Dynamics Of The Coherent Post-Newtonian Lagrangian Equation

The Post-Newtonian approximation is the scientific method of using general relativity to study the laws of motion of celestial bodies and their mechanics.The higher-order post-Newtonian approximation theory provides theoretical support for the study of relativistic astrodynamics and the detection of gravitational waves.The dynamical equations describing the system are the post-Newtonian Lagrangian system and the post-Newtonian Hamiltonian system.The post-Newtonian system terms mainly include Newtonian terms,first order post-Newtonian terms,second order post-Newtonian terms,spin-orbit coupling terms,spin-spin coupling terms,and other post-Newtonian higherorder terms.The Lagrangian is equivalent to Hamiltonian if its contain only Newtonian terms.When post-Newtonian terms are considered,the equivalence relationship between these two and the corresponding dynamics is worth studying.In this paper,we derive the post-Newtonian Lagrangian coherent equations of motion based on the Euler-Lagrange equations,evaluate the applicability of the coherent equations of motion by comparing the orbital dynamics of the Lagrangian coherent equations of motion,the approximate equations of motion,and the Hamiltonian canonical equations and illustrate the equivalence between the post-Newtonian Lagrangian and the Hamiltonian system.Meanwhile,the orbital dynamics of post-Newtonian spinning compact binary systems and post-Newtonian three-body problems are studied based on the post-Newtonian Lagrangian coherent equations of motion.The differential equation for the velocity can be obtained directly by using the Euler-Lagrangian equation of the post-Newtonian Lagrangian system.This process requires truncating the higher order terms in the equation,so that the equations of motion can only approximate the conservation of the conserved quantities of the system.The equations of motion derived in this way are an approximate description of the state of the dynamical system and are therefore called the approximate equations of motion.The approximate equation of motion has been widely used in the study of the dynamics of Lagrangian systems.In fact,the differential equation of generalized momentum and the relationship between generalized momentum and velocity can be obtained by using the Euler-Lagrange equation without any truncation of higher order terms.This equation of motion is called the coherent equation of motion,which can accurately maintain the conservation of the conserved quantities of the system,and its conservation accuracy is also much higher than the approximate equation of motion.Thus,the coherent equations of motion for post-Newtonian Lagrangian systems can be used to study the orbital dynamics of post-Newtonian spin compact binary systems and post-Newtonian three-body problems.The dynamics of binary systems with quadrupole moments is studied by using the coherent equations of motion and the approximate equations of motion.Numerical results show that the system conserved quantities are more accurate in the coherent equations of motion than in the approximate equations of motion.In addition to considering the accuracy of the conserved quantities,the coherent equations of motion are used to study the chaotic dynamics of the system,especially those containing quadrupole moments.When the quadrupole moment increases,the differences of the orbits are obvious in the approximate and the coherent equations of motion.Although the contribution of the quadrupole moment to the system is smaller than that of the spin effect,the effect of the quadrupole moment on the orbital chaotic characteristics is similar to that of the spin-spin orbital coupling.With the increase of quadrupole moment,chaotic orbits are easily obtained in the coherent equations of motion,but chaos is not obvious in the approximate equations of motion.It indicates that there is a difference between the coherent equations of motion and the approximate equations of motion in describing the dynamics of spinning compact binary systems.The search for periodic orbits of three-body systems is a difficult problem in celestial mechanics because of the lack of a sufficient number of first integrals to make the system integrable and there is no analytical solution.The figureeight orbit is a well known periodic orbit in the three-body system.The coherent equations of motion,the approximate equations of motion,and the canonical equations are used to find the periodic orbits of the three-body problem and to analyze the differences in the dynamics of each equation of motion.The numerical results show that the energy of the three-body system remains approximately conserved in the approximate equations of motion,while it remains exactly conserved in the coherent equations of motion and the canonical equations.Although the three-body system is not equivalent in the three equations of motion,the dynamics of the system is approximately related in the three equations of motion.When the initial separation is large,the initial conditions for satisfying the figure-eight orbit in the approximate equations of motion are also applicable to the coherent equations of motion and the canonical equations.When the initial separation is small,the initial conditions for satisfying the figure-eight orbit in either equation of motion do not apply to the other two equations of motion.In order to find the initial conditions to satisfy the figure eight orbit with small initial separation,the initial velocity of the third object is scanned in three equations of motion in this paper.Numerical results show that the figure-eight orbits at smaller separations are extremely sensitive to the initial conditions in the three equations of motion.In this paper,the scanning velocity method is used to find other periodic orbits of the three-body problem.The gravitational wave radiation of the three-body problem is worth studying.The expression of gravitational waves for the post-Newtonian threebody problem is derived from the basic formula of gravitational waves.The gravitational waveforms are calculated by using the coherent equation of motion proposed in this paper.Differences in the waveforms of the figure-eight orbits of the post-Newtonian three-body problem are studied in the approximate equations of motion,the coherent equations of motion and the canonical equations.Meanwhile,the characteristics of gravitational waveforms of other periodic orbits in the three equations of motion are compared.The amplitude of gravitational waves increases when the trajectories of celestial bodies are very close and do not collide.Such gravitational wave signals could potentially be captured by laser interferometric gravitational wave observatories in the future,which would open a window to use gravitational waves to explore three-body systems in the universe.