Logarithmic Hamiltonian Explicit Symmetric Method In Extended Phase Space And Its Application

The symplectic algorithm has the stability of long-term evolution in the integration of celestial body motion,the energy remains unchanged and the symplectic structure is maintained.The symplectic algorithm is suitable for qualitative research in Hamiltonian system of astrodynamics.The symplectic algorithm can be expressed in two forms: one is explicit symplectic algorithm,which is efficient and stable,but only suitable for separable Hamiltonian systems,the other is implicit symplectic algorithm.The implicit symplectic algorithm can be used to solve any Hamiltonian system with high accuracy,but the implicit symplectic algorithm usually takes a lot of time to get the result.Therefore,the researchers want to combine the advantages of explicit symplectic algorithm and implicit symplectic algorithm,which can not only satisfy the high accuracy but also save time.With the development of celestial mechanics,a lot of the method of studying the movement of celestial bodies has appeared.The research methods not only require high precision,but also improve the requirement of time or step size.For the high eccentricity,it is relatively convenient to use variable step size.The logarithmic Hamiltonian method is one of the variable step method and has some advantages.For an inseparable Hamiltonian,this problem can be solved by using the phase space expansion method.The logarithmic Hamiltonian explicit symmetry method,which combines the logarithmic Hamiltonian method with the phase space method to enlarge the phase space,can solve many problems and has a very obvious advantage.The Hamiltonian system consists of a separable Hamiltonian system and an inseparable Hamiltonian system.When coordinates and coordinates in logarithmic Hamiltonian system can be separated,explicit symplectic algorithm can be used.But when the coordinate and momentum can not be separated,the logarithmic Hamiltonian method of expanding phase space is a good choice.In this article I improve the logarithmic Hamiltonian method.On the basis of mikkola and tanikawa,a constant or function is added to improve its accuracy.Combining coordinate permutation and momentum permutation in extended phase space,midpoint permutation and optimized midpoint permutation,a series of logarithmic Hamiltonian explicit symmetry methods in extended phase space are obtained.These methods are applied to circular restricted three-body problems,third-order Newtonian spin dense binary systems and Ernst-Schwarzschild black holes,especially in highly eccentric orbits and chaotic orbits.In the Ernst-Schwarzschild black hole,if there is gravitational action,the gravitational effect will destroy the integrability of the primordial space-time,so the chaotic phenomenon may occur in the system.And chaos increases with the increase of energy or magnetic field.Therefore,when charged particles move in electromagnetic field and space-time geometry,electromagnetic force plays an important role in enhancing orsuppressing the degree of chaos caused by magnetic field gravitational effect.In the process of numerical simulation,the logarithmic Hamiltonian explicit symmetry method with extended phase space is of high accuracy and high efficiency.