Dr. Wang Xingbo Fall ， 2005 Mathematical Mechanical Method in Mechanical Engineering 1/3304/03/2016 23:41

Dr. Wang XingboDr. Wang Xingbo

FallFall ，， 20052005

Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering

1/3308/05/23 23:12

:عناوینمقدمه ای ازبهینه سازی معادالت جبری، 1.

chapter5_Calculus-of-Variations 3 اسالید 4 تا 1صفحات Functional Euler's equation.

فانکشنال و مسئله بهینه سازی دینامیکی: 2..1 مطلب فایل ترکیبی 10 تا 3صفحات

، خانم نجیمی؟Matlab فایل m.نگاهی به 3.

ادامه با استفاده از اسالید جاری.4.

و چند strictتعمیم، بعینه سازی قطعی، 5. 1 فایل ترکیبی و 18 تا 16متغیره صفحات

Calculus of Variations - PowerPoint Presentation به بعد16صفحه 08/05/23 23:12 2

:سلسله مراتب بهینه سازی

• V-1 Maximum and Minimum of Functions• V-2 Maximum and Minimum of Functionals• V-3 The Variational Notation• V-4 Constraints and Lagrange Multiplier

08/05/23 23:12 3

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

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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

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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

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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)

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of y(Variation x) y(x)

consider the increment of J[y(x)] consider the increment of J[y(x)] caused by δy(x)caused 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)]L[y(x), δy(x)]+β[y(x),δy(x)]••max|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)]

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1.Rules for permissible functions1.Rules for permissible functions yy((xx) and variable ) and variable xx..



)()( ydxd

dxdy

δx=dxδx=dx

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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 1

δ(J 1/ J 2)=( J 2δJ 1- J 1δJ 2)/ J 22

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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 yy y

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



Let Let J J be a functional defined on be a functional defined on CC22[a,b] with [a,b] with JJ[y(x)] given by [y(x)] given by

dxxyxyxFxyJb

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

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

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The Euler-Lagrange Equation



dxxyxyxFxyJb

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

( ) 0'

F d Fy dx y

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To find y(x) to minimize J of:

The following equation must be satisfied:

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)()(

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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 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 P and Q.

Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering


dxxyLb

a 2)]('[1

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2/32

2/32

22

2

2/1222

2'

))]('[1()("

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

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

))]('[1

)('(0

xyxy

xyxyxyxy

xyxyxyxyxyxy

xy

xydxdF

dxdF

y y

0)(" xy y(x)=ax+b 17/3308/05/23 23:12


Beltrami Identity.

If then the Euler-Lagrange equation is simplified and is equivalent to:

0Fx

''

FF y Cy

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The Brachistochrone Problem



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 :



b

a vdsxyF )]([

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

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By Newton's second law ,we obtain:



))()(()]([21 2 xyaymgxvm

b

adx

xyaygxyxyF

))()((2|)('|1)]([2

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• Euler-Lagrange Equation



Cxgyxy

xyxgyxy

xgyxy

)()(')

)]('[1)(2)(('

21

)(2)]('[1

2

2

22

2

21)())]('[1( k

gCxyxy

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The solution of the above equation is a cycloid curve

)cos1(21)(

)sin(21)(

2

2

ky

kx

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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))(( ' yFdxd

y CyFy )'(' ' 1y C

1 2y C x C 24/3308/05/23 23:12


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))(( ' yFdxd

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

dtCtfCy )1,(225/3308/05/23 23:12


Integration of the Euler-Lagrange EquationIntegration of the Euler-Lagrange EquationCase 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 The Euler-Lagrange Equation of VariationalVariational Notation Notation

b

aFdxJ

'' yy FyyFF

b

adxyyxF 0)',,(

b

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

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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

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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 GFGFdxd

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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

ydxd

1'1

'2

cxy

y

Leads to 22 )1(

)1('cx

cxy

222 )1()2( cxcy 30/3308/05/23 23:12


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 dxFdxdFy

ii

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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' iFdxdF

ii yy

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Higher Derivatives Higher Derivatives

b

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

0"2

2

' yyy FdxdF

dxdF

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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"2 2

2

ydxdk

dcxbxaxy 23 3 2y x x

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Class is Over! Class is Over!

See you!See you!


35/3308/05/23 23:12

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Dr. Wang Xingbo Fall ， 2005 Mathematical Mechanical Method in Mechanical Engineering 1/3304/03/2016 23:41