# Calculus of Variations: Important Results and Using the Euler-Lagrange Equation

## Faculty Mentor

Frank Lynch

## Document Type

Oral Presentation

## Start Date

10-5-2023 10:20 AM

## End Date

10-5-2023 10:40 AM

## Location

PUB 317

## Department

Mathematics

## Abstract

Calculus of Variations is an area of math where some of the ideas and concepts from differential and integral calculus are applied to functional equations (or functions of functions) to find minimum and maximum functions for a given functional. This paper starts by looking at some of the theory behind Calculus of Variations, including the Fundamental Lemma of Calculus of Variations, which is a key result that is then applied, along with setting the first variation of a functional equal to zero to derive the Euler-Lagrange equation, a differential equation whose solutions are minimizers of a functional J(y). We then illustrate the use of Calculus of Variations through the Euler-Lagrange equation by obtaining a differential equation modeling a particular system, then solving (or approximating the solution) of the differential equation. Where appropriate, the second variation is calculated to check whether the solution of the differential equation is in fact a minimizer of the functional in question. Alternative forms of the Euler-Lagrange equation are also briefly explored, such as the case where the Lagrange Function inside the functional is not an explicit function of x and the case where a system has damping and an external force.

## Recommended Citation

Quindlen, Michael, "Calculus of Variations: Important Results and Using the Euler-Lagrange Equation" (2023). *2023 Symposium*. 3.

https://dc.ewu.edu/srcw_2023/res_2023/os1_2023/3

## Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.

Calculus of Variations: Important Results and Using the Euler-Lagrange Equation

PUB 317

Calculus of Variations is an area of math where some of the ideas and concepts from differential and integral calculus are applied to functional equations (or functions of functions) to find minimum and maximum functions for a given functional. This paper starts by looking at some of the theory behind Calculus of Variations, including the Fundamental Lemma of Calculus of Variations, which is a key result that is then applied, along with setting the first variation of a functional equal to zero to derive the Euler-Lagrange equation, a differential equation whose solutions are minimizers of a functional J(y). We then illustrate the use of Calculus of Variations through the Euler-Lagrange equation by obtaining a differential equation modeling a particular system, then solving (or approximating the solution) of the differential equation. Where appropriate, the second variation is calculated to check whether the solution of the differential equation is in fact a minimizer of the functional in question. Alternative forms of the Euler-Lagrange equation are also briefly explored, such as the case where the Lagrange Function inside the functional is not an explicit function of x and the case where a system has damping and an external force.