Development of innovative numerical integrators which preserving all of the conserved quantities

辛积分和保能差分方法在哈密顿动力系统中有着广泛的应用。然而，由于这样的数值积分器不保留所有的守恒量，它们的行为可能与哈密顿系统的轨迹相当不同。事实上，这些积分器不能描述开普勒引力二体运动的长时间行为。本课题的研究目的是发展一种新的数值积分器TCI，即完全守恒积分器，它能保持哈密顿系统的所有守恒量。Minesaki和中村在2006年提出了一种新的积分器，该积分器能保持引力三体运动的所有守恒量。由于一般的三体运动是一个混沌动力学系统，这样的积分器的候选人制定的情况下，拉格朗日的等边三角形的解决方案在一个平原。积分器的基本设计是将运动方程分解为二体方程和三体方程的相互作用方程，然后分别离散。一个关键的想法是使用一个间隙的时间变量出现在离散的2体运动，以保持相对坐标为零的总和。因此，它表明，所得的数值积分保持Langrage的等边三角形的解决方案。这种显著的性质在数值模拟中被观察到，除了在计算机中的舍入误差之外。新的积分器被证明是上级比Stormer-Verlet的辛积分器的三体运动。计算结果表明，该积分器对三体运动的“8”解具有良好的积分性能，且相应的能量在较长时间内保持恒定。

Symplectic integrators and energy preserving difference methods have been widely used for Hamiltonian dynamical systems. However, since such numerical integrators do not preserve all of the conserved quantities, their behavior may be rather different from the trajectory of Hamiltonian systems. Indeed, these integrators can not describe a long-time behavior of the gravitational 2-body motion of Kepler. The purpose of the research project is to develop a new numerical integrator named TCI, totally conservative integrator, which preserving all of the conserved quantities of Hamiltonian systems. It was shown in a paper by Minesaki and Nakamura that a long-time behavior of the Kepler motion is completely preserved by the TCI.In 2006, the last year of the project, Minesaki and Inoue developed a new integrator which preserving all the conserved quantities of the gravitational 3-body motion. Since the general 3-body motion is a chaotic dynamical system, a candidate of such an integrator is formulated for the case of Langrage's equilateral triangle solution on a plain. The basic design of the integrator is to divide the equations of motion into 2-body parts and interaction parts of 3-body and then discretize them, individually. A key idea is a use of a gap in a time variable which appears in the discrete 2-body motion to keep the sum of relative coordinates zero. Consequently, it is shown that the resulting numerical integrator preserves Langrage's equilateral triangle solution exactly. Such a remarkable property is viewed in numerical simulation except for a round-off error in computer. The new integrator is shown to be superior than Stormer-Verlet's symplectic integrator for 3-body motion. It is also verified that the new integrator behaves well for the letter "8" solution of the 3-body motion and the corresponding energy is kept constant for a long period.