Mathematical Sciences: The Variational Bicomplex: Theory and Applications

首席研究员将继续研究 变分双复形这一理论最早被引入 大约15年前，提供了一个统一的方法， 变分法中的多重积分问题。的 双复形也被用来解决微积分中的反问题 的变化。变分双复形在 微分方程的几何理论 在有限维空间中的deRham复形所起的作用 流形这一主题将在研究中得到发展， 具体例子的应用是动机的主要来源。 创建一个“演算”或形式主义， 可以应用于一个类的大量问题。的 本研究中所研究的形式主义适用于 这些问题被称为变分问题。出现这些问题 在自然科学中的很多情况下都是如此。 一个简单的例子是，在所有问题中， 连接两点的曲面上的曲线， 最短长度

The principal investigator will continue his research on the variational bicomplex. This theory was first introduced approximately fifteen years ago to provide a uniform approach to multi-integral problems in the calculus of variations. The bicomplex has also been used to solve inverse problems in calculus of variations. The variational bicomplex plays a role in the geometric theory of differential equations somewhat analogous to the role played by the deRham complex in finite dimensional manifolds. This theme will be developed in the research with application to specific examples a primary source of motivation. It is often useful to create a "calculus" or formalism which can be applied to a large number of problems of a class. The formalism being studied in this research applies to the class of problems known as variational problems. These problems arise naturally in a very wide range of circumstances in natural science. A simple example would be the problem of determining, among all curves on a surface connecting two points, the one with the shortest length.