论文标题
当动作是针对具有行动依赖性拉格朗日的系统的行动时
When action is not least for systems with action-dependent Lagrangians
论文作者
论文摘要
根据Herglotz的变异原理，可以通过计算动作依赖性动作的第一个变化来得出某些非保守和耗散系统的动力学。这与通常用于得出保守系统动力学的汉密尔顿的变异原理直接类似。正如保守系统动作的第二个变化可以用于推断该系统的可能轨迹是否在动态上稳定一样，也可以使用该系统的第二个变化来推断非保存和耗散系统的可能轨迹是否动态稳定。在本文中，我显示的是，对保守系统动作的第二个变化进行了概括，如何计算动作依赖性动作的第二个变化以及如何将其应用于两个物理上重要的系统：一种独立于时间依赖的谐波振荡器和时间依赖性的谐波振荡器。
The dynamics of some non-conservative and dissipative systems can be derived by calculating the first variation of an action-dependent action, according to the variational principle of Herglotz. This is directly analogous to the variational principle of Hamilton commonly used to derive the dynamics of conservative systems. In a similar fashion, just as the second variation of a conservative system's action can be used to infer whether that system's possible trajectories are dynamically stable, so too can the second variation of the action-dependent action be used to infer whether the possible trajectories of non-conservative and dissipative systems are dynamically stable. In this paper I show, generalizing earlier analyses of the second variation of the action for conservative systems, how to calculate the second variation of the action-dependent action and how to apply it to two physically important systems: a time-independent harmonic oscillator and a time-dependent harmonic oscillator.
