Orbital Mechanics near a Rotating Asteroid

This study investigates the different novel forms of the dynamical equations of a particle orbiting a rotating asteroid and the effective potential, the Jacobi integral, etc. on different manifolds. Nine new forms of the dynamical equations of a particle orbiting a rotating asteroid are presented, and the classical form of the dynamical equations has also been found. The dynamical equations with the potential and the effective potential in scalar form in the arbitrary body-fixed frame and the special body-fixed frame are presented and discussed. Moreover, the simplified forms of the effective potential and the Jacobi integral have been derived. The dynamical equation in coefficient-matrix form has been derived. Other forms of the dynamical equations near the asteroid are presented and discussed, including the Lagrange form, the Hamilton form, the symplectic form, the Poisson form, the Poisson-bracket form, the cohomology form, and the dynamical equations on the K\"ahler manifold and another complex manifold. Novel forms of the effective potential and the Jacobi integral are also presented. The dynamical equations in scalar form and coefficient-matrix form can aid in the study of the dynamical system, the bifurcation, and the chaotic motion of the orbital dynamics of a particle near a rotating asteroid. The dynamical equations of a particle near a rotating asteroid are presented on several manifolds, including the symplectic manifold, the Poisson manifold, and complex manifolds, which may lead to novel methods of studying the motion of a particle in the potential field of a rotating asteroid.