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

辛积分器和能量守恒差分方法已被广泛应用于哈密顿动力系统。然而，由于这样的数值积分器并不保持所有的守恒量，它们的行为可能与哈密顿系统的轨迹有很大的不同。事实上，这些积分器不能描述开普勒引力二体运动的长期行为。研究项目的目的是开发一种新的数值积分器TCI，完全守恒积分器，它保持了哈密顿系统的所有守恒量。在Minesaki和Nakamura的一篇论文中，TCI完全保留了开普勒运动的一个长期行为。2006年，也就是该项目的最后一年，Minesaki和Inoue开发了一个新的积分器，它保持了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，对于平面上的朗格朗日等边三角形解，给出了这种积分器的一种候选形式。积分器的基本设计是将运动方程分解为二体部分和三体相互作用部分，然后分别离散。一个关键的想法是利用离散两体运动中出现的时间变量的间隙来保持相对坐标之和为零。结果表明，所得到的数值积分器完全保持了朗格朗日等边三角形解。除了计算机的舍入误差外，这种显著的特性在数值模拟中是可以看到的。对于三体运动，新积分器优于Stormer-Verlet辛积分器。并验证了新积分器对三体运动的字母“8”解表现良好，且相应的能量长期保持不变。

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.