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.

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关于牛顿系统的解析表示

我们表明自伴随微分方程理论可用于为经典力学中的逆变分问题提供令人满意的解决方案。将牛顿方程转换为自伴随形式时，可以为其找到合适的拉格朗日表示（直接解析表示）。另一方面，同一个牛顿方程及其伴随方程为系统构造不同的拉格朗日表示（间接解析表示）提供了基础。我们从运动方程的自伴随形式得到阻尼谐振子的时间相关拉格朗日量，同时用所谓的贝特曼图像方程识别方程的伴随方程，以构建一个时间-独立的间接拉格朗日表示。我们提供了许多案例研究来证明我们得出的方法的有用性。我们还通过使用拉格朗日函数的积分表示为许多非线性微分方程提供了类似的结果，并提出了一些有用的评论。