ChapterZero

Calculus of Variations

The calculus of variations is the mathematical theory concerned with finding functions which are stationary with respect to a given functional (a mapping from functions to reals). For example, it deals with the classical problem of finding the boundary which maximizes the enclosed area, given that the boundary must have a fixed length. Incidentally, the answer to this problem is given by the Isoperimetric inequality:

$\displaystyle A({\rm int}(\vfun{\gamma})) \leq \frac{l(\vfun{\gamma})^2}{4\pi},$

where $\vfun{\gamma}$ is a simple closed curve, $l(\vfun{\gamma})$ is the length (period) of the curve, and $A({\rm int}(\vfun{\gamma}))$ is the area of the interior of the curve, and equality holds only if $\vfun{\gamma}$ is a circle.

Three problems characteristic of the calculus of variations are:

What plane curve connecting two given points has the shortest length?
Given two points and in a vertical plane, find the path which the movable particle will traverse in shortest time, assuming that its acceleration is due only to gravity.
Find the minimum surface of revolution passing through two given fixed points, and .

The first problem in the calculus of variations is to find a function y(x) which minimizes/maximizes the integral $I = \int_a^b F\left(x,y,\frac{dy}{dx}\right) dx$ . By analogy to the case of finding the point at which the maximum of a continuous function occurs by finding the critical points, we can find a differential equation which must satisfy in order to be a relative maximum/minimum of this integral expression.

To do so, we will assume that $F(x,y,y^\prime)$ possess the relavant partial derivatives necessary for application of the MVT for functions of several variables. We will only consider arcs from to for which can be determined, or admissible . The problem is further proscribed by requiring that the values of at the endpoints a,b be predetermined.

Assume s(x) is a function which minimizes/maximizes , also known as a stationary point. Let $y(x)=s(x)+\epsilon t(x)$ be another admissible arc, where $\epsilon$ is an arbitrary constant independent of x,y and is a function independent of $\epsilon$ . With that restriction, is said to be subjected to weak variation. The important point of weak variations can be seen from the fact that $\frac{dy}{dx} = s^\prime(x) + \epsilon t^\prime(x)$ , so as $\epsilon$ approaches , $y(x) \rightarrow s(x) , \; \frac{dy}{dx} \rightarrow s^\prime(x)$ .

Define

$\displaystyle I_s = \int_a^b F(x, s, s^\prime) dx$

$\displaystyle I_s + \delta I_s = \int_a^b F(x, s + \epsilon t, s^\prime + \epsilon t^\prime) dx$

Now apply the MVT:

$\displaystyle F(x, s+\epsilon t, s^\prime + \epsilon t^\prime) = F(x, s, s^\prime)+ \epsilon \left( t \frac{\partial F}{\partial s} + t^\prime \frac{\partial F}{\partial s^\prime}) \right) + \frac{\epsilon^2}{2!} \left( t^2 \frac{\partial^2 F}{\partial s^2} + 2tt^\prime \frac{\partial^2 F}{\partial s \partial s^\prime} + t^{\prime 2} \frac{\partial^2 F} {\partial s^{\prime 2}} \right) + O(\epsilon^3)$

more to come

References

A Brief Survey of the History of Calculus of Variations and its Applications. James Ferguson. University of Victoria.
An Introduction to the Calculus of Variations. Charles Fox.

Possibly relevant posts:

Greens Theorem (9/5/2004)
A cool proof of what exactly? (4/29/2005)
Cauchy’s theorem via Homotopy (8/8/2007)

Calculus of Variations

References

Possibly relevant posts:

Leave a comment

Sticky Posts

Archives

intelligently designed

light and not filling