The Calculus of Variations for Double Integrals and It's Application to Theories of Membranes and Elastic Plates

二重积分变分法及其在膜和弹性板理论中的应用

• 批准号: 8709401
• 负责人: Clifford Truesdell
• 金额: $ 1.8万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1989-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8709401&HistoricalAwards=false
• 关键词: Calculus; Variations; Double; Integrals; Application

It is well recognized and increasingly studied that scientific instruments have transformed the nature of science. The telescope and microscope, for example, literally opened up whole new worlds for scientific examination. Another kind of instrument has also transformed science; but, because of the difficulties of examination of these developments, these instruments are rarely examined with any kind of historical sophistication. These instruments are new techniques of mathematics. While Newton's Principia was not written using the new calculus (so his contemporaries could understand his arguments) Newton's theory could not have been fully developed with out calculus. Similarly, Einstein's theory of general relativity could not have been developed without new mathematical tools which he had to learn in the early years of the second decade of this century. These are the obvious examples, but there are many, many more which simply have not been examined. Professor Truesdell, an award winning mathematician and scientist, has the technical skills and a unique historical sensitivity to carry out the kind of analysis of the role of mathematics in the development of science. Under this grant, he will examine a previously completely ignored area: the history of the calculus of variations for double integrals. Variations of multiple integrals are essential tools of analysis for contemporary theories of fluids, elastic solids and electromagnetism. The use of multiple integrals in the mechanics of continua enables us to comprehend and in part control the world around us today. Studies thus are possible of the oceans, rivers, lakes, tides and winds, acoustic vibrations, flight in the atmosphere, and earthquakes just to give a few examples. The beginnings and the earliest application of variational principles to double and triple integrals are due to Lagrange in his work between 1762 and 1811. Lagrange's studies affected the development of the general theory of elastic deformation in the 1830's by Navier, Poisson, and Cauchy. They had an obvious effect on the research of George Green and, through him, have influenced every branch of mathematical physics. Professor Truesdell will examine this history up through about 1830, tracing the mathematical development in detail and describing its early application to mechanics and differential geometry by Lagrange, Poisson, and Rodrigues. This study will greatly enhance our understanding of the critical interaction of mathematics and science--an interaction which has literally reshaped every one of the sciences.

人们普遍认识到， 仪器改变了科学的本质。 望远镜 和显微镜，例如，从字面上打开了全新的世界， 科学检查。 另一种仪器也 改变了科学;但是，由于检查的困难， 在这些发展中，这些工具很少被任何人审查， 历史的复杂性。 这些仪器是新的 数学技术。 虽然牛顿的《原理》并没有写成 使用新的微积分（这样他的同时代人就可以理解他的 牛顿的理论不可能完全发展， 出微积分。 同样，爱因斯坦的广义相对论 如果没有新的数学工具， 在第二个十年的早期， 世纪。 这些都是明显的例子，但还有很多很多 更多的只是没有被检查。 特鲁斯德尔教授，一位 屡获殊荣的数学家和科学家，拥有技术技能， 和独特的历史敏感性来进行这种分析 数学在科学发展中的作用 根据本 格兰特，他将研究一个以前完全被忽视的领域： 重积分变分法的历史。 多重积分的变化是分析的基本工具， 当代流体、弹性固体和 电磁学 多重积分在弹性力学中的应用 连续统使我们能够理解并部分地控制世界 今天在我们周围。 因此，研究海洋、河流、 湖泊，潮汐和风，声振动， 大气和地震只是举几个例子。 的 变分原理的起源和最早应用 二重和三重积分是由于拉格朗日在他的工作之间 1762年和1811年。 拉格朗日的研究影响了 19世纪30年代Navier提出的弹性变形的一般理论， 泊松和柯西。 他们对研究的影响是显而易见的。 乔治绿色，并通过他，影响了每一个分支的 数学物理 特鲁斯戴尔教授将研究这段历史 一直到1830年，详细追踪数学的发展， 并描述了它在力学和微分学中的早期应用 拉格朗日，泊松和罗德里格斯几何。 本研究将 极大地增强了我们对 数学和科学--一种真正重塑了 每一门科学