#### 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.

#### Recommended Citation

Whitney, Erin, "The Calculus of Variations" (2012). *Honors Theses*. 920.

https://digitalworks.union.edu/theses/920