dr. wang xingbo fall ， 2005

Dr. Wang XingboDr. Wang Xingbo

FallFall ，， 20052005

Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering

Functional and Calculus of Variation Functional and Calculus of Variation


Introduction to Calculus of Variations 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

The arclength isThe arclength is


Introduction to Calculus of VariationsIntroduction to Calculus of Variations

dxxyb

a 2)]('[1

The problem is to minimize the above integral

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



dxxyxyxFxyJb

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

A functional can have more general form

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

We will only focus on functional with integral We will only focus on functional with integral



A increment of y(x) is called variation of y(x), denoted as δy(x)

P

variation of y(x):δy(x)

y(x)

consider the increment of J[y(x)] causeconsider the increment of J[y(x)] caused by δy(x)d by δy(x)

ΔJ [y(x)]= J [y(x)+δy(x)]- J [y(x)]ΔJ [y(x)]= J [y(x)+δy(x)]- J [y(x)]ΔJ [y(x)]=ΔJ [y(x)]=L[y(x), δy(x)]+β[y(x),δy(x)]•max|δy(x)| L[y(x), δy(x)]+β[y(x),δy(x)]•max|δy(x)|



If β[y(x),δy(x)] is a infinitesimal of δy(x), then L is called variation of J[y(x)] with the first order, or simply variation of J[y(x)],denoted by δJ[y(x)]

1.Rules for permissible functions1.Rules for permissible functions y y((xx) an) and variable d variable xx..



)()( ydx

d

dx

dy

δx=dxδx=dx

Rules for functional Rules for functional J.J.



δ2 J =δ(δJ), …, δkJ =δ(δk-1J)δ(J1+ J 2)= δJ 1+δJ 2

δ(J 1 J 2)= J 1δJ 2+ J 2δJ 1δ(J 1/ J 2)=( J 2δJ 1- J 1δJ 2)/ J 2

2

Rules for functional Rules for functional JJ and and FF



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

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

''

F FJ F y y

y y

If J[y(x)] reaches its maximum (or If J[y(x)] reaches its maximum (or minimum) at yminimum) at y00(x), then δ(x), then δJJ[[yy00((xx)]=0.)]=0.



Let Let J J be a functional defined on Cbe a functional defined on C22

[a,b] with [a,b] with JJ[y(x)] given by [y(x)] given by dxxyxyxFxyJ

b

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

How do we determine the curve How do we determine the curve yy((xx) which produces such a minimu) which produces such a minimum (maximum) value for m (maximum) value for JJ? ?

The Euler-Lagrange Equation



dxxyxyxFxyJb

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

( ) 0'

F d F

y dx y

Let M(x) be a continuous function on the interval [a,b], Suppose that for any continuous function h(x) with h(a) = h(b) = 0 we have


Fundamental principle of variations Fundamental principle of variations

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

b

adxxhxM 0)()(

choose h(x) = -M(x)(x - a)(x - b) Then M(x)h(x)≥0 on [a, b]


Fundamental principle of variations 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

Example :Prove that the shortest curve connecting planar point P and Q is the straight line connected P and Q

Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering


dxxyLb

a 2)]('[1



2/32

2/32

22

2

2/1222

2'

))]('[1(

)("

))]('[1(

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

)]('[1

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

))]('[1

)('(0

xy

xy

xy

xyxyxy

xy

xyxyxyxyxy

xy

xy

dx

dF

dx

dF

y y

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


Beltrami Identity.

If then the Euler-Lagrange equation is equivalent to:

0F

x

''

FF y C

y

The Brachistochrone Problem



Find a path that wastes the least time for a bead travel from P to Q

P

Q

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



b

a v

dsxyF )]([

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

By Newton's second law we obtain



))()(()]([2

1 2 xyaymgxvm

b

adx

xyayg

xyxyF

))()((2

|)('|1)]([

2

• Euler-Lagrange Equation



Cxgy

xy

xy

xgyxy

xgy

xy

)(

)(')

)]('[1

)(2)(('

2

1

)(2

)]('[12

2

22

2

2

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

gCxyxy



The solution of the above equation is a cycloid curve

)cos1(2

1)(

)sin(2

1)(

2

2

ky

kx


Integration of the Euler-Lagrange Equation Integration of the Euler-Lagrange Equation

Case 1. F(x, y, y’) = F (x)

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

Case 3. F (x, y, y’) = F (y’) :

0))(( ' yFdx

dy CyFy )'(' ' 1y C

1 2y C x C


Integration of the Euler-Lagrange EquationIntegration 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))(( ' yFdx

dy 1)',(' CyxFy )1,(' Cxfy

dtCtfCy )1,(2


Integration of the Euler-Lagrange EquationIntegration of the Euler-Lagrange Equation

Case 6 F (x, y, y’) = F (y, y’)

)',()',(' yyFyyFdx

dyy

yy FyFy '''

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

'"'' yy FyFyF

')''( ' FFy y

CFFy y ''


The Euler-Lagrange Equation of Variational NotatThe Euler-Lagrange Equation of Variational Notation ion

b

aFdxJ

'' yy FyyFF

b

adxyyxF 0)',,(

b

a yy dxyyxFyyyxyF 0))',,(')',,(( '


The Lagrange Multiplier Method for the Calculus The Lagrange Multiplier Method for the Calculus of Variations of Variations

dxxyxyxFxyJb

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

ldxxyxyxGb

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

Conditions Conditions

The minimize problem of following functional is equal to the conditional ones.


The Lagrange Multiplier Method for the Calculus The Lagrange Multiplier Method for the Calculus of Variationsof Variations

where λ is chosen that y(a)=A, y(b)=B

b

a

b

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

ldxxyxyxGb

a ))('),(,(

0)()( '' yyyy GFGFdx

d

ExampleExample


The Lagrange Multiplier Method for the Calculus of VariationsThe Lagrange Multiplier Method for the Calculus of Variations

E-L equation is

2

0)(][ dyxyyJ

2

0 2 3))('(1 dxxyunder

1)'1

'(

2

y

y

dx

d

1'1

'2

cxy

y

Leads to 22 )1(

)1('

cx

cxy

222 )1()2( cxcy


Variation of Multi-unknown functions Variation of Multi-unknown functions

b

adxxyxyxyxyxFxyxyJ ))('),('),(),(,()](),([ 212121

0)',',,,( 2121 dxyyyyxFb

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

b

a

0)'( '

2

1

dxFyFyii yi

b

ai

yi

0}{2

1'

i

b

a yyi dxFdx

dFy

ii

The Euler-Lagrange equation for a functional The Euler-Lagrange equation for a functional with two functions with two functions yy11((xx),),yy22((xx) are ) are


Variation of Multi-unknown functionsVariation of Multi-unknown functions

2,1,0' iFdx

dF

ii yy


Higher Derivatives Higher Derivatives

b

adxxyxyxyxFxyJ ))("),('),(,()]([

0"2

2

' yyy Fdx

dF

dx

dF

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

clamped so thatclamped so that y y (0) = (0) = yy (1) = (1) = yy’ (0) = 0 ’ (0) = 0 and yand y’ (1) = 1.’ (1) = 1.


Example Example

1

0

2)"(][ dxykyJ 0"22

2

ydx

dk

dcxbxaxy 23 3 2y x x

Class is Over! Class is Over!

See you!See you!


dr. wang xingbo fall ， 2005

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