Faculty Bibliography Spotlight
Manning, Robert (2009). “Conjugate points revisited and Neumann-Neumann problems.” SIAM Review 51(1): 193-212
This article is unusual in that it merges new research with a pedagogical argument. It addresses a central question in the “calculus of variations”, a field within mathematics that considers functions of functions, i.e., instead of a function like G(x) = 2x2+3 that one sees in introductory calculus, which takes a number x as its input and outputs a number G(x), a function in the calculus of variations takes a function f as its input and outputs a number G(f). The basic problem is familiar from calculus---find a local minimum of G---but harder to picture since instead of a one-dimensional “x-axis” we now have an infinite-dimensional “f-axis”. The strategy for solving this problem is also familiar from calculus: find “critical points” and then categorize each critical point as a local minimum or not. This categorization was developed by Carl Jacobi in the 1830’s. Jacobi associated to each critical point a new function, whose zeroes he called “conjugate points”, and if there are no conjugate points, then the critical point is a local minimum.
This classical Jacobi conjugate point theory has been a fairly static part of the calculus of variations for many decades, both in terms of the types of problems that it can handle and how it is presented to students. Over the past decade, I have worked with collaborators to extend this theory to new problems, by applying a modern mathematical framework to better understand why the theory works. One landmark in this process was a paper I wrote with George Bulman HC ’02 on an elastic rod buckling into a wall; this was my first verification that the extended conjugate point theory could solve a problem outside the scope of the classical theory. The SIAM Review (2009) article distills the extended theory into as simple a form as I could manage, and argues that the modern mathematical framework makes conjugate points both more comprehensible and applicable to a wider range of problems. As an illustration, it fills in a glaring hole in the classical theory: that Jacobi’s conjugate points did not apply when imposing Neumann boundary conditions on the function f. Such Neumann conditions are common in applied math; indeed, my discovery of how to handle them arose from their presence in experiments modeled in the Bulman and Manning paper.
SIAM Review, as the flagship journal of SIAM, the largest professional society of applied mathematicians, is mailed directly to every SIAM member. Thus, I hoped that publishing this article in SIAM Review would serve to both advance the field of calculus of variations and also influence the way that it is taught. Because the presentation of Jacobi’s theory is fairly opaque in most textbooks, I believe that it is omitted (or maybe treated minimally) in most calculus of variations courses. In contrast, the framework presented in this article is more aesthetic, and more obviously related to other core ideas in applied mathematical training, and thus should be more appealing to include in such courses (as I did recently in “Advanced Topics in Applied Math” at Haverford).
Robert Manning is William H. and Johanna A. Harris Computational Science Professor and Associate Professor of Mathematics at Haverford College.
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