具有反二次非线性项的非中心振子高阶超可积性的新证明
A new proof of the higher-order superintegrability of a noncentral oscillator with inversely quadratic nonlinearities
摘要：
研究了具有非线性“离心”项的有理数谐振子（具有有理数频率比的非中心谐振子）的超可积性。在第一部分中，系统直接研究在欧几里德平面;高阶超可积性（运动积分的高阶比2的动量）的存在性被证明通过引入一个变形的二次复方程的线性系统。明确地获得了非线性系统的运动常数。在第二部分中，反问题进行了分析，在一般情况下的$n$度的自由，开始与一般的哈密顿量$H$，并引入适当的条件，获得超可积性，特别是“离心”的非线性。
英文摘要：
The superintegrability of a rational harmonic oscillator (non-central harmonic oscillator with rational ratio of frequencies) with non-linear "centrifugal" terms is studied. In the first part, the system is directly studied in the Euclidean plane; the existence of higher-order superintegrability (integrals of motion of higher order than 2 in the momenta) is proved by introducing a deformation in the quadratic complex equation of the linear system. The constants of motion of the nonlinear system are explicitly obtained. In the second part, the inverse problem is analyzed in the general case of $n$ degrees of freedom; starting with a general Hamiltonian $H$, and introducing appropriate conditions for obtaining superintegrability, the particular "centrifugal" nonlinearities are obtained.
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