收敛弦方法：Hamilton边值问题的存在性和逼近
A convergent string method: Existence and approximation for the Hamiltonian boundary-value problem
摘要：
本文研究一类Hamilton边值问题长时间解的存在性及其相容数值逼近。例如，这样的边值问题在分子动力学中是常见的，其中人们的目标是找到将给定的初态与最终状态连接的动态轨迹，其中演化由经典（哈密顿）动力学控制。这里所考虑的设置是足够一般的，因此，只要适当地选择总能量$E$，就可以包括连接两种构型的长时间转捩轨迹。特别是，这里提出的配方可以用来检测两个稳定的盆地之间的过渡路径，从而证明存在的长期轨迹。出发点是制定的经典力学的运动方程的框架中的雅可比原理;曲线缩短程序的启发伯克霍夫的方法，然后开发找到测地线的解决方案。这种方法可以看作是一种字符串方法。
英文摘要：
This article studies the existence of long-time solutions to the Hamiltonian boundary value problem, and their consistent numerical approximation. Such a boundary value problem is, for example, common in Molecular Dynamics, where one aims at finding a dynamic trajectory that joins a given initial state with a final one, with the evolution being governed by classical (Hamiltonian) dynamics. The setting considered here is sufficiently general so that long time transition trajectories connecting two configurations can be included, provided the total energy $E$ is chosen suitably. In particular, the formulation presented here can be used to detect transition paths between two stable basins and thus to prove the existence of long-time trajectories. The starting point is the formulation of the equation of motion of classical mechanics in the framework of Jacobi's principle; a curve shortening procedure inspired by Birkhoff's method is then developed to find geodesic solutions. This approach can be viewed as a string method.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
