arXiv 0905.16272009-05-11math-phmath.MP

局部路径拟合：变分积分的一种新方法

Local Path Fitting: A New Approach to Variational Integrators

D S Vlachos, O T Kosmas

备注：: 15 pages, 4 figures, 2 tables

摘要：

在这项工作中，我们提出了一种新的方法来建设变分积分。在一般情况下，在时间间隔$[q_k,q_{k+1}]$的作用积分的估计被用来构建一个辛映射$(q_k,q_{k+1})\to (q_{k+1},q_{k+2})$。这里的基本思想是，在一般理论中只需要拉格朗日量的作用量积分的估计的偏导数。这些衍生物的分析计算，提出了一个新的积分，它不依赖于拉格朗日，但对欧拉-拉格朗日向量，在连续和精确的情况下，消失。由于这种新的积分只能通过基于一些内部网格点的数值方法计算，我们可以通过要求欧拉-拉格朗日矢量在这些网格点处消失来局部拟合精确曲线。因此积分为零，并且该过程显著地简化了高阶近似的计算。在高偏心率（高达0.99）的二体问题和外太阳系的高阶解的新技术进行了测试。

英文摘要：

In this work, we present a new approach to the construction of variational integrators. In the general case, the estimation of the action integral in a time interval $[q_k,q_{k+1}]$ is used to construct a symplectic map $(q_k,q_{k+1})\to (q_{k+1},q_{k+2})$. The basic idea here, is that only the partial derivatives of the estimation of the action integral of the Lagrangian are needed in the general theory. The analytic calculation of these derivatives, give raise to a new integral which depends not on the Lagrangian but on the Euler--Lagrange vector, which in the continuous and exact case vanishes. Since this new integral can only be computed through a numerical method based on some internal grid points, we can locally fit the exact curve by demanding the Euler--Lagrange vector to vanish at these grid points. Thus the integral vanishes, and the process dramatically simplifies the calculation of high order approximations. The new technique is tested for high order solutions in the two-body problem with high eccentricity (up to 0.99) and in the outer solar system.

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摘要：

英文摘要：

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