A new way to use nonlocal symmetries to determine first integrals of second-order nonlinear ordinary differential equations|非线性方程数据集|对称性分析数据集

Finding first integrals of second-order nonlinear ordinary differential equations (nonlinear 2ODEs) is a very difficult task. In very complicated cases, where we cannot find Darboux polynomials (to construct an integrating factor) or a Lie symmetry (that allows us to simplify the equations), we sometimes can solve the problem by using a nonlocal symmetry. In [1], [2], [3] we developed (and improved) a method (S-function method) that is successful in finding nonlocal Lie symmetries to a large class of nonlinear rational 2ODEs. However, even with the nonlocal symmetry, we still need to solve a 1ODE (which can be very difficult to solve) to find the first integral. In this work we present a novel way of using the nonlocal symmetry to compute the first integral with a very efficient linear procedure.

寻找二阶非线性常微分方程（非线性2ODEs）的第一积分是一项极具挑战性的任务。在极其复杂的情况下，当无法找到达布多项式（以构造积分因子）或李对称性（允许我们简化方程）时，我们有时可以通过使用非局部对称性来解决问题。在文献[1]、[2]、[3]中，我们（并对其进行了改进）开发了一种方法（S函数方法），该方法在寻找大量非线性有理2ODEs的非局部李对称性方面取得了成功。然而，即使有了非局部对称性，我们仍需解决一个一阶常微分方程（可能非常难以解决）以找到第一积分。在本工作中，我们提出了一种新颖的方法，利用非局部对称性通过一个非常高效的线性程序来计算第一积分。