  The study of Kerr geodesics has a long history, particularly for those
occurring within the equatorial plane, which is generally well-understood.
However, upon comparison with the classification introduced by one of us
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.105.024075 , it becomes
apparent that certain classes of geodesics, such as trapped orbits, are still
lacking analytical solutions. Thus, in this study, we provide explicit
analytical solutions for equatorial timelike geodesics in Kerr spacetime,
including solutions of trapped orbits, which capture the characteristics of
special geodesics, such as the positions and conserved quantities of circular
orbits, bound orbits, and deflecting orbits. Specifically, we determine the
precise location at which retrograde orbits undergo a transition from
counter-rotating to prograde motion due to the strong gravitational effects
near the rotating black hole. Interestingly, we observe that for orbits with
negative energy, the trajectory remains prograde despite the negative angular
momentum. Furthermore, we investigate the intriguing phenomenon of deflecting
orbits exhibiting an increased number of revolutions around the black hole as
the turning point approaches the turning point of the trapped orbit.
Additionally, we find that only prograde marginal deflecting geodesics are
capable of traversing through the ergoregion. In summary, our findings present
explicit solutions for equatorial timelike geodesics and offer insights into
the dynamics of particle motion in the vicinity of a rotating black hole.
