Logarithmic Hamiltonian Algorithm And Its Application

Celestial mechanics numerical method as one of the important areas of celestial mechanics get long-term development after symplectic algorithm was put forward,which symplectic algorithm could preserve a symplectic structure and do not show secular errors in energy and angular momentum integrals. Symplectic algorithm appropriate research the long-term qualitative evolution of a Hamiltonian system but also has the numerical accuracy is not high and explicit symplectic algorithms require constant time-steps. Normally, for the motion of objects close encounters and in highly eccentric orbits need to shorten the time-steps to overcome the sharp increase acceleration that due to excessive gravity, adaptive time-steps will lose the advantage of keep the symplectic structure of symplectic algorithm, considering the thinking of time transformation, the original time variables take adaptive step size and the new time are fixed step size, then not only adjust time-steps but also and can keep the inherent advantage of symplectic algorithm. The main content of this article for considering different logarithmic Hamiltonian algorithm to the different Hamiltonian and demonstration while applied in improving numerical accuracy and ensure effective chaotic discriminant result.According to the different Hamiltonian system structured different forms of time transformation symplectic algorithm. For the Hamiltonian function that only separated into kinetic energy and potential energy section, just contain generalized momentum and generalized coordinates respectively, that can construct the time transformation function form of two parts of different functions but equivalent get explicit logarithmic Hamiltonian method, in which time transformation on the Hamiltonian function, this paper constructs the explicit logarithmic Hamiltonian Yoshida fourth-order method that composed of three second-order leapfrog operator.For the system that kinetic energy part of cross terms of generalized momentum and generalized coordinates but the potential energy part only contains position variables structure explicit implicit mixed logarithmic Hamiltonian method, applied implicit point method for kinetic energy part. While for a more general system structure implicit logarithmic Hamiltonian method. Implicit method has a wider application but also due to included the iterative algorithm take more computer time to reduce the computational efficiency.This paper demonstrates the explicit logarithmic Hamiltonian method comparedwith not time transformation symplectic algorithm have advantage in numerical accuracy when applied to Newton circular restricted three body problem and relativity circular restricted three body problem. And in the previous system that precision advantage independently of the change of the orbital eccentricity. For this phenomenon latter one fails to happen but numerical accuracy is obviously superior to the conventional symplectic algorithm. Especially for high eccentricity orbits, the have not time transformation algorithm to get the false chaos discriminant indicators,such as the Lyapunov exponents and fast Lyapunov indicators(FLI). And by the logarithmic Hamiltonian method,reliably qualitative analysis results can be obtained,thoroughly solve the excessive estimation of Lyapunov exponents of high eccentricity orbit and the FLI agreed to quickly increase in post-Newtonian circular restricted three-body problem. After demonstration, logarithmic Hamiltonian method is applied in discusses the influence the the change of the dynamic parameters that the distance between the two main bodies to the transformation of order and chaos of dynamic systems. Through numerical simulation demonstrated that logarithmic Hamiltonian method has higher accuracy and reliable qualitative research results can be obtained,and it is suitable for the qualitative analysis and quantitative calculation of high eccentricity problem. Lead celestial mechanics research opens up a new way of thinking. Reflect the dynamic nature of integral dynamic evolution of an actual motion of objects close encounters object.