Extension And Application Of Force Gradient Symplectic Algorithm

It is very clear that numerical methods and chaos indicators are the main tools for invertigating celestial mechanics and nonlinear dynamics. 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.It was reported that gradient methods are more accurate than their corresponding usual non-gradient ones. Fourth-order force gradient symplectic algorithms were developed by Chin (Chinl997,2007; Omelyan et al.2002). We also develop the fourth-order force gradient symplectic algorithm at the basis of Chin. All of them are applicated in the H'enon-Heiles system and the Newtonian Core-shell System. We find that the new fourth-order force gradient symplectic algorithm has the better numerical performance. At the same time, the restricted three-body problem is discussed for using the force gradient symplectic algorithm in the degree thesis. Some details are as follows:First, an operator associated with third-order potential derivatives and a force gradient operator corresponding to second-order potential derivatives are used together to design a number of new fourth-order explicit symplectic integrators for the natural splitting of a Hamiltonian into both the kinetic energy with a quadratic from of momenta and the potential energy as a function of position coordinates, including those of Chin and coworkers. The new fourth-order force gradient symplectic algorithms are applicated to study the regular orbit and the chaotic orbit of H'enon-Heiles system and Newtonian Core-shell System. Numerical tests show that the new fourth-order force gradient symplectic algorithms have given much better results than non-force gradient symplectic algorithms of Forest-Ruth, the optimal algorithms have good demonstrations of the accuracy of energy calculations. it is worth recommending for practical computation.Second, the restricted three-body problem is one very important model in celestial mechanics. The oblateness influences equilibium points of the restricted three-body problem. This paper studies out-of-plane equilibrium points and their stability in the restricted three-body problem with oblateness of the J2 and J3 terms. The Hamiltonian of the restricted three-body problem can be splitted into both the kinetic enery and the potential energy as two integral function. The possibility of application is discussed and the effect of the accuracy of energy is assessed by force gradient symplectic algorithms. Finally we studied the relationship between the properties of the order or the chaos orbits and the dynamic parameters.In brief, the main jobs of the degree thesis are that the new force gradient symplectic algorithms are expand from pre-existing fourth-order force gradient symplectic algorithms. That the feasibility of application in the restricted three-body problem for using force gradient symplectic algorithms and the relationship between the chaos and the dynamic parameters are discussed.