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Klein-Gordon equation for Screened Manning-Rosen potential combined with Poschl-Teller potential and Kepler problem in hypersphere non-central potential was solved by hypergeometric method. Klein-Gordon equation was divided into a radial part, an angular part, and an azimuthal part by using a variable separation method. The radial part used Screened Manning-Rosen potential while the angular part used Poschl-Teller potential, and the azimuthal part used Kepler problem in hypersphere potential. The relativistic energy spectrum was calculated numerically. The value of the relativistic energy increases as the quantum number n and parameter potential increase. The wave function was expressed in hypergeometric equation term. [ABSTRACT FROM AUTHOR]

The ‘captured’ 2-body Kepler problem considering relativity is solved by using an iterative integral equation. The rationale for this approach is to increase the accuracy of the limits for a satellite’s motion and potentially provide a test to evaluate different gravitational laws. Moreover, this effort might provide additional insights to resolve other anomalies such as the flyby anomaly, the Faint Young Sun paradox, the Pioneer anomaly and other inconsistencies that potentially may be used to validate Einstein’s Theory of Relativity. The mathematical solution solves a nonlinear Volterra integral equation using an iterative fashion, which reveals a correction factor for treating a given closed orbit. However. This correction factor is not a constant value but rather a function of the elliptical or circular orbit angular displacement. This function may be insignificant during portions of the trajectory, say at apogees or perigees. Nonetheless, these results are encouraging where relativity effects may or may not exist to understand and resolve several of the other variances or gravitational anomalies currently related to our solar system. [ABSTRACT FROM AUTHOR]

The aim of this research is to study the efficiency of symplectic and non-symplectic Runge-Kutta methods in solving Kepler problem. The numerical behavior of the Runge-Kutta (RK) methods that are symmetric such as the implicit midpoint rule (IMR), implicit trapezoidal rule (ITR), 2-stage and 2-stage Gauss (G2) method are compared with the non-symmetric Runge-Kutta methods such as the explicit and implicit Euler (EE and IE), explicit midpoint rule (EIMR), explicit trapezoidal rules (EITR), explicit 4-stage Runge-Kutta (RK4) method and 2-stage Radau IIA method (R2A). Kepler problem is one type of nonlinear Hamiltonian problem that describes the motion in a plane of a material point that is attracted towards the origin with a force inversely proportional to the distance squared. The exact solutions phase diagram produces a unit circle. The non-symplectic methods only reproduce a unit circle at certain time intervals while the symplectic methods do produce a unit circle at any time intervals. Some phase diagram show spiral in or spiral out patterns which means the solutions are running away from the unit circle. This also means that the absolute error will be increasing in long time integration. The numerical experiments for the Kepler problem are given for many time intervals and the results show that the most efficient method is G2 of order-4 and surprisingly RK4 seems to be efficient too although it is not a symplectic nor a symmetric method. The numerical results on Kepler problem concluded that, the higher the order of the method, the most efficient the method can be in solving Kepler problem despite whether they are explicit or implicit or symmetric and symplectic. [ABSTRACT FROM AUTHOR]

The presence of errors in the initial conditions and the different physical parameters affect to numerical integration of equations of motion necessarily. It is interesting to determine which the causes of the possible phase errors are and/or the dumping/dissipative terms that can appear. The modified equation can be useful to reduce numerical instability errors which may be confused by geometrical or physical errors. This is a necessary first step to detect another kind of errors related to the mathematical/physical model and not to the integration method. [ABSTRACT FROM AUTHOR]

This paper studies the topology of the phase flow of a Keplerian motion in a rotating reference frame, with help of some analytical and graphics tools. By means of Liouville-Arnold theorem and some specifics techniques, we obtained a complete topological classification of the phase flow associated to this system. [ABSTRACT FROM AUTHOR]