耗散形式下后牛顿拉格朗日和哈密顿动力学性质与引力波形比较

On the basis of the equivalence between the conservative post-Newtonian (PN) Lagrangian and Hamiltonian for compact binaries, this project mainly focuses on the dynamics of dissipative systems when the gravitational radiation is included, and the corresponding theories about the construction and analysis of geometric numerical integrators. In this project, we will analyze the difference between the approximation motion equation and the exact motion equation for the PN Lagrangian. Then the evolution law of the two polarizations of gravitational waves respectively in PN Lagrangian and Hamiltonian formulation will be compared. Especially, numerical integrators that preserve the radiation will be constructed for the dissipative system including the gravitational radiation. By developing explicit geometric numerical integrators of high efficiency for PN compact binaries, we further investigate the dynamical behavior and gravitational waveform template of the dissipative compact binaries by quantitative research. By means of this project, we will clarify that if the approximation motion equation of the PN Lagrangian causes its difference from the PN Hamiltonian at the same order. Then we can form a relatively complete system of numerical integrators for the dissipative system of compact binaries, when the gravitational radiation effect is included. This project will be helpful to the development of PN theories in compact object system, and provide more precise theoretical waveform template for the detection of gravitational waves. Finally, it will promote the development of celestial mechanics in our country.

本项目在保守系统下的后牛顿拉格朗日和哈密顿等价性研究的基础上，着重研究引入引力辐射项时后牛顿致密天体动力学性质和相应保结构算法的分析构造理论。分析致密双星系统下后牛顿拉格朗日近似运动方程与准确运动方程的差异；比较两种构型的引力波形基本偏振态的演化规律；研究保持引力辐射效应的数值积分方法的构造；发展可高效求解后牛顿致密双星系统的显式算法；进而通过所得到的数值方法定量研究耗散形式下致密双星系统的动力学性质及引力波形模板。通过本项目的开展，将彻底澄清运动方程的近似是否是造成同阶后牛顿拉格朗日与哈密顿动力学性质差异的根源；在具有引力辐射项的耗散系统框架下，形成比较完整的系统动力学数值研究方法。这项研究对于后牛顿理论的进一步发展有重要意义，将为引力波的探测提供更为精准的理论模板，从而促进后牛顿天体力学的发展。

本项目主要研究后牛顿近似下的拉格朗日和哈密顿系统动力学及相关数值计算方法，旨在通过理论分析及数值分析的手段探讨两种不同近似方法的有效性及后牛顿效应对于动力学性质的影响。项目主要成果如下：1. 提出了求解后牛顿（非封闭形式）拉格朗日系统的非截断积分方法，避免隐式运动方程的截断；进一步，基于速度与广义动量的实时坐标转换，构建了后牛顿拉格朗日系统的保辛、保能量方法。2. 对于只能构造隐式辛算法的不可分后牛顿哈密顿系统，考虑到不可分的后牛顿项具有较小量级这一事实构建了具有更高计算效率的显式近似辛算法。3. 在四精度环境下设计并验证了12阶全隐辛算法在太阳系后牛顿三体问题中的长期稳定计算及高精度优点，为后续后牛顿太阳系长期动力学的研究奠定数值算法基础。4. 对于半线性的哈密顿系统，从两种不同途径解决指数型保能量方法的构造问题，拓宽了保结构算法的学科前沿。