论文标题
革命的可整合系统与Zoll的指标
superintegrable systems versus Zoll metrics of revolution
论文作者
论文摘要
Koenigs在Momenta中构建了一个具有线性和两个二次积分的二维共聚（SI）模型的家族（1,2）。最近，Matveev和Shevchishin表明，这种结构确实概括为具有一个线性和两个立方积分的模型，即$（1,3）$，直到解决非线性普通微分方程的解决方案。我们对该方程式的明确解决方案允许构建这些SI系统，并导致证明在S^2上全球定义的系统是Zoll。对于任何n \ geq 2，我们将将这些结果推广到案例（1，n）。我们的方法再次具有建设性，并表明在n奇怪的情况下，在s^2上定义的指标的存在确实是ZOLL（在参数的适当限制下）的存在（如果n是n是n是衡量的，我们都不会在s^2上构建s^2，而构建的模型已经是ko的。
Koenigs constructed a family of two dimensional superintegrable (SI) models with one linear and two quadratic integrals in the momenta, shortly (1,2). More recently Matveev and Shevchishin have shown that this construction does generalize to models with one linear and two cubic integrals i.e. $(1,3)$, up to the solution of a non-linear ordinary differential equation. Our explicit solution of this equation allowed for the construction of these SI systems and led to the proof that the systems globally defined on S^2 are Zoll. We will generalize these results to the case (1,n) for any n \geq 2. Our approach is again constructive and shows the existence, when n is odd, of metrics globally defined on S^2 which are indeed Zoll (under appropriate restrictions on the parameters), while if n is even the metrics we found are never globally defined on S^2, as it is already the case for the (1,2) models constructed by Koenigs.
