| Spinning compact binaries consisting of neutron stars or black holes, as a high nonlinear and nonintegrable relativistic two-body problem, not only have richer dynamic phenomena of resonance and chaos, but also are the most promising source for detecting gravitational waves. The polarization waveforms h+and h×can be determined by the second derivative of mass quadrupole moment of the system and positions of an observer, and the total energy radiated is proportional to square of the second derivative of mass quadrupole moment. This implies that the dynamic character of the gravitational body can be hidden in the gravitational waveforms. Therefore, the chaoticity of celestial bodies is a challenge for gravitational wave detection, and provides a good chance to be observed. This dissertation deals mainly with the dynamics of a conservative post-Newtonian (PN) Hamiltonian formulation of spinning compact binaries by means of manifold correction methods (or symplectic integrators) and the fast Lyapunov indicators. Relationships between the dynamics and gravitational waves are considered, too.As we have known, the conservative spinning compact binary system holds six conserved qualities, involving the total energy, three components of the total angular momentum and two constant lengths of spins. The conventional numerical integration schemes like Runge-Kutta type integrators will yield artificial dissipation in long time numerical integration such that the qualitative conclusion becomes unreliable. The scaling methods of manifold correction can avoid this problem. The single-scaling method and dual-scaling method are designed in terms of the least-squares correction. A fifth order Runge-Kutta (RK5), as a basic integrator, studies the effects of the new manifold corrections and the Nacozy's approach, according to the PN contributions, the spin-orbit coupling at 1.5PN order, the spin-spin coupling at 2PN order, and the classification of orbits (including small and large eccentric regular orbits and chaotic eccentric orbit). They are all nearly equivalent to correct the integrals at the level of the machine epsilon for the pure Kepler problem. Once the third-order post-Newtonian contributions are added to the pure orbital part, these corrections have explicit differences on controlling the errors of these integrals. Specially, when the spin effects are also included, the effectiveness of the Nacozy's approach becomes further weaken, even gets useless for the chaotic case. In all cases tested, the new scaling scheme shows always the optimal performance. It requires a little but not much expensive additional computational cost when the spin effects exist and several time-saving techniques are used. As an interesting case, the efficiency of the correction to chaotic eccentric orbits is generally better than one to quasicircular regular orbits. With the aid of the dual-scaling method and the fast Lyapunov indicators, initial conditions associated to the dynamical structure of order and chaos are given.From a qualitative point of view, symplectic integration schemes are the best suitable integrator for the PN Hamiltonian of spinning compact binaries. Lubich et al. designed a symplectic integrator for the PN Hamiltonian of spinning compact binaries. As a point to note, it isn't a globally structure-preserving algorithm for the Hamiltonian of spinning compact binaries due to the use of the noncanoncial spin variables except one of the position and momentum coordinates. To solve this problem, Wu and Xie designed a pair of canonical spin variables to rewrite the spin Hamiltonian part in 2010. This will be a good chance for the application of symplectic integrators to the PN Hamiltonian of spinning compact binaries. On the other hand, the mixed symplectic integrators with the semi-implicit Euler method, in which there is a symmetric composite of the analytical solution from the Kepler part and the spin effects and the numerical solution from the PN pure orbital contributions, has a defect of poor numerical stability. In fact, our numerical tests confirmed that the Euler mixed integrator is inferior to the midpoint mixed method in the numerical stability. In view of the two cases, we improved the work of Lubich et al. by constructing the second-order and the fourth-order fixed symplectic integrators, where the second-order symplectic implicit midpoint rule and its symmetric compositions are together used to integrate a PN Hamiltonian formulation with the canonical spin variables. Some details of implementation are as follows. The Hamiltonian can be split into the Kepler part and the perturbation part involving the pure orbital PN, spin-orbit and spin-spin contributions. The Kepler flow can be solved analytically, and the perturbation flow can be integrated numerically by the implicit midpoint rule. Then the analytical and numerical solutions are required to symmetrically compose a mixed algorithm. As a result, many numerical tests show that the mixed leapfrog integrator is always superior to the midpoint rule in the energy accuracy, while both of them are almost equivalent in the computational efficiency. Particularly, the optimized fourth-order algorithm compared with the mixed leapfrog scheme provides a good precision and needs no expensive additional computational time. Although the spin contributions included have a great effect on the quality of these symplectic integrators, there are no secular changes of the energy errors. This is just a basic character of these symplectic integrators. In addition, the chaoticity of the system can have iterative convergence and improve the computational efficiency, synchronously. Finally, the optimized fourth-order algorithm and the fast Lyapunov indicators are recommended to scan the structures of the global phase space.In short, with the help of both the manifold corrections added to a certain low-order integration algorithm as a fast and high-precision device and the symplectic integrators as well as the fast Lyapunov indicators of two nearby trajectories, phase space scans for chaos in spinning compact binaries show that the dynamics of the spinning compact binaries can not be determined uniquely by the dynamical parameters, initial conditions and initial spin angles. Instead, a combination of them is a sourse for causing chaos. The results support ones of Wu and Xie.On the other hand, we shall do some theoretical analysis. As was claimed by Wu and Xie, the construction of the canonical spin variables provides the theoretical support for the application of symplectic integrators to a canonical PN Hamilton for spinning compact binaries, and can directly determine the integrability of the canonical Hamiltonian system. The conservative PN Hamiltonian formulation of spinning compact binaries having one spinning body with a pure orbital part to arbitrary PN order or two spins restricted to the leading order spin-orbit interaction are integrable according to the relationship between the integrability of a PN Hamilton of spinning compact binaries and the canonical spin variables. Without question, the conservative PN Hamiltonian formulation of spinning compact binaries with the next-to-leading order spin-orbit interactions is also explicitly integrable and regular because there are 5 independent exact isolating integrals in the 10-dimensional phase space. With the help of symplectic integrators and the fast Lyapunov indicators of two nearby trajectories, numerical investigations also support the absence of chaos.Finally, in the case of the radiation-reaction effects turned off, let us evaluate how the dynamical parameters, the spin-orbit coupling, the spin-spin coupling and the classification of orbits exert influences on the gravitational waveforms. The numerical results show that the gravitational waveforms of order orbits sound regular with time, while those of chaotic orbits are stochastic. In particular, the chaotic dynamic behavior can enhance the strength of emission power. In addition, the magnitude of the spin parameters does exert significant influences on the gravitational waveforms.
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