Explicit Symplectic Algorithms In Relativity Theory And Their Applications

Most of the relativistic systems in celestial mechanics are nonlinear systems.Except for a few systems whose analytical solutions can be solved,other systems are very complex,and it is generally difficult to obtain analytical solutions for nonlinear systems.The most classic method for studying the short-term motion characteristics of celestial bodies is the Runge-Kutta(RK)method.The RK method can also be divided into two types: the explicit RK method has a fast calculation speed but the evolution of the calculation accuracy will increase with time,and is not suitable for long-term integration,but the high-order explicit RK method can be used to integrate the motion of short-term celestial bodies.The implicit RK method needs to be iterative and satisfy the symplectic matrix,and the calculation error will not increase with time evolution,but it is time-consuming.Numerical methods is used to solve nonlinear systems instead of analytical solutions,such as symplectic algorithms.There are two existing symplectic algorithms: the explicit symplectic algorithm,which has the advantages of fast computation and stable structure,but has a limited scope of application and can only be used for separable Hamiltonian equations.The implicit symplectic algorithm is applicable to a wide range,as long as it is a Hamiltonian equation,but it takes a long time to calculate.In order to save time in previous research,a method of explicit and implicit cross-use was proposed.There is also a kind of explicit symplectic algorithm constructed by expanding the phase space.There are also people who have constructed a fully apparent symplectic algorithm in relativistic space-time by ingenious methods.Numerical methods are fundamental tools for studying nonlinear dynamics.In this paper,we transform the Hamiltonian of particles surrounding the deformed Schwarzschild black hole in the magnetic field into an equivalent new Hamiltonian by extending the time transformation method of phase space variables.The new Hamiltonian can construct a second-order and fourth-order Hamiltonian.Explicit symplectic form of order.At the same time,the variational equation of the particle equation of the deformed Schwarzschild black hole moving outside the black hole is constructed.With the help of Lyapunov indicators(LECs),fast Lyapunov indicators(FLI),smaller alignment indicators(SALI)and generalized alignment indicators(GALIs)discriminant studies the dynamics(chaotic and ordered orbits)of charged particles moving around a deformed Schwarzschild black hole in an external magnetic field.The smaller alignment index is employed to investigate how the parameters affect the dynamical transition from order to chaos.The main results are given here.For appropriate choices of initial conditions and other parameters,chaos is strengthened when the energy,positive magnetic field parameter,and the magnitude of the negative deformation parameter increase from small to large.However,chaos is weakened as the positive deformation parameter,angular momentum,and the absolute value of the negative magnetic field parameter increase.This paper is not simply to study the numerical calculation method,nor simply to discuss the construction process of the model,but to organically combine the two-sided.