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

科学仪器已经改变了科学的本质，这一点得到了充分的认识和越来越多的研究。例如，望远镜和显微镜为科学研究打开了一个全新的世界。另一种仪器也改变了科学；但是，由于检查这些发展的困难，很少用任何历史复杂性来检查这些工具。这些工具是新的数学技术。虽然牛顿的《原理》不是用新的微积分写成的（这样他的同时代人就可以理解他的论点），但如果没有微积分，牛顿的理论就不可能得到充分发展。同样，如果没有新的数学工具，爱因斯坦的广义相对论也不可能发展起来，而这些工具是他在本世纪第二个十年的早期不得不学习的。这些都是显而易见的例子，但还有很多很多的例子根本没有被研究过。特鲁斯戴尔教授是一位屡获殊荣的数学家和科学家，他具有专业技能和独特的历史敏感性，可以对数学在科学发展中的作用进行分析。在这项资助下，他将研究一个以前完全被忽视的领域：二重积分变分学的历史。多重积分的变分是当代流体、弹性固体和电磁学理论分析的重要工具。在连续力学中使用多重积分使我们能够理解并在一定程度上控制我们周围的世界。因此，研究海洋、河流、湖泊、潮汐和风、声波振动、大气中的飞行和地震都是可能的，这只是举几个例子。变分原理最早应用于二重积分和三重积分是由拉格朗日在1762年至1811年间的工作中提出的。拉格朗日的研究影响了19世纪30年代由纳维尔、泊松和柯西提出的广义弹性变形理论的发展。它们对乔治·格林的研究产生了明显的影响，并通过他影响了数学物理的各个分支。Truesdell教授将研究这段历史，直到1830年左右，详细追溯数学的发展，并描述拉格朗日、泊松和罗德里格斯在力学和微分几何方面的早期应用。这项研究将极大地增强我们对数学和科学之间关键的相互作用的理解——这种相互作用实际上重塑了每一门科学。

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.