Equilibria, periodic orbits around equilibria, and heteroclinic connections in the gravity field of a rotating homogeneous cube

This paper investigates the dynamics of a particle orbiting around a rotating homogeneous cube, and shows fruitful results that have implications for examining the dynamics of orbits around non-spherical celestial bodies. This study can be considered as an extension of previous research work on the dynamics of orbits around simple shaped bodies, including a straight segment, a circular ring, an annulus disk, and simple planar plates with backgrounds in celestial mechanics. In the synodic reference frame, the model of a rotating cube is established, the equilibria are calculated, and their linear stabilities are determined. Periodic orbits around the equilibria are computed using the traditional differential correction method, and their stabilities are determined by the eigenvalues of the monodromy matrix. The existence of homoclinic and heteroclinic orbits connecting periodic orbits around the equilibria is examined and proved numerically in order to understand the global orbit structure of the system. This study contributes to the investigation of irregular shaped celestial bodies that can be divided into a set of cubes.

💡 Research Summary

This paper presents a comprehensive dynamical study of a test particle moving in the gravity field of a uniformly rotating homogeneous cube. Using the polyhedral method of Werner and Scheeres, the authors derive an analytical expression for the gravitational potential of a cube with edge length 2a and constant density σ. The potential contains coupled logarithmic, polynomial, and arctangent terms and exhibits symmetry with respect to the coordinate planes and the diagonal planes of the cube.

In a synodic (rotating) reference frame where the cube spins about its z‑axis with constant angular velocity ω, an effective potential W(x,y)=U(x,y)+½ω²(x²+y²) is introduced. By scaling lengths with a and time with 1/ω, a dimensionless parameter R=Gσ/(ω²a) is defined; the study focuses on the case R=1. The equations of motion reduce to a planar, autonomous Hamiltonian system with a conserved energy integral C, whose zero‑velocity curves delimit admissible regions of motion.

Setting the gradient of W to zero yields eight equilibrium points in the xy‑plane, symmetrically placed with respect to the axes and the lines y=±x. Numerical refinement provides their precise coordinates (Table 1). Linearization about each equilibrium leads to a fourth‑order characteristic equation λ⁴+(W_xx+W_yy)λ²+(W_xxW_yy−W_xy²)=0. Four equilibria (E₁, E₃, E₅, E₇) possess purely imaginary eigenvalues and are linearly stable (LS); the other four (E₂, E₄, E₆, E₈) have a pair of real eigenvalues and are linearly unstable (U).

To explore the local dynamics, the authors compute periodic orbits around each equilibrium using the classical differential correction method. Initial guesses are taken from the linear solutions ξ(t)=A₁cos(λ₁t)+A₂cos(λ₂t) (and analogous expressions for η). Two families of orbits are generated by exciting either the x‑direction (A₁≠0, A₂=0) or the y‑direction (A₁=0, A₂≠0). The monodromy matrix of each periodic orbit is evaluated, and its stability index k=tr(M)−2 is used as a diagnostic: |k|<2 indicates linear stability, |k|>2 indicates instability, and |k|=2 marks a critical case. For the linearly stable equilibria, the index remains within (−2, 2) for amplitudes up to 0.2, confirming stable quasi‑elliptical periodic orbits. Conversely, all periodic orbits around the unstable equilibria exhibit |k|>2 for the same amplitude range, confirming their instability.

The global phase‑space structure is investigated by constructing the two‑dimensional stable (Ws) and unstable (Wu) invariant manifolds associated with each periodic orbit. A Poincaré section defined by the plane y₁=0, x₁<0 (where (x₁,y₁,z₁) is a rotated coordinate system aligned with the cube’s symmetry axes) is used to detect manifold intersections. For the periodic orbit around E₂ (amplitude A=0.07), the first intersection of Ws and Wu yields twelve crossing points: two lie on the x₁‑axis, corresponding to symmetric homoclinic connections, while the remaining ten correspond to asymmetric homoclinic orbits. Representative symmetric and asymmetric homoclinic trajectories are visualized in Figures 8 and 9.

Heteroclinic connections are also identified. By intersecting the stable manifold of the E₂ periodic orbit with the unstable manifold of the E₆ periodic orbit (and their symmetric counterparts), the authors demonstrate the existence of heteroclinic trajectories that link distinct equilibria. These connections provide natural pathways for low‑energy transfers between regions of the rotating cube’s potential field.

The paper concludes that even the simplest Platonic solid, the cube, generates a rich set of nonlinear dynamical phenomena, including multiple equilibria, families of stable and unstable periodic orbits, and intricate homoclinic/heteroclinic networks. The results are directly relevant to the modeling of irregularly shaped small bodies (asteroids, comets) that can be approximated as assemblies of cubes or tetrahedra. Future work is suggested on nonlinear stability analysis, extension to multi‑cube configurations, and validation against high‑fidelity shape models derived from spacecraft observations.