# Functionals In The Calculus Of Variations

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## Authors

Kao, Tsai-tse

## Date of Issue

1970-04-01

## Type

thesis

## Language

## Subject Keywords

## Other Titles

## Abstract

This thesis deals with an introduction to the functionals in the calculus of variations. Since calculus of variations is a relatively advanced topic in mathematics, this thesis is written with the attempt to introduce some of the oasis notions with regard to what the calculus of variations is. To do this, an attempt is made to try to draw an analogy between the calculus of variations and ordinary maximum and minimum theory of functions which a reader may be very familiar with, and see how some of the ideas and methods used in ordinary maximum and minimum problems are carried over to the calculus of variations.
The thesis begins with the consideration of maximum and minimum theory of functions. This is done by two examples. This is followed by a consideration of a problem that is caused by the point of inflection in ordinary maximum and minimum theory. With all these considerations, one hopes to shed some light as to how some of these ideas are used in the calculus of variations. Next one shall see what the calculus of variations is and see how a problem in calculus of variations is dealt with. To do this, an example concerning the finding of a shortest distance curve Joining two given fixed points in a two dimensional Euclidean space is considered.In this example, one shall see in fact how one can change a problem in the calculus of variations into an ordinary maximum and minimum problem, and try to draw some analogy between the two.
Following this, one shall see how the Euler’s differential equation is derived in general for a fixed end points problem in two dimensional Euclidean space. Here because of the limitations of this thesis, only some specific functions, namely, functions that are continuous and twice differentiable are dealt with. Some remarks about Euler’s differential equation will be given. The form of Euler’s equation for n dependent variables and one independent variable will be stated without proof here. A simple application of Euler’s equation to physics problem will be given. Finally this thesis will be concluded with an invitation to more and deeper studies of the calculus of variations. Because of the limitations of this thesis, no attempt has Mr been made to deal with other necessary conditions other than Euler’s differential equation. Also no attempt has been made with regard to the consideration of existence and sufficiency proof. The notion of strong variations and second or higher variations are also not considered here. However, a list of bibliography is given at the end of this thesis for the interested reader who wants to go deeper into the theory of functionals in the calculus of variations.
The reader of this thesis is expected to have as part of his permanent knowledge most of the concerts and techniques learned in his first year calculus course. Knowledge other than this will be given in the thesis. All the mathematics has been reduced down to an elementary level. The central idea of this thesis is not to present a complete theory of functionals in the calculus of variations, but to try to give the reader some notions of the calculus of variations and to get him interested in the subject, and to see how come of the concepts and techniques that he learned in his first year calculus course can be carried over to the studies of the calculus of variations. This thesis deals with an introduction to the functionals in the calculus of variations. Since calculus of variations is a relatively advanced topic in mathematics, this thesis is written with the attempt to introduce some of the oasis notions with regard to what the calculus of variations is. To do this, an attempt is made to try to draw an analogy between the calculus of variations and ordinary maximum and minimum theory of functions which a reader may be very familiar with, and see how some of the ideas and methods used in ordinary maximum and minimum problems are carried over to the calculus of variations.
The thesis begins with the consideration of maximum and minimum theory of functions. This is done by two examples. This is followed by a consideration of a problem that is caused by the point of inflection in ordinary maximum and minimum theory. With all these considerations, one hopes to shed some light as to how some of these ideas are used in the calculus of variations. Next one shall see what the calculus of variations is and see how a problem in calculus of variations is dealt with. To do this, an example concerning the finding of a shortest distance curve Joining two given fixed points in a two dimensional Euclidean space is considered.In this example, one shall see in fact how one can change a problem in the calculus of variations into an ordinary maximum and minimum problem, and try to draw some analogy between the two.
Following this, one shall see how the Euler’s differential equation is derived in general for a fixed end points problem in two dimensional Euclidean space. Here because of the limitations of this thesis, only some specific functions, namely, functions that are continuous and twice differentiable are dealt with. Some remarks about Euler’s differential equation will be given. The form of Euler’s equation for n dependent variables and one independent variable will be stated without proof here. A simple application of Euler’s equation to physics problem will be given. Finally this thesis will be concluded with an invitation to more and deeper studies of the calculus of variations. Because of the limitations of this thesis, no attempt has Mr been made to deal with other necessary conditions other than Euler’s differential equation. Also no attempt has been made with regard to the consideration of existence and sufficiency proof. The notion of strong variations and second or higher variations are also not considered here. However, a list of bibliography is given at the end of this thesis for the interested reader who wants to go deeper into the theory of functionals in the calculus of variations.
The reader of this thesis is expected to have as part of his permanent knowledge most of the concerts and techniques learned in his first year calculus course. Knowledge other than this will be given in the thesis. All the mathematics has been reduced down to an elementary level. The central idea of this thesis is not to present a complete theory of functionals in the calculus of variations, but to try to give the reader some notions of the calculus of variations and to get him interested in the subject, and to see how come of the concepts and techniques that he learned in his first year calculus course can be carried over to the studies of the calculus of variations.