论文标题
关于牛顿系统的分析表示
On the analytic representation of Newtonian systems
论文作者
论文摘要
我们表明，可以使用自相关差分方程的理论来提供令人满意的解决方案，以使经典力学中的逆变异问题解决。牛顿方程转换为自我偶像形式时，人们可以为其找到适当的拉格朗日表示（直接分析表示）。另一方面，同一牛顿方程与其伴随相同，为系统构建了不同的拉格朗日表示（间接分析表示）提供了一个基础。我们从运动方程式的自我参与形式获得了阻尼的谐波振荡器的时间依赖性的拉格朗日，同时识别出与所谓的bateman图像方程的方程式的伴奏，并以构建时间独立的间接拉格格兰语表示。我们提供了许多案例研究，以证明我们得出的方法的有用性。我们还通过使用Lagrangian函数的积分表示并发表一些有用的评论，为许多非线性微分方程提供了相似的结果。
We show that the theory of self-adjoint differential equations can be used to provide a satisfactory solution of the inverse variational problem in classical mechanics. A Newtonian equation when transformed to the self-adjoint form allows one to find an appropriate Lagrangian representation (direct analytic representation) for it. On the other hand, the same Newtonian equation in conjunction with its adjoint provides a basis to construct a different Lagrangian representation (indirect analytic representation) for the system. We obtain the time-dependent Lagrangian of the damped Harmonic oscillator from the self-adjoint form of the equation of motion and at the same time identify the adjoint of the equation with the so called Bateman image equation with a view to construct a time-independent indirect Lagrangian representation. We provide a number of case studies to demonstrate the usefulness of the approach derived by us. We also present similar results for a number of nonlinear differential equations by using an integral representation of the Lagrangian function and make some useful comments.
