应用数学中的无穷小与无穷大数
Infinitesimal and infinite numbers in applied mathematics
AI总结：
本文回顾广义光滑函数理论，用严格的无穷小与无穷大语言处理连续与广义函数，并通过奇异动力系统、变长摆、阻尼摆、钢材应力-应变、奇异拉格朗日量及量子力学等模型展示其应用价值。
摘要：
在应用中，描述非线性系统对脉冲刺激的突变或响应这一需求无处不在。同时，非正式地使用无穷小和无穷大量仍然是一种方法，用于在著名的J. von Neumann所提出的相当广泛的可应用范围内构建理想化但可处理的模型。我们回顾广义光滑函数理论，将其作为同时满足这两类需求的候选方案：它提供了一种严格但简单的无穷小与无穷大量的语言，并且使得连续函数和广义函数可以像光滑映射一样处理：具有逐点值、自由复合以及由此带来的非线性运算、所有经典微积分定理、良好的积分理论，以及微分方程新的存在性结果。我们通过若干奇异动力系统模型来例示该理论的应用：推广到广义函数的热方程和波动方程的推导、一个缠绕在平行六面体上的奇异变长摆、由不同介质阻尼的摆的振动、钢材的非线性应力-应变模型、光学中使用的奇异拉格朗日量，以及一些来自量子力学的例子。
英文摘要：
The need to describe abrupt changes or response of nonlinear systems to impulsive stimuli is ubiquitous in applications. Also the informal use of infinitesimal and infinite quantities is still a method used to construct idealized but tractable models within the famous J. von Neumann reasonably wide area of applicability. We review the theory of generalized smooth functions as a candidate to address both these needs: a rigorous but simple language of infinitesimal and infinite quantities, and the possibility to deal with continuous and generalized function as if they were smooth maps: with pointwise values, free composition and hence nonlinear operations, all the classical theorems of calculus, a good integration theory, and new existence results for differential equations. We exemplify the applications of this theory through several models of singular dynamical systems: deduction of the heat and wave equations extended to generalized functions, a singular variable length pendulum wrapping on a parallelepiped, the oscillation of a pendulum damped by different media, a nonlinear stress-strain model of steel, singular Lagrangians as used in optics, and some examples from quantum mechanics.
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