  The rosette-shaped motion of a particle in a central force field is known to
be classically solvable by quadratures. We present a new approach of describing
and characterizing such motion based on the eccentricity vector of the two body
problem. In general, this vector is not an integral of motion. However, the
orbital motion, when viewed from the nonuniformly rotating frame defined by the
orientation of the eccentricity vector, can be solved analytically and will
either be a closed periodic circulation or libration. The motion with respect
to inertial space is then given by integrating the argument of periapsis with
respect to time. Finally we will apply the decomposition to a modern central
potential, the spherical Hernquist-Newton potential, which models dark matter
halos of galaxies with central black holes.
