Application Of Force Gradient Symplectic Integrators To The Circular Restricted Three-body

It is very clear that chaos indicators and numerical methods are the main tools for invertigating nonlinear dynamics and celestial mechanics. At present, it is an important topic to find the reliable numerical methods and chaos identification methods. Our works are extension and application of force gradient symplectic algorithm in the degree thesis.Since the kinetic energy of the circular restricted three-body problem in a rotating frame has contributions from the non-inertial rotating frame, it is not a standard positive quadratic function with respect to momenta. In this case, there may be a difficulty in the application of 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 associated with 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, as the force gradient operator on the circular restricted three-body problem in an inertial frame is. Therefore, it is reasonable to apply 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 the best. Because of this, the optimized gradient scheme is recommended to calculate chaos indicators, such as Lyapunov exponents of and fast Lyapunov indicators of two nearby trajectories. This is helpful to obtain a true description of dynamically qualitative properties.