$E^2$中自治动力系统的三次第一积分的算法研究
Cubic first integrals of autonomous dynamical systems in $E^2$ by an algorithmic approach
摘要：
在最近的一篇论文（A。Mitsopoulos和M. Tsamparlis，J.Geom.Phys.170，104383，2021），给出了一个一般定理，该定理提供了一种算法方法，用于根据由系统的动力学方程定义的动力学度量的对称性来计算自治动力系统的第一积分（FI）。在目前的工作中，我们应用这个定理来计算三次FI自治保守牛顿动力系统的两个自由度。我们表明，已知的结果，这一主题，这已经通过各种不同的方法，并在这项工作中得到的额外的可以得到由该定理提供的单一算法的方法。结果被收集在四个表中，这些表可以用作这种类型的可积和超可积势的更新参考。我们所得到的结果是针对自由参数的特殊值，因此，利用本文所发展的方法，其他研究者通过不同的参数选择，将能够找到新的可积势和超可积势.
英文摘要：
In a recent paper (A. Mitsopoulos and M. Tsamparlis, J. Geom. Phys. 170, 104383, 2021), a general theorem is given which provides an algorithmic method for the computation of first integrals (FIs) of autonomous dynamical systems in terms of the symmetries of the kinetic metric defined by the dynamical equations of the system. In the present work, we apply this theorem to compute the cubic FIs of autonomous conservative Newtonian dynamical systems with two degrees of freedom. We show that the known results on this topic, which have been obtained by means of various different methods, and additional ones derived in this work can be obtained by the single algorithmic method provided by this theorem. The results are collected in four Tables which can be used as an updated reference of this type of integrable and superintegrable potentials. The results we find are for special values of free parameters; therefore, using the methods developed here, other researchers by different suitable choice of the parameters will be able to find new integrable and superintegrable potentials.
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