We mainly focus on the effects of small changes of parameters on the dynamics of charged particles around Kerr black holes surrounded by an external magnetic field, which can be considered as a tidal environment. The radial motions of charged particles on the equatorial plane are studied via an effective potential. It is found that the particle energies at the local maxima values of the effective potentials increase with an increase in the black hole spin and the particle angular momenta, but decrease with an increase of one of the inductive charge parameter and magnetic field parameter. The radii of stable circular orbits on the equatorial plane also increase, whereas those of the innermost stable circular orbits decrease. On the other hand, the effects of small variations of the parameters on the orbital regular and chaotic dynamics of charged particles on the non-equatorial plane are traced by means of a time-transformed explicit symplectic integrator, Poincaré sections and fast Lyapunov indicators. It is shown that the dynamics sensitivity depends on small variations in the inductive charge parameter, magnetic field parameter, energy, and angular momentum. Chaos occurs easily as each of the inductive charge parameter, magnetic field parameter, and energy increases but is weakened as the angular momentum increases. When the dragging effects of the spacetime increase, the chaotic properties are not always weakened under some circumstances.

ABSTRACT A Kepler solver is an analytical method used to solve a two-body problem. In this paper, we propose a new correction method by slightly modifying the Kepler solver. The only change to the analytical solutions is that the obtainment of the eccentric anomaly relies on the true anomaly that is associated with a unit radial vector calculated by an integrator. This scheme rigorously conserves all integrals and orbital elements except the mean longitude. However, the Kepler energy, angular momentum vector, and Laplace–Runge–Lenz vector for perturbed Kepler problems are slowly varying quantities. However, their integral invariant relations give the quantities high-precision values that directly govern five slowly varying orbital elements. These elements combined with the eccentric anomaly determine the desired numerical solutions. The newly proposed method can considerably reduce various errors for a post-Newtonian two-body problem compared with an uncorrected integrator, making it suitable for a dissipative two-body problem. Spurious secular changes of some elements or quasi-integrals in the outer Solar system may be caused by short integration times of the fourth-order Runge–Kutta algorithm. However, they can be eliminated in a long integration time of 108 yr by the proposed method, similar to Wisdom–Holman second-order symplectic integrator. The proposed method has an advantage over the symplectic algorithm in the accuracy but gives a larger slope to the phase error growth.

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The discovery of numerous close-in planets has updated our knowledge of planet formation. The tidal interaction between planets and host stars has a significant impact on the orbital and rotational evolution of the close planets. Tidal evolution usually takes a long time and requires reliable numerical methods. The manifold correction method, which strictly satisfies the integrals dissipative quasiintegrals of the system, exhibits good numerical accuracy and stability in the quasi-Kepler problem. Different manifold correction methods adopt different integrals or integral invariant relations to correct the numerical solutions. We apply the uncorrected five- and six-order Runge–Kutta–Fehlberg algorithm [RKF5(6)], as well as corrected by the velocity scaling method and Fukushima’s linear transformation method to solve the tidal evolution of exoplanet systems. The results show that Fukushima’s linear transformation method exhibits the best performance in the accuracy of the semimajor axis and eccentricity. In addition, we predict the tidal timescale of several current close exoplanetary systems by using this method.

Equations of motion for a general relativistic post-Newtonian Lagrangian approach mainly refer to acceleration equations, i.e. differential equations of velocities. They are directly from the Euler-Lagrangian equations, and usually have higher-order terms truncated when they remain at the same post-Newtonian order of the Lagrangian. In this sense, they are incoherent equations of the Lagrangian and approximately conserve constants of motion in this system. In this paper, we show that the Euler-Lagrangian equations can also yield the equations of motion for consistency of the Lagrangian in the general case. The coherent equations are the differential equations of generalized momenta rather than those of the velocities, and have no terms truncated. The velocities are not integration variables, but they can be solved from the algebraic equations of the generalized momenta with an iterative method. Taking weak relativistic fields in the Solar System and strong relativistic fields of compact objects as examples, we numerically evaluate the accuracies of the constants of motion in the two sets of equations of motion. It is confirmed that these accuracies well satisfy the theoretical need if the chosen integrator can provide a high enough precision. The differences in the dynamical behavior of order and chaos between the two sets of equations are also compared. Unlike the incoherent post-Newtonian Lagrangian equations of motion, the coherent ones can theoretically, strictly conserve all integrals in some post-Newtonian Lagrangian problems, and therefore are worth recommending.