AI 中文总结
本研究提出一类由二阶常微分方程交换因式分解的参数形变生成的混合Liénard型非自治非线性振子，推导了幂律因式分解下的等时波形闭式解，刻画了其变分结构，并得到位置相关质量形式的运动方程。
AI 中文摘要
我们提出了一类混合Liénard型非自治非线性振子方程，它源自对具有周期解的二阶常微分方程所应用的交换因式分解过程的参数形变。针对因式分解函数取幂律形式$\phi(x)=kx^{q}$（其中$k\in \mathbb{R}$，$q\in\mathbb{N}$，对应所谓的modified Emden振子）的情况，我们通过Riccati约化方案得到了其解，尤其是等时波形的闭式解，且这些解不依赖于任意形变参数$A_1$。该方程的变分结构由一个拉格朗日量和一个广义瑞利耗散函数共同表征，后者包含一个关于$\dot{x}$的非标准三次项。我们还证明，将运动方程乘以雅可比乘子$M(x)=x^{-2A_1}$后，可得到具有相关摩擦力和回复力的位置相关质量（PDM）形式。
英文摘要
We introduce a class of nonautonomous nonlinear oscillator equations of mixed Liénard type that arises from a parametric deformation of the commutative factorization procedure applied to second-order ordinary differential equations with periodic solutions. Their solutions, in particular the isochronous waveforms, are obtained in closed form through a Riccati reduction scheme for the power-law choice of the factorization function, $ϕ(x)=kx^{q}$, $k\in \mathbb{R}$, and $q\in\mathbb{N}$, corresponding to the so-called modified Emden oscillators, and do not depend on the arbitrary deformation parameter $A_1$. The variational structure of the equation is characterised by a Lagrangian supplemented with a generalised Rayleigh dissipation function that contains a non-standard cubic term in $\dot{x}$. We also show that multiplying the equation of motion by the Jacobi multiplier $M(x)=x^{-2A_1}$ a position-dependent-mass (PDM) form is obtained with related friction and restoring force.
Comments9 pages, 3 figures, 22 references
