三类保辛摄动及其数值比较

摄动法近似应当保辛．本文指出，有限元位移法自动保辛，有限元混合能表示也保辛．摄动法的刚度阵Taylor级数展开能证明保辛；混合能的Taylor级数展开摄动也证明了保辛．但传递辛矩阵的Taylor级数展开摄动却不能保辛．辛矩阵只能在乘法群下保辛，故传递辛矩阵的保辛摄动必须采用正则变换的乘法．虽然刚度阵加法摄动、混合能矩阵加法摄动与传递辛矩阵正则变换乘法摄动都保辛，但这3种摄动近似并不相同．最后通过数值例题给出了对比．

Numerical comparison for three different symplectic perturbation methods

The perturbation approximation should be symplectic conservative. The FEM displacement method is naturally symplectic conservative, the mixed energy representation is also shown symplectic conservative. The stiffness matrix, as well as the mixed energy matrix, perturbations based on the Taylor series expansion are all proved symplectic conservative. However, the Taylor series expansion perturbation for the symplectic transfer matrix is not able to reach symplectic conservative. The symplectic conservative perturbation for a transfer symplectic matrix should be based on the canonical transformation method. Although, the perturbation methods of the stiffness matrix, of the mixed energy matrix and of the transfer symplectic matrix multiplicative perturbation are all symplectic conservative, however, they are not identical. Therefore, numerical comparison is given for an example.