Dynamic Comparison Between Lagrangian And Hamiltonian Systems At The Post-newtonian Approximation

For Lagrangian and Hamiltonian systems in recent years,many researchers were studied,the research work are roughly around the equivalence between the two system what conditions under equivalent or inequitble.Therefore,this article will follow the footsteps of predecessors to further study.According to the present research,the compact binaries tend to be strong gravitational system and need to use Einstein’s general theory of relativity.Usually the Einstein field equation has no analytical solutions,the Newtonian approximation method is widely used to find the solution.The Lagrangian function and the Hamiltonian function are the two most common forms of expression in Newtonian mechanics,as in Newtonian mechanics,which is the most common.but because of the fact that the latter has regular variables and the nature of the regular dynamical system,Convert the former into the latter to study.There is doubt that the two are equivalent in Newtonian mechanics and do not need to argue at all,but the question of equivalent in the framework of Newtonian mechanics is difficult to answer.In fact,the analysis of compact binaries systems,the Lagrange first begins to study its corresponding Hamiltonian.For a variety of model structure,using Legendre transformation first let Lagrange convert to hamiltonian,in the case of coupling will produce some new Hamiltonian.Through theoretical and numerical analysis in combination with several,first: under the same order Lagrangian and Hamiltonian inequitable;The second: spin-spin orbit coupling is the main cause of chaos;Third: do not contain spin orbit coupling of Lagrange system than even contain spin orbit coupling were more likely to lead to chaos.For the dynamics of the high-order spin-orbit coupled compact bimodals of linear momentum,the Lagrangian part is mainly studied in the case of one Hamiltonian.Since some of the high-order spin-spin coupling terms are generated for the Hamiltonian form with spin orbit and Newtonian terms converted to Lagrangian by Legendre,it is also necessary to strictly compute.This part is mainly about how the new generation of these high-order spin-spin items will change the system dynamics.At the same time in the study with the equivalent of Hamilton,where the approximate equivalent Hamiltonian can be compared with the original,and discuss Hamilton’s integrability.And then to consider the Hamiltonian spin-spin with the effect of the spin,the resulting Lagrangian will be what changes,which is a focus of the study.