Abstract

The classical Lagrange problem for dynamical systems introduces a Lagrangian action functional defined for any dynamical process that is envisioned to take place over a fixed interval of time with its state at each time lying on an unknown, but prescribed, configuration between two given end points in an \(n\) -dimensional state space \(\mathbb{R}^{n}\) . It is proposed that the fundamental dynamical field equation that characterizes the dynamical process and determines the precise motion between the two given end points is the Euler–Lagrange equation related to the stationarity of the Lagrangian action functional, expressed as the integral of a particularly formulated action density over the fixed time interval, among all admissible configurations that span the two given end points. Thus stated, this variational calculus problem introduces variations of a configuration that carries a dynamical process, and emphasizes the novelty and need to express explicitly how the configuration influences the state of that process. At each time during a dynamical process the state is subjected to an extrinsic force (classically taken to be conservative) which must be transmitted to the configuration that carries the process and, by action-reaction the configuration responds with a configuration contact force on the state of equal magnitude but opposite direction. This allows the Lagrangian action functional for a dynamical process to be interpreted as the difference between the average kinetic energy of the dynamical process that is carried by that configuration and the average configurational work done by the configuration contact force on the moving state as the state traverses that configuration during the fixed time interval. The aim in the Problem of Lagrange is to extremize this difference over all admissible configurations. The implication is that given a time interval and initial and final end points in the space of all states, the dynamical process of physical interest must follow a configuration that optimizes the gap between the average expended kinetic energy and the average expended configurational work. When the optimal condition is met and the dynamical process is so restricted, the difference between these average expenditures of energy and work will be at a local maximum, a local minimum, or a saddle point known as a condition of “least action”.

Herein, we investigate the optimization implications of this novel interpretation of the action functional for the Problem of Lagrange for dynamical systems for a general, possibly non-conservative, state-dependent extrinsic force field. We show that only a conservative state-dependent extrinsic force field is allowable within the statement of the problem and, thus, reaffirm the predominant classical hypothesis of restricting attention to conservative extrinsic force fields.

We close with a section which covers an analogous investigation of the Problem of Lagrange for nonlinear elastodynamics.

中文翻译：

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解决动力系统和非线性弹性动力学拉格朗日问题的新方法

摘要

动力系统的经典拉格朗日问题引入了拉格朗日作用函数，该函数定义为任何动态过程，该过程预计在固定的时间间隔内发生，其每次状态都位于两个给定端点之间的未知但规定的配置上。一个\(n\)维状态空间\(\mathbb{R}^{n}\)。提出表征动力过程并确定两个给定端点之间的精确运动的基本动力场方程是与拉格朗日作用泛函的平稳性相关的欧拉-拉格朗日方程，表示为特定公式化作用的积分在跨越两个给定端点的所有允许的配置中，固定时间间隔内的密度。因此，这种变分微积分问题引入了承载动态过程的配置的变化，并强调新颖性和需要明确表达配置如何影响该过程的状态。在动态过程中的每次，状态都会受到外在力（通常被认为是保守的）的影响，该外力必须传递到承载该过程的构型，并且通过作用-反作用，构型以构型接触力对状态做出响应大小相等但方向相反。这使得动态过程的拉格朗日作用函数可以被解释为该配置所承载的动态过程的平均动能与状态遍历时配置接触力对移动状态所做的平均配置功之间的差值在固定时间间隔内进行该配置。拉格朗日问题的目的是在所有可接受的配置上将这种差异最大化。这意味着，给定时间间隔以及所有状态空间中的初始和最终端点，物理兴趣的动力学过程必须遵循优化平均消耗动能和平均消耗构型功之间差距的配置。当满足最佳条件并且动态过程受到如此限制时，这些平均能量消耗和功之间的差异将处于局部最大值、局部最小值或称为“最小作用”条件的鞍点。

在此，我们研究了这种对拉格朗日问题的作用泛函的新颖解释的优化含义，该解释适用于一般的、可能非保守的、依赖于状态的外力场的动力系统。我们表明，在问题的陈述中只允许保守的依赖于状态的外力场，因此，重申了限制对保守外力场的关注的主要经典假设。

我们最后一节涵盖了非线性弹性动力学拉格朗日问题的类似研究。