AI 中文总结
本文针对含速度比例摩擦力的耗散线性动力学系统，提出一种改进的分数阶微积分方法，解决了经典平稳作用原理无法推导其运动方程的问题，得到了正确的相关方程、能量变化及能量耗散的几何解释。
AI 中文摘要
长期以来，人们已知常系数耗散线性动力学系统的运动方程无法从经典平稳作用原理推导得出，因为运动方程中与速度成比例的项会导致拉格朗日量中出现半阶时间导数。因此，人们采用了分数阶微积分方法，但这些方法在数学和物理层面均存在缺陷。本文提出一种基于分数阶微积分的改进方案，可得到正确的欧拉-拉格朗日方程、最终的哈密顿方程、运动物体的能量变化，并尝试对能量耗散的几何解释进行探索。
英文摘要
It has been known for a long time that the equation of motion for dissipative linear dynamical systems with constant coefficients cannot be derived from the classical principle of stationary action because the term proportional to velocity in the equation of motion leads to the time derivative of order one half in the Lagrangian. Thus, approaches utilising fractional calculus have been used; however, they suffer from deficiencies both from mathematical and physical points of view. We here present our version of such fractional calculus based approach that provides correct Euler-Lagrange and, ultimately, the Hamilton equations, energy change of the moving body, and an attempt for a geometric interpretation of how energy dissipates.
