非线性动力系统的变分反问题
Inverse variational problem for nonlinear dynamical systems
摘要：
在本文中，我们选择了两种不同的方法来解决变分法的反问题。第一种方法是基于拉格朗日函数的积分表示，其使用运动运动方程的第一积分，而第二种方法依赖于众所周知的Noether定理的推广，并直接从运动运动方程构造拉格朗日函数。作为Lagrange函数积分表示的应用，我们首先对修正的Emden型方程的Lagrange函数给出了一些有用的注记，然后得到了（i）五阶Duffing振子，（ii）Liénard型振子和（iii）Mathews-Lakshmanan振子的Lagrange函数的结果.与修正的埃姆登型方程一样，这些振子被发现具有非标准拉格朗日量的特征，除了人们也可以将标准拉格朗日量分配给Duffing振子。我们使用第二种方法找到了三个速度相关方程的间接解析（拉格朗日）表示（iv）亚伯拉罕-洛伦兹振荡器，（v）洛伦兹振荡器和（vi）范德波尔振荡器。对于（i）-（vi）中的每一个动力系统，我们计算了Jacobi积分的结果，从而提供了一种在不求助于使用所谓的Legendre变换的情况下获得Hamilton函数的方法。
英文摘要：
In this paper we have chosen to work with two different approaches to solving the inverse problem of the calculus of variation. The first approach is based on an integral representation of the Lagrangian function that uses the first integral of the equation of motion while the second one relies on a generalization of the well known Noether's theorem and constructs the Lagrangian directly from the equation of motion. As an application of the integral representation of the Lagrangian function we first provide some useful remarks for the Lagrangian of the modified Emden-type equation and then obtain results for Lagrangian functions of (i) cubic-quintic Duffing oscillator, (ii) Liénard-type oscillator and (iii) Mathews-Lakshmanan oscillator. As with the modified Emden-type equation these oscillators were found to be characterized by nonstandard Lagrangians except that one could also assign a standard Lagrangian to the Duffing oscillator. We used the second approach to find indirect analytic (Lagrangian) representation for three velocity-dependent equations for (iv) Abraham-Lorentz oscillator, (v) Lorentz oscillator and (vi) Van der Pol oscillator. For each of the dynamical systems from (i)-(vi) we calculated the result for Jacobi integral and thereby provided a method to obtain the Hamiltonian function without taking recourse to the use of the so-called Legendre transformation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
