论文标题
使用Liegroup/代数方法对时间无关的哈密顿系统的代数离散化
Algebraic discretization of time-independent Hamiltonian systems using a Lie-group/algebra approach
论文作者
论文摘要
在本文中，与时间无关的汉密尔顿体系通过lie group/代数形式主义进行了研究。与哈密顿量相关的（未知的）解决方案被认为是初始数据的底盘转换，其中组参数充当时间。时间进化发生器（即与组转换相关的LIE代数）是在代数级别构建的，因此避免了离散情况的时间衍生物的离散化。这种形式主义使得针对时间无关的汉密尔顿系统的连续和离散版本的时间，并且不需要有关系统的其他信息（除了哈密顿本身和解决方案的初始条件之外）。当独立于时间的汉密尔顿系统在liouville的意义上是可以集成的时，可以使用动作角度坐标来拉直时间进化发生器并构建精确的方案（即没有错误的方案）。此外，还提供了一种分析近似/数值方案误差的方法。这些考虑因素应用于与一维谐波振荡器相关的知名示例。
In this paper, time-independent Hamiltonian systems are investigated via a Lie-group/algebra formalism. The (unknown) solution linked with the Hamiltonian is considered to be a Lie-group transformation of the initial data, where the group parameter acts as the time. The time-evolution generator (i.e. the Lie algebra associated to the group transformation) is constructed at an algebraic level, hence avoiding discretization of the time-derivatives for the discrete case. This formalism makes it possible to investigate the continuous and discrete versions of time for time-independent Hamiltonian systems and no additional information on the system is required (besides the Hamiltonian itself and the initial conditions of the solution). When the time-independent Hamiltonian system is integrable in the sense of Liouville, one can use the action-angle coordinates to straighten the time-evolution generator and construct an exact scheme (i.e. a scheme without errors). In addition, a method to analyse the errors of approximative/numerical schemes is provided. These considerations are applied to well-known examples associated with the one-dimensional harmonic oscillator.
