PHY 711 Fall 2015 -- Lecture 5 19/4/2015

PHY 711 Classical Mechanics and Mathematical Methods

10-10:50 AM MWF Olin 103

Plan for Lecture 5:Start reading Chapter 3 – First focusing on the “calculus of variation”

PHY 711 Fall 2015 -- Lecture 59/4/2015 2

PHY 711 Fall 2015 -- Lecture 5 39/4/2015

In Chapter 3, the notion of Lagrangian dynamics is developed; reformulating Newton’s laws in terms of minimization of related functions. In preparation, we need to develop a mathematical tool known as “the calculus of variation”.

Minimization of a simple function

0dxdV

local minimum

global minimum

PHY 711 Fall 2015 -- Lecture 5 49/4/2015

Minimization of a simple function

0dxdV

local minimum

global minimum

0 :conditionNecessary

.maximized)(or minimized is )(for which of value(s) thefind ,)(function aGiven

dxdV

xVxxV

PHY 711 Fall 2015 -- Lecture 5 59/4/2015

Functional minimization

0 :conditionNecessary

.,),( extremizes which )(function theFind

.,),(function a and )( and )(

points end with the,)( functions offamily aConsider

L

xdxdyxyLxy

xdxdyxyLyxyyxy

xy

ffii

1,1

0,0

22

:Example

dydxL

PHY 711 Fall 2015 -- Lecture 5 69/4/2015

1

0

2

1,1

0,0

22

1

:Example

dxdxdy

dydxL

4789.141 )(

4142.1211 )(

4789.1411 )(

:functions Sample

1

0

222

1

02

1

01

dxxL xxy

dxL xxy

dxx

L xxy

PHY 711 Fall 2015 -- Lecture 5 79/4/2015

Calculus of variation example for a pure integral functions

0 :conditionNecessary

.,),(,),( where

,),( extremizes which )(function theFind

L

dxxdxdyxyfx

dxdyxyL

xdxdyxyLxy

f

i

x

x

./

:Formally

)()( )(

)()()(let ,any At

,,

dxdxdy

dxdyfy

yfL

dxxdy

dxxdy

dxxdy

xyxyxyx

f

i

x

x yxdxdyx

PHY 711 Fall 2015 -- Lecture 5 89/4/2015

After some derivations, we find

fiyx

dxdyx

fi

x

x yxdxdyx

x

x yxdxdyx

xxxdxdy

fdxd

yf

xxxdxydxdy

fdxd

yf

dxdxdy

dxdyfy

yfL

f

i

f

i

allfor 0 /

allfor 0/

/

,,

,,

,,

PHY 711 Fall 2015 -- Lecture 5 99/4/2015

0

/1

/

0 /

1,),( 1

1)1(;0)0( -- points End :Example

2

,,

21

0

2

dxdy

dxdydxd

dxdyf

dxd

yf

dxdyx

dxdyxyfdx

dxdyL

yy

yxdxdyx

xxy

K

KKdxdy K

dxdy

dxdy

)(

1'

/1

/

:Solution

22

PHY 711 Fall 2015 -- Lecture 5 109/4/2015

y

x

xi yi

xf yf

0

/1

/

0 /

1,),( 12

:Example

2

,,

22

dxdy

dxxdydxd

dxdyf

dxd

yf

dxdyxx

dxdyxyfdx

dxdyxA

yxdxdyx

x

x

f

i

PHY 711 Fall 2015 -- Lecture 5 119/4/2015

21

212

2

1

12

2

ln)(

1

1/1

/

0/1

/

KxxKKxy

Kxdx

dy

Kdxdy

dxxdy

dxdy

dxxdydxd

PHY 711 Fall 2015 -- Lecture 5 129/4/2015

Another example:(Courtesy of F. B. Hildebrand, Methods of Applied Mathematics)

a

xaxy

aydx

yd

adxaydxdyI

yyxy

sinsin)(

0

:equation Lagrange-Euler

0constant for

:integral theminimizethat 11 and 00 with curves allConsider

2

2

1

0

22

PHY 711 Fall 2015 -- Lecture 5 139/4/2015

0 /

: extremize tocondition necessary a

,,),(for :Review

,,

yxdxdyx

x

x

dxdyf

dxd

yf

dx,xdxdyy(x),f

xdxdyxyf

f

i

xf

dxdy

dxdyff

dxd

xf

dxdy

dxd

dxdyf

dxdy

dxdyf

dxd

xf

dxdy

dxd

dxdyf

dxdy

yf

dxdf

xdxdyxyf

/

//

/

,,),(for that Note

PHY 711 Fall 2015 -- Lecture 5 149/4/2015

Brachistochrone problem: (solved by Newton in 1696) http://mathworld.wolfram.com/BrachistochroneProblem.html

A particle of weight mg travels frictionlessly down a path of shape y(x). What is the shape of the path y(x) that minimizes the travel time fromy(0)=0 to y()=-2 ?

PHY 711 Fall 2015 -- Lecture 5 159/4/2015

0

1

1

0/

1,),(

because 2

1

2

2

221

2

dxdyy

dxd

dxdy

dxdyff

dxd

ydxdy

xdxdyxyf

mgymvdxgydxdy

vdsT

f

i

ff

ii

x

x

yx

yx

0

1

1

21

:dcomplicate more isequation aldifferenti

0, /

:equation Lagrange-Eulerof form original for the that Note

23

2

,,

dxdyy

dxdy

dxd

ydxdy

dxdyf

dxd

yf

yxdxdyx

PHY 711 Fall 2015 -- Lecture 5 169/4/2015

ydxdy

xdxdyxyf

mgymvdxgy

dxdy

vdsT

f

i

ff

ii

x

x

yx

yx

2

221

2

1,),(

because 2

1

Alternative relationships for extremization:

0 /

f d fy dx dy dx

xf

dxdy

dxdyff

dxd

/

PHY 711 Fall 2015 -- Lecture 5 179/4/2015

aKdxdyy

dxdyy

dxd

xf

dxdy

dxdyff

dxd

ydxdy

xdxdyxyf

21 0

1

1

/

1,),(

2

2

2

PHY 711 Fall 2015 -- Lecture 5 18

Parametric equations for Brachistochrone:

1cos

sin

ayax

9/4/2015

dx

yady

ya

dxdy

aKdxdyy

12

12

212

0

22

22

22

sin''cos1

1sin22

cossin2

12

1cossin2Let

adax

dx

aa

da

yady

aay

PHY 711 Fall 2015 -- Lecture 5 199/4/2015