$n$维非线性PDM振子：线性化、不变性和精确可解性
Isochronous $n$-dimensional nonlinear PDM-oscillators: linearizability, invariance and exact solvability
摘要：
在标准拉格朗日设置（即，（动能和势能之间的差异），我们讨论并报道了一些$n$维非线性位置依赖质量（PDM）振子的等时性、线性化性和精确可解性.在这个过程中，PDM势力场的负梯度被证明不再与正则动量的时间导数$\mathbf{p}% =m\left( r\right) \mathbf{\dot{r}}$相关，而是与伪动量的时间导数$\mathbf{π}\left( r\right) =\sqrt{% m\left( r\right) }\mathbf{\dot{r}}$相关（即，Noether momentum）。此外，利用点变换方法，我们证明了$n$维非线性PDM振子的线性化仅对$n=1$是可能的，而对$n\geq 2$是不可能的。在PDM设置下，$n\geq 2$的欧拉-拉格朗日不变性不足/不完整。因此，我们要寻找替代的不变性。这种不变性，像穆斯塔法\cite{42}的\emph{Newtonian invariance}一样，有效地授权使用一个系统的精确解来找到另一个系统的解。一个样本的等轴$n$维非线性PDM振荡器的例子报告。
英文摘要：
Within the standard Lagrangian settings (i.e., the difference between kinetic and potential energies), we discuss and report isochronicity, linearizability and exact solubility of some $n$-dimensional nonlinear position-dependent mass (PDM) oscillators. In the process, negative the gradient of the PDM-potential force field is shown to be no longer related to the time derivative of the canonical momentum, $\mathbf{p}% =m\left( r\right) \mathbf{\dot{r}}$, but it is rather related to the time derivative of the pseudo-momentum, $\mathbf{π}\left( r\right) =\sqrt{% m\left( r\right) }\mathbf{\dot{r}}$ (i.e., Noether momentum). Moreover, using some point transformation recipe, we show that the linearizability of the $n$-dimensional nonlinear PDM-oscillators is only possible for $n=1$ but not for $n\geq 2$. The Euler-Lagrange invariance falls short/incomplete for $n\geq 2$ under PDM settings. Alternative invariances are sought, therefore. Such invariances, like \emph{Newtonian invariance} of Mustafa \cite{42}, effectively authorize the use of the exact solutions of one system to find the solutions of the other. A sample of isochronous $n$-dimensional nonlinear PDM-oscillators examples are reported.
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