Most elementary numerical schemes found useful for solving classical trajectory problems are {\it canonical transformations}. This fact should be make more widely known among teachers of computational physics and Hamiltonian mechanics. From the perspective of advanced mechanics, there are no bewildering number of seemingly arbitrary elementary schemes based on Taylor's expansion. There are only {\it two} canonical first and second order algorithms, on the basis of which one can comprehend the structures of higher order symplectic and non-symplectic schemes. This work shows that, from the most elementary first-order methods to the most advanced fourth-order algorithms, all can be derived from canonical transformations and Poisson brackets of advanced mechanics.

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数值算法的结构和高级力学

大多数用于解决经典轨迹问题的基本数值方案是 {\it 规范变换}。这个事实应该在计算物理和哈密顿力学的教师中得到更广泛的了解。从高级力学的角度来看，基于泰勒展开式的看似随意的基本方案并没有令人眼花缭乱的数量。只有 {\it two} 规范的一阶和二阶算法，在此基础上可以理解高阶辛和非辛格式的结构。这项工作表明，从最基本的一阶方法到最先进的四阶算法，都可以从高级力学的规范变换和泊松括号中推导出来。