Dynamical Difference Between Second-order Post-newtonian Lagrangian Configuration And'hamiltonian Configuration On Spinning Compact Binary

In the study of relativistic celestial mechanics,the study of post-Newtonian theory is one in every of its main tasks.When there is a great progress during this field,most of the analysis work primarily discusses the subsequent two aspects.The primary is that the dynamic study of the post-Newtonian Hamiltonian system and the post-Newtonian Lagrangian system.The second is that the study of whether or not the two descriptions are equivalent,that is,whether or not the two descriptions totally different within the method of characterizing the physical system.The analysis on the equivalence of the two includes a polar role in uranology.It's accepted that Einstein's theory of relativity could be a sensible theory of gravity.However,in a strong gravitational system like compact binary bodies,Einstein's field equations are often not analytically processed,therefore post-Newtonian theory could be a powerful tool during this case.There is no doubt that the Lagrangian and Hamiltonian descriptions are fully equivalent inside Newtonian mechanics,but their equivalence has been under deep investigation in post-Newtonian theory.For this reason,in the present work,I choose the compact binary system with spin and investigate the orbital dynamics performance of both Lagrangian and Hamiltonian descriptions in this strong gravitational system under the post-Newtonian approximation,respectively.According to the two representations,the corresponding multiple comparison models are constructed separately.In the construction of the models,the effects of the second-order post-Newtonian terms with spin-orbit coupling terms and their spin-spin coupling terms on the dynamical behavior of the system are considered successively.In terms of the choice of method,we use two different algorithms for the post-Newtonian Lagrangian formulation.One is an approximate equation of motion that has been used in the past to approximately preserve energy,and the other is a more recent coherent equations of motion that is truncation-free and can precisely preserve energy and constants of motion,a consideration that makes the dynamics investigation results more accurately.After numerical calculations for the corresponding same-order models under the two formulations with the same initial conditions,the analytical results shows that our numerical results indicates the following three points: First,the non-truncated coherent approach of the post-Newtonian Lagrangian is more convincing than the approximate Lagrangian approach when discussing the dynamics of the two formulations of the same post-Newtonian order Hamiltonian and Lagrangian.Second,for the same order post-Newtonian Hamiltonian form and the coherent post-Newtonian Lagrangian form,they exhibits different dynamical behaviors,which suggests that the two formulations are not equivalent.Third,the spin-orbit coupling in the Hamiltonian part is the main cause of chaos.Moreover,in the post-Newtonian Hamiltonian model,the second-order post-Newtonian part plays an important role in causing chaos under the post-Newtonian Hamiltonian form of the system.