Subjects: Astronomy submitted time 2023-08-02 Cooperative journals: 《天文学报》
Abstract： The Hamiltonian describing the motion of charged particles around the Schwarzschild black hole immersed in an external magnetic field is nonintegrable. Such relativistic Hamiltonian systems do not have two splitting parts with analytical solutions as explicit functions of time. This leads to the difficulty in the construction and application of explicit symplectic algorithms to the relativistic systems. Recently, Chinese scholars have published a series of works in the Astrophysical Journal, where explicit symplectic methods are successfully designed for these relativistic Hamiltonians split into three or more explicit integrable parts. There are two questions of whether the numbers of splitting these Hamiltonians affect the numerical accuracy and which of the explicit symplectic integrators shows the best performance. Our latest work in the Astrophysical Journal answered the two questions, and shows that the fourth-order optimal Partitioned-Runge-Kutta (PRK$_{6}4$) explicit symplectic algorithms with the three-part splitting method as the least number of splitting these Hamiltonians performs the best accuracy. This paper applies such an integrator to obtain Poincar\'{e} cross-section, maximum Lyapunov indicators and fast Lyapunov indicators (FLIs), which distinguish between the regular and chaotic dynamical properties of charged particles moving near the magnetized Schwarzschild black hole. For given specific values of the particle energy and angular momentum, a small magnetic field does not induce chaos, whereas a large positive magnetic field parameter easily causes the occurrence of chaos. The strength of chaos increases with the magnetic field increasing. Chaos is also strengthened as the particle energy increases. However, it is weakened when a negative magnetic field parameter and the particle angular momentum increase.
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