二次Li $\acute{\textbf{e}}$ nard型方程$\ddot{x}+f(x)\dot{x}^2+g(x)=0$的Lie点对称分类
Classification of Lie point symmetries for quadratic Li$\acute{\textbf{e}}$nard type equation $\ddot{x}+f(x)\dot{x}^2+g(x)=0$
摘要：
本文对二次Li $\acute{e}$ nard型方程$\ddot {x} + f(x){\dot {x}}^{2} + g(x)= 0$的Lie点对称群进行了完全分类，其中$f(x)$和$g(x)$是$x$的任意函数.对称性分析分为两种情况，$(i)$最大（八参数）对称群和$(ii)$非最大（三，二和一个参数）对称群。我们确定的最一般形式的二次李$\acute{e}$纳德方程在这些情况下。在八参数对称群的情况下，所确定的一般方程既可线性化又可等时。我们提出了具体的例子物理兴趣。对于非最大的情况下，所确定的方程都是可积的，包括几个物理上有趣的例子，如马修斯-拉克什曼南振荡器，粒子上的旋转抛物井等，我们还分析了基本的等价变换。
英文摘要：
In this paper we carry out a complete classification of the Lie point symmetry groups associated with the quadratic Li$\acute{e}$nard type equation, $\ddot {x} + f(x){\dot {x}}^{2} + g(x)= 0$, where $f(x)$ and $g(x)$ are arbitrary functions of $x$. The symmetry analysis gets divided into two cases, $(i)$ the maximal (eight parameter) symmetry group and $(ii)$ non-maximal (three, two and one parameter) symmetry groups. We identify the most general form of the quadratic Li$\acute{e}$nard equation in each of these cases. In the case of eight parameter symmetry group, the identified general equation becomes linearizable as well as isochronic. We present specific examples of physical interest. For the nonmaximal cases, the identified equations are all integrable and include several physically interesting examples such as the Mathews-Lakshmanan oscillator, particle on a rotating parabolic well, etc. We also analyse the underlying equivalence transformations.
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