Application of force gradient symplectic integrators to the circular restricted three-body problem

The kinetic energy of the circular restricted three-body problem in a rotating frame is no longer a standard positive quadratic function of moment, owing to the additional part in the non-inertial rotating frame, which leads to a difficulty in using force gradient symplectic integrators. To address this problem, we show through the calculation of Lie operators that the force gradient operator on the system is still related to the gradient of the gravitational forces from the two main objects rather than that of the resultant force of both the gravitational forces and the non-inertial force exerted by the rotating frame, just as the force gradient operator on the circular restricted three-body problem in an inertial frame. Therefore, it is reasonable to use the gradient symplectic integrators for integrating the circular restricted three-body problem in the rotating frame from a theoretical point of view. Numerical simulations describe that a fourth-order force gradient symplectic method is always greatly superior to the non-gradient Forest-Ruth algorithm in the numerical accuracy, and its optimized version is best. Because of this, the optimized gradient scheme is recommended for calculating chaos indicators, such as Lyapunov exponents of and fast Lyapunov indicators of two nearby trajectories, which is conductive to obtaining a true description of dynamically qualitative properties.