ON THE HAMILTONIAN INTERPOLATION OF NEAR-TO-THE-IDENTITY SYMPLECTIC MAPPINGS WITH APPLICATION TO SYMPLECTIC INTEGRATION ALGORITHMS
ON THE HAMILTONIAN INTERPOLATION OF NEAR-TO-THE-IDENTITY SYMPLECTIC MAPPINGS WITH APPLICATION TO SYMPLECTIC INTEGRATION ALGORITHMS
复制标题
DOI：
10.1007/bf02188219
复制
发表时间：
1994-03-01
影响因子：
1.6
通讯作者：
GIORGILLI, A
中科院分区：
文献类型：
--
作者：
BENETTIN, G;GIORGILLI, A
我们重新考虑辛映射的哈密顿插值问题。根据Moser的方案，我们证明了对于任意映射PSI(epsilon)，解析映射和接近恒等的映射，存在一个解析自治哈密顿系统H(epsilon)，使得它的时间- 1映射PHI（Hepsilon）与PSI（epsilon）相差一个指数小的量（1/epsilon）。该结果特别适用于用辛算法对哈密顿系统进行数值积分的问题；事实证明，当使用s阶的解析辛算法来整合哈密顿系统K时，实际上是“精确地”遵循，即在计算机舍入误差范围内，插值哈密顿函数H（epsilon）的轨迹，或者相当于重新缩放的哈密顿函数K(epsilon) = epsilon-1 H（epsilon）的轨迹，它与K不同，但结果是接近于它。特别注意散射问题的数值积分。
We reconsider the problem of the Hamiltonian interpolation of symplectic mappings. Following Moser's scheme, we prove that for any mapping PSI(epsilon), analytic and epsilon-close to the identity, there exists an analytic autonomous Hamiltonian system, H(epsilon) such that its time-one mapping PHI(Hepsilon) differs from PSI(epsilon) by a quantity exponentially small in 1/epsilon. This result is applied, in particular, to the problem of numerical integration of Hamiltonian systems by symplectic algorithms; it turns out that, when using an analytic symplectic algorithm of order s to integrate a Hamiltonian system K, one actually follows ''exactly,'' namely within the computer roundoff error, the trajectories of the interpolating Hamiltonian H(epsilon), or equivalently of the rescaled Hamiltonian K(epsilon) = epsilon-1 H(epsilon), which differs from K, but turns out to be epsilon(s) close to it. Special attention is devoted to numerical integration for scattering problems.