论文标题
牛顿方程揭示了具有振荡力的特殊极限和振动力
Distinguished Limits and Vibrogenic Force revealed by Newton's Equation with Oscillating Force
论文作者
论文摘要
在本文中，我们分析了振动动力学的基本思想和两种成熟的方法。为了使我们的分析最具启发性，我们选择了一般振荡力的牛顿方程式。我们在高频限制下处理其渐近解决方案。我们的治疗很简单，但一般。我们研究的目标是\ emph {特殊极限}和\ emph {通用振动力}。 \ emph {特殊极限过程}的目的是确定如何在方程式中出现小参数。小参数的正确出现导致\ emph {有效的连续近似}，尤其是平均方程式的封闭系统。我们表明，只有两个区别的限制。这意味着牛顿方程式具有高频强迫，具有两种有趣的渐近解决方案。所有区别限制的平均方程中的关键项是\ emph {唯一的振动力}。该领域的当前最新技术是：许多特定的示例是众所周知的，有效和先进的一般方法（例如Krylov-Bogolyubov方法）。但是，提出的一般和简单的分析是对杰出极限和作为紧凑的实用指南配制的振动力是新颖的。我们处理的一个优点是它可以直接用于具有振荡系数的各种ODE和PDE。
In this paper, we analyse the basic ideas of Vibrodynamics and the two-timing method. To make our analysis most instructive, we have chosen the Newton's equation with a general oscillating force. We deal with its asymptotic solutions in the high frequency limit. Our treatment is simple but general. The targets of our study are \emph{the distinguished limits} and \emph{the universal vibrogenic force}. The aim of \emph{the distinguished limit procedure} is to identify how the small parameter can appear in an equation. The proper appearance of a small parameter leads to \emph{valid successive approximations}, and, in particular, to closed systems of averaged equations. We show, that there are only two distinguished limits. This means that Newton's equation, with high-frequency forcing, has two types of interesting asymptotic solutions. The key item in the averaged equations for all distinguished limits is \emph{the unique vibrogenic force}. The current state-of-the-art in this area is: a large number of particular examples are well-known, effective and advanced general methods (like the Krylov-Bogolyubov approach) are well developed. However, the presented general and simple analysis of distinguished limits and the vibrogenic force, formulated as a compact practical guide, is novel. An advantage of our treatment is the possibility of its straightforward use for various ODEs and PDEs with oscillating coefficients.
