  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.
