混合二次线性Li $\acute{\textbf{e}}$ nard型方程$\ddot{x}+f(x)\dot{x}^2+g(x)\dot{x}+h(x)=0$的完全Lie点对称分类
On the complete Lie point symmetries classification of the mixed quadratic-linear Li$\acute{\textbf{e}}$nard type equation $\ddot{x}+f(x)\dot{x}^2+g(x)\dot{x}+h(x)=0$
摘要：
在本文中，我们开发了一个系统的和自洽的程序的基础上的一组兼容性条件，以确定所有的最大（八个参数）和非最大（一个和两个参数）的对称群与混合二次线性Li $\acute{e}$纳德型方程，$\ddot {x} + f(x){\dot {x}}^{2} + g(x)\dot{x}+h(x)= 0$，其中$f(x),\,g(x)$和$h(x)$是$x$的任意函数。借助于这个过程，我们证明了对称函数$b(t)$在非极大值情况下为零，而在极大值情况下则不然。在此基础上，将对称性分析分为两种情况：最大对称群$(b\neq0)$的$(i)$和非最大对称群$(b=0)$的$(ii)$。然后，我们确定的最一般形式的混合二次线性李$\acute{e}$纳德型方程在这些情况下。在八参数对称群的情况下，所确定的一般方程变为可线性化的。我们提出一个物理兴趣的具体例子。在非极大对称群的情况下，所确定的方程都是可积的。通过给出通解或构造与时间无关的哈密顿量，证明了所有方程的可积性。我们还分析了潜在的等价变换。
英文摘要：
In this paper we develop a systematic and self consistent procedure based on a set of compatibility conditions for identifying all maximal (eight parameter) and non-maximal (one and two parameter) symmetry groups associated with the mixed quadratic-linear Li$\acute{e}$nard type equation, $\ddot {x} + f(x){\dot {x}}^{2} + g(x)\dot{x}+h(x)= 0$, where $f(x),\,g(x)$ and $h(x)$ are arbitrary functions of $x$. With the help of this procedure we show that a symmetry function $b(t)$ is zero for non-maximal cases whereas it is not so for the maximal case. On the basis of this result the symmetry analysis gets divided into two cases, $(i)$ the maximal symmetry group $(b\neq0)$ and $(ii)$ non-maximal symmetry groups $(b=0)$. We then identify the most general form of the mixed-quadratic linear Li$\acute{e}$nard type equation in each of these cases. In the case of eight parameter symmetry group, the identified general equation becomes linearizable. We present a specific example of physical interest. In the case of non-maximal symmetry groups the identified equations are all integrable. The integrability of all the equations is proved either by providing the general solution or by constructing time independent Hamiltonians. We also analyse the underlying equivalence transformations.
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