Fourth-order Force Gradient Symplectic Algorithms For A Perturbation Splitting Of Hamiltonians

To truly describe the chaotic motion of the Hamiltonian system, we need reliable a numerical method as well as chaos indicators. In this paper, the development and utilization of the numerical method are considered particularly.According to an idea of fourth-order force gradient symplectic algorithms proposed by Chin and coworkers, we construct two types of explicit fourth-order force gradient symplectic algorithms. They are extended to solve an n-body Hamiltonian problem with a perturbation decomposition form of Wisdom and Holman in the solar system dynamics. Also, one of the developed force gradient symplectic integrators is chosen to study the chaotic motion of a charged particle in a planetary magnetosphere. Some details are as follows.First, by adding force gradient operators to symmetric compositions, we build a set of explicit fourth-order force gradient symplectic algorithms, including those of Chin and coworkers, for a separable Hamiltonian system with quadratic kinetic energy T and potential energy V. They are extended to solve a gravitational n-body Hamiltonian system that can be split into a Keplerian part Ho and a perturbation part H1 in Jacobi coordinates. It is found that the accuracy of each gradient scheme is greatly superior to that of the standard fourth-order Forest-Ruth symplectic integrator in T+V-type Hamiltonian decomposition, but they are both almost equivalent in the mean longitude and the relative position for H0+H1-type decomposition. At the same time, there are no typical differences between the numerical performances of these gradient algorithms, either in the splitting of T+V or in the splitting of H0+H1. In particular, compared with the former decomposition, the latter can dramatically improve the numerical accuracy. Because this extension provides a fast and high-precision method to simulate various orbital motions of n-body problems, it is worth recommending for practical computations.Second, we numerically investigate the chaotic motion of a charged particle in a planetary magnetosphere by means of a new proposed fourth-order force gradient symplectic algorithm with the H0+H1-type perturbation decomposition. The mentioned physical model can be simplified as a perturbation two-body Hamiltonian problem. On the equatorial plane, several kinds of the phase portraits of the charged particle should be determined by two dynamical parameters of the ratio charge to mass and the z-component of the angular momentum. Certainly, besides the two parameters, the energy also yields an influence on the motion of the particle out the equatorial plane. It is found that increasing the energy or the absolute value of the ratio does always cause the extent of chaos. However, chaos becomes weaker for a larger magnitude of the angular momentum. Some qualitative interpretations to the dynamical transition of the charged particle with the variation of the dynamical parameters are given, too.