基于对偶变量变分原理和两端位移独立变量的保辛方法

将广义位移和动量同时用拉格朗日多项式近似,并选择积分区间两端位移为独立变量,然后基于对偶变量变分原理导出了哈密顿系统的离散正则变换和对应的数值积分保辛算法。当位移和动量的拉格朗日多项式近似阶数满足一定条件时,可以自然导出保辛算法的不动点格式。通过数值算例分析了位移和动量采用不同阶次插值所需最少Gauss积分点个数,并讨论了位移插值阶数、动量插值阶数以及Gauss积分点个数对保辛算法精度的影响,说明了上述不动点格式恰好是一种最优格式。

Symplectic method based on dual variational principle and independent displacement variables at two ends

In this paper, the generalized displacements and momentum are approximated by Lagrange polynomial and the displacements at the two ends of time interval are taken as the independent variables, then the discrete Hamilton canonical equations and the corresponding symplectic method are derived based on the dual variable principle. A fixed point iteration formula can be derived when the order of the approximate polynomials of displacements and momentum satisfy some certain conditions. In the numerical examples part, the minimum number of Gauss integration pointrequired for different order of the approximate polynomials of displacements and momentum is discussed, and also the numerical precision of the proposed symplectic method for different orders of the approximate polynomials of displacements and momentum and numbers of Gauss integration point is discussed. It demonstrates that the fixed point iteration formula is the optimal one.