and the model complexity is substantially greater than that of the classical two-body problem. Owing to the relatively small separation between the two asteroids in a binary system and the modest difference in their masses, each binary system can be modeled as a rigid dumbbell, thereby establishing a double-dumbbell full two-body system as shown in Fig. 1. The system consists of two dumbbell-shaped rigid bodies, each composed of two asteroids connected by a massless rod. To describe the configuration of the dumbbells, the study employs the special Euclidean group SE(3) to uniformly represent position and attitude, and introduces a body-fixed coordinate frame attached to the second dumbbell, reducing the system motion to the evolution of the relative position and attitude of the first dumbbell with respect to the second. This modeling approach avoids the singularity issues inherent in Euler angle representations and naturally captures the coupling effects between translation and rotation.
Second, based on the discrete Hamilton's principle, the paper constructs a Hamel variational integrator tailored for the double-dumbbell system. The core idea of the Hamel variational integrator is to utilize the body-frame motion framework provided by the left-invariant vector fields on the Lie group, expressing both the continuous and discrete equations of motion entirely on the Lie algebra. Fig. 2 illustrates the computational procedure for the double-dumbbell system: first, the continuous Euler–Lagrange equations and Hamilton equations are derived from Hamilton's principle; subsequently, the discrete Hamilton's principle is employed to construct the discrete Lagrangian, yielding the discrete equations of motion on the Lie algebra. Since all equations are formulated at the Lie algebra level, the computational complexity of the implicit equations is lower than that of Lie group variational integrators based on Lie group elements. Through the discrete Legendre transform, an explicit iterative scheme in the Hamiltonian form is established; within each time step, the state update is accomplished by solving the implicit equations on the Lie algebra, sequentially obtaining the evolution of the relative position, relative attitude, and each momentum variable. This method preserves both the symplectic and Lie group structures of the system, while maintaining long-term exact conservation of total energy and angular momentum.
Finally, this paper systematically validates the performance of the Hamel variational integrator through numerical simulations of both regular-shaped and irregular-shaped dumbbell models. Fig. 3 presents the energy evolution over time, in which the kinetic and potential energies exhibit periodic interconversion; at the ninth unit of time, the two dumbbells reach their closest distance, corresponding to minimum potential energy and maximum kinetic energy, while the total energy remains constant. Figs. 4 and 5 compare the three methods in terms of energy error and orthogonality error, respectively: both the Hamel variational integrator and the Lie group variational integrator outperform the Runge–Kutta method, with the Hamel variational integrator demonstrating superior structure-preserving performance. The CPU time comparison in Fig. 6 reveals that the Hamel variational integrator achieves slightly higher computational efficiency than the Lie group variational integrator, as its implicit equations are formulated on the Lie algebra rather than on Lie group elements. In the irregular-shaped examples, the Runge–Kutta method fails to preserve the orthogonality of the rotation matrix, leading to significant accumulation of force and torque computation errors over time, whereas the Hamel variational integrator consistently maintains low errors. By comparing the barycentric trajectories of the two dumbbell models, it is found that the irregular shape induces a noticeable deviation in the y-direction of the barycentric trajectory, indicating that the irregular geometry of celestial bodies exerts a non-negligible influence on full two-body motion. This study provides a numerical method that balances structure preservation and computational efficiency for long-term high-precision orbital evolution of binary asteroid systems.
