The Plateau Problem of Michell Trusses and Orthogonality in Springs

Abstract

Given finitely many pointed forces in the plane. Suppose that these forces sum up to zero and their net torques also sum up to zero. One can show that there exists a system of springs whose boundary forces exactly counter-balance these pointed forces. We will generalize to higher dimensions using the Cauchy stress tensor for elastic materials. Given a system of springs, we can multiply the length of each spring with its corresponding spring constant and then sum these products up. The result is called the total mass of the system. We are interested in the Plateau problem of the existence of the minimal spring system given a boundary condition. This minimization problem was first introduced in 1904 by A. Michell. He showed that a minimizer could smear out. The Michell Truss became known in mechanical engineering. It raised attention in optimal design, such as minimizing costs in building bridges. In 1960s and 1970s, the problem was developed using PDE and convex analysis by introducing an equivalent dual maximization problem. In 2008, Bouchitt\'{e}, Gangbo, and Sppecher introduced lines of principal actions to generalize Hencky-Prandtle net to higher dimensional duality and proved that the minimizer can be found provided that it exists. In the unpublished notes of Gangbo, he also showed that if springs of the same kind are optimal. In this paper, we are going to solve the Plateau problem using two different tools in GMT: first, a minimizer can be viewed as a flat chain complex; second, a minimizer can also be viewed as a current. At the end, we are going to show one progress in discovering the topological properties of minimizers: compressed and stretched springs must be perpendicular to each other at non-boundary points. I appreciate my advisor Prof. Robert Hardt for communicating with me regularly on this problem.