The Calculus of Variations: An Introduction

The Calculus of Variations: An Introduction

By Kolo Sunday Goshi

Some Greek Mythology 

Queen Dido of Tyre – –



Iarbas’ (King of Libya) offer –



Fled Tyre after the death of her husband Arrived at what is present day Libya “Tell them, that this their Queen of theirs may have as much land as she can cover with the hide of an ox.”

What does this have to do with the Calculus of Variations?

What is the Calculus of Variations 

 

“Calculus of variations seeks to find the path, curve, surface, etc., for which a given function has a stationary value (which, in physical problems, is usually a minimum or maximum).” (MathWorld Website) Variational calculus had its beginnings in 1696 with John Bernoulli Applicable in Physics

Calculus of Variations  

Understanding of a Functional Euler-Lagrange Equation –



Proving the Shortest Distance Between Two Points –





In Euclidean Space

The Brachistochrone Problem –



Fundamental to the Calculus of Variations

In an Inverse Square Field

Some Other Applications Conclusion of Queen Dido’s Story

What is a Functional? 



The quantity z is called a functional of f(x) in the interval [a,b] if it depends on all the values of f(x) in [a,b]. Notation b

z    f  x   a

–

Example 1 2

  x    cos  x 2  dx 0 0

1

Functionals 

The functionals dealt with in the calculus of variations are of the form   f  x     F  x, y( x), y( x)  dx a b

 

The goal is to find a y(x) that minimizes Г, or maximizes it. Used in deriving the Euler-Lagrange equation

Deriving the Euler-Lagrange Equation 

I set forth the following equation: y  x   y  x    g  x 

Where yα(x) is all the possibilities of y(x) that extremize a functional, y(x) is the answer, α is a constant, and g(x) is a random function.

y1 y(b)

y0 = y

y(a)

y2

a

b

Deriving the Euler-Lagrange Equation   f  x    F  x, y( x), y( x)  dx a b



Recalling



It can now be said that:   y    F  x, y , y  dx b

a



At the extremum yα = y0 = y and



The derivative of the d functional with respect  to α must be evaluated d and equated to zero

d 0 d  0

   F x , y , y a       dx b


The mathematics b  d involved    F  x, y , y   dx a d    b  F y d F y      dx a d  y  y  

–



Recalling

y  x   y  x    g  x 

So, we can say b  F b F b F dg d F    g g dx   gdx   dx a a a d y  y y dx  y

Deriving the Euler-Lagrange Equation b F b F dg d  gdx   dx a a d y y dx 

Integrate the first part by parts and get



b

a

d  F g  dx  y

  dx 



So



Since we stated earlier that the derivative of Г with respect to α equals zero at α=0, the extremum, we can equate the integral to zero

 F b d d  F  g   a d  y dx  y

   dx 


So

0

b

a



 F d  F   g     dx  y0 dx  y0  

We have said that y0 = y, y being the extremizing function, therefore

y1 y0 = y y2



Since g(x) is an arbitrary function, the quantity in the brackets must equal zero

0

b

a

 F d  F   g     dx  y dx  y  

The Euler-Lagrange Equation 

We now have the Euler-Lagrange Equation

F d  F    0 y dx  y   When F  F  y, y , where x is not included,

the modified equation is F Fy C y

The Shortest Distance Between Two Points on a Euclidean Plane 

What function describes the shortest distance between two points? –

Most would say it is a straight line  



Logically, this is true Mathematically, can it be proven?

The Euler-Lagrange equation can be used to prove this

Proving The Shortest Distance Between Two Points 

Define the distance to be s, so s   ds b

ds

dy dx

a



Therefore

s   dx 2  dy 2


Factoring a dx2 inside the square root and taking its square root we obtain s   dx  dy 2

2

s

b

a

 dy  1   dx  dx 

dy  Now we can let y  dx 

so

s

b

a

2

1  y 2 dx  


Since



b

a

1  y 2 dx

  f  x     F  x, y( x), y( x)  dx a b



And we have said that



we see that

F  1 y2

therefore

F 0 y



F y  y 1 y2


Recalling the Euler-Lagrange equation

F d  F    0 y dx  y 



Knowing that

F 0 y



A substitution can be made



Therefore the term in brackets must be a constant, since its derivative is 0.

F y  y 1 y2

d  y    0 2 dx  1  y 


More math to reach the solution

y 1 y2

C

y 2  C 2 1  y 2  y 2 1  C 2   C 2 y2  D yM


Since

yM

We see that the derivative or slope of the minimum path between two points is a constant, M in this case. The solution therefore is:

y  Mx  B

The Brachistochrone Problem 

Brachistochrone –

Derived from two Greek words  



The problem –

Find the curve that will allow a particle to fall under the action of gravity in minimum time. 



brachistos meaning shortest chronos meaning time

Led to the field of variational calculus

First posed by John Bernoulli in 1696 –

Solved by him and others

The Brachistochrone Problem 

The Problem restated –



Find the curve that will allow a particle to fall under the action of gravity in minimum time.

The Solution –

–

A cycloid Represented by the parametric equations

D x    2  sin 2  2 D y  1  cos 2  2 

Cycloid.nb

The Brachistochrone Problem In an Inverse Square Force Field 

The Problem –

–

Find the curve that will allow a particle to fall under the action of an inverse square force field defined by k/r2 in minimum time. Mathematically, the force is defined as

k Fr   2 r

y

1

2

r0 F 

k rˆ r2

x




Since the minimum time is being considered, an expression for time must be determined

An expression for the velocity v must found and this can be done using the fact that KE + PE = E

t

2

1

ds v

1 k 2 mv   E 2 r


The initial position r0 is known, so the total energy E is given to be –k/r0, so

An expression can be found for velocity and the desired expression for time is found

1 k k 2 mv    2 r r0

v

2k  1 1     m  r r0 

m 2 ds t 2k 1  1 1      r r0 

The Brachistochrone Problem In an Inverse Square Force Field Determine an expression for ds rdΘ r

ds

dr r + dr

ds   dr   r  d  2

2

2

2




We continue using a polar coordinate system

An expression can be determined for ds to put into the time expression

ds   dr   r  d  2

2

2

2

2   2  dr  2 2 ds   d    r   d  

ds  r 2  r 2 d




Here is the term for time t

The function F is the term in the integral

rr0 (r  r ) r0  r 2

m t 2k 1

2

rr0 (r  r ) F r0  r 2

2

2


Using the modified Euler-Lagrange equation

F F r C r

rr0 (r 2  r 2 )  r2 r0  r

rr0 C 2 2  r0  r  (r  r )


More math involved in finding an integral to be solved

r (r 2  r 2 )  r2 r0  r r2 r 2 2 r  r ( r  r ) 0 

D

r D 2 2  r0  r  (r  r )

r5 G 2 2  r0  r  (r  r )




Reaching the integral

Solving the integral for r(Θ) finds the equation for the path that minimizes the time.

r 5  r 2G(r0  r ) dr r  d G (r0  r )



G(r0  r ) dr    d 5 2 r  r G (r0  r )


Challenging Integral to Solve –



Where to then? – –



Brachistochrone.nb Use numerical methods to solve the integral Consider using elliptical coordinates

Why Solve this? –

Might apply to a cable stretched out into space to transport supplies

Some Other Applications 

The Catenary Problem –

–

–



Derived from Greek for “chain” A chain or cable supported at its end to hang freely in a uniform gravitational field Turns out to be a hyperbolic cosine curve

Derivation of Snell’s Law

n1 sin i  n2 sin 2

Conclusion of Queen Dido’s Story  

Her problem was to find the figure bounded by a line which has the maximum area for a given perimeter Cut ox hide into infinitesimally small strips –

– –



Used to enclose an area Shape unknown City of Carthage

Isoperimetric Problem – –

Find a closed plane curve of a given perimeter which encloses the greatest area Solution turns out to be a semicircle or circle

References  

  

Atherton, G., Dido: Queen of Hearts, Atherton Company, New York, 1929. Boas, M. L., Mathematical Methods in the Physical Sciences, Second Edition, Courier Companies, Inc., United States of America, 1983. Lanczos, C, The Variational Principles of Mechanics, Dover Publications, Inc., New York, 1970. Ward, D., Notes on Calculus of Variations Weinstock, R., Calculus of Variations, Dover Publications, Inc., New York, 1974.