Date of Award

6-2012

Document Type

Open Access

Degree Name

Bachelor of Science

Department

Mathematics

First Advisor

Christina Tonnesen-Friedman

Language

English

Keywords

functions, calcululs, mathematics, equations, applications

Abstract

The Calculus of Variations is a highly applicable and advancing field. My thesis has only scraped the top of the applications and theoretical work that is possible within this branch of mathematics. To summarize, we began by exploring a general problem common to this field, finding the geodesic be-tween two given points. We then went on to define and explore terms and concepts needed to further delve into the subject matter. In Chapter 2, we examined a special set of smooth functions, inspired by the Calabi extremal metric, and used some general theory of convex functions in order to de-termine the minimizer of the functions. Finally, we looked into Hamilton’s equations and touched upon symplectic geometry. For those people looking to investigate further into the calculus of variations, there are many different paths to take. A particularly interesting one would be to explore the specific applications the field has to other disciplines, like physics and economics. We briefly explored the applications in physics, but did not explore applications in economics at all. One could learn about the problems of exhaustible resources, which demand knowledge of the calculus of variations. In this specific application, the extremum that we are looking for is the maximal profit. This branch of mathematics is useful to a variety of other disciplines, and it would be beneficial to explore how the fields coincide in order to solve real-world problems. We have explored some theoretical mathematics of this field, so this would be the natural next step. There is also the possibility that applications for the Calculus of Variations have not yet been realized, leaving the potential for a large amount of growth and progress.

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