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Dr. Wang Xingbo

Fall 2005

Mathematical & Mechanical

Method in Mechanical Engineering

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Functional and Calculus of Variation

Introduction to Calculus of Variations

P=(a,y(a))

Q=(b,y(b))

Find the shortest curve connecting P= (a, y(a))and Q= (b, y(b)) in XY plane

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Introduction to Calculus of Variations

dxxyb

a

2)]('[1

The problem is to minimize the above integral

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A function like J is actually called a functional . y(x)

is call a permissible function

Introduction to Calculus of Variations

dxxyxyxFxyJb

a

))('),(,()]([

A functional can have more general form

))('),(,()]([ xyxyxfxyJ

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We will only focus on functional with integral

Introduction to Calculus of Variations

A increment ofy(x) is called variation of y(x),

denoted as y(x)

P

variation ofy(x):y(x)

y(x)

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Introduction to Calculus of Variations

If [y(x),y(x)] is a infinitesimal of y(x), then L is calledvariation ofJ[y(x)] with the first order, or simply variation of

J[y(x)],denoted by J[y(x)]

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Introduction to Calculus of Variations

)()( ydx

d

dx

dy

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Introduction to Calculus of Variations

2 J=(J), , kJ=(k-1J)

(J1+ J2)= J1+J2

(J1J

2)= J

1J

2+ J

2J

1

(J1/J2)=( J2J1- J1J2)/J22

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Rules for functionalJandF

Introduction to Calculus of Variations

( , , ') ( , , ')b b

a aJ F x y y dx F x y y dx

''

F FJ F y yy y

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If J[y(x)] reaches its maximum (or

minimum) at y0(x), then J[y0(x)]=0.

Introduction to Calculus of Variations

Let Jbe a functional defined onC2[a,b] with J[y(x)] given by

dxxyxyxFxyJb

a

))('),(,()]([

How do we determine the curve y(x)

which produces such a minimum

(maximum) value forJ?

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Introduction to Calculus of Variations

dxxyxyxFxyJb

a ))('),(,()]([

( ) 0'

F d Fy dx y

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Let M(x)be a continuous function on the interval[a,b],

Suppose that for any continuous function h(x)withh(a) = h(b) = 0we have

Fundamental principle of variations

Then M(x) is identically zero on [a, b]

b

a

dxxhxM 0)()(

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choose h(x) = -M(x)(x - a)(x - b)

Then M(x)h(x) 0 on [a, b]

Fundamental principle of variations

0 = M(x)h(x) = [M(x)]2[-(x - a)(x - b)]

M(x)=0

If the definite integral of a non-negative function is zero

then the function itself must be zero

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Example :Prove that the shortest curve connecting

planar point P and Q is the straight line connected Pand Q

Introduction to Calculus of Variations

dxxyLb

a

2)]('[1

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Introduction to Calculus of Variations

2/32

2/32

22

2

2/1222

2'

))]('[1(

)("))]('[1(

)(")]('[))]('[1(

)]('[1

))]('[1)((")]('[)(")]('[1

))]('[1

)('(0

xy

xyxy

xyxyxy

xy

xyxyxyxyxy

xy

xy

dx

dF

dx

dF

yy

0)(" xy y(x)=ax+b

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Beltrami Identity.

If then the Euler-Lagrange equation isequivalent to:

0Fx

''

FF y Cy

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The Brachistochrone Problem

Introduction to Calculus of Variations

Find a path that wastes the least time for a bead

travel from P to Q

P

Q

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Let a curve y(x) that connects P and Q represent the wire

The Brachistochrone Problem

b

a v

dsxyF )]([

21 | '( ) | , '( )ds y x dx v y x

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By Newton's second law we obtain

The Brachistochrone Problem

))()(()]([2

1 2 xyaymgxvm

b

adx

xyayg

xyxyF

))()((2

|)('|1)]([

2

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Euler-Lagrange Equation

The Brachistochrone Problem

Cxgy

xy

xy

xgyxy

xgy

xy

)(

)(')

)]('[1

)(2)(('

2

1

)(2

)]('[12

2

2

2

2

2

1)())]('[1( k

gCxyxy

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The Brachistochrone Problem

The solution of the above equation is a cycloid

curve

)cos1(2

1)(

)sin(2

1)(

2

2

ky

kx

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Integration of the Euler-Lagrange Equation

Case 1. F(x, y, y) = F(x)Case 2. F(x, y, y) = F(y) :Fy(y)=0

Case 3.F(x, y, y) = F(y) :

0))(( ' yFdxd

y CyFy )'(' ' 1y C

1 2y C x C

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Integration of the Euler-Lagrange Equation

Case 4.F(x, y, y) = F(x, y)

Fy (x, y) = 0 y = f (x)

Case 5.F(x, y, y) = F(x, y)

0))(( ' yFdxd

y 1)',(' CyxFy )1,(' Cxfy

dtCtfCy )1,(2

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Integration of the Euler-Lagrange Equation

Case 6 F(x, y, y) = F(y, y)

)',()',(' yyFyyFdxd yy

yy FyFy '''

)'('")''( ''' yyy FyFyFy

'"'' yy FyFyF

')''( ' FFy y

CFFy y ''

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The Euler-Lagrange Equation of Variational

Notation

b

aFdxJ

'' yy FyyFF

b

adxyyxF 0)',,(

b

ayy dxyyxFyyyxyF 0))',,(')',,(( '

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The Lagrange Multiplier Method for the Calculus

of Variations

dxxyxyxFxyJb

a ))('),(,()]([

ldxxyxyxGb

a ))('),(,( BbyAay )(,)(Conditions

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The minimize problem of following functional is

equal to the conditional ones.

The Lagrange Multiplier Method for the Calculus

of Variations

where is chosen that y(a)=A,y(b)=B

b

a

b

adxxyxyxGdxxyxyxF ))('),(,())('),(,(

ldxxyxyxGb

a ))('),(,(

0)()( '' yyyy GFGFdx

d

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Example

The Lagrange Multiplier Method for the Calculus of Variations

E-L equation is2

0 )(][ dyxyyJ 2

02 3))('(1 dxxyunder

1)'1

'(

2

y

y

dx

d

1'1

'

2cx

y

y

Leads to

22 )1(

)1('

cx

cxy

222

)1()2(

cxcy

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Variation of Multi-unknown functions

b

a

dxxyxyxyxyxFxyxyJ ))('),('),(),(,()](),([ 212121

0)',',,,( 2121 dxyyyyxFb

a 0)',',,,( 2121 dxyyyyxF

b

a

0)'( '

2

1 dxFyFy ii yi

b

ai

yi

0}{2

1

' i

b

ayyi dxF

dx

dFy

ii

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The Euler-Lagrange equation for a functionalwith two functionsy1(x),y2(x) are

Variation of Multi-unknown functions

2,1,0' iFdx

dF

ii yy

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Higher Derivatives

b

a

dxxyxyxyxFxyJ ))("),('),(,()]([

0"2

2

' yyy F

dx

dF

dx

dF

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What is the shape of a beam which is bent and which is

clamped so that y (0) =y (1) =y (0) = 0 and y (1) = 1.

Example

1

0

2)"(][ dxykyJ 0"2

2

2

ydx

dk

dcxbxaxy 233 2y x x

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Class is Over!

See you!