PHY 711 Fall 2013 -- Lecture 5 19/6/2013

PHY 711 Classical Mechanics and Mathematical Methods

10-10:50 AM MWF Olin 103

Plan for Lecture 6:

Start reading Chapter 3 –

First focusing on the “calculus of variation”

PHY 711 Fall 2013 -- Lecture 59/6/2013 2

PHY 711 Fall 2013 -- Lecture 5 39/6/2013

Comment on HW #3

PHY 711 Fall 2013 -- Lecture 5 49/6/2013

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

0dx

dV

local minimum

global minimum

PHY 711 Fall 2013 -- Lecture 5 59/6/2013

Minimization of a simple function

0dx

dV

local minimum

global minimum

0 :conditionNecessary

.maximized)(or minimized is )(for which

of value(s) thefind ,)(function aGiven

dx

dV

xV

xxV

PHY 711 Fall 2013 -- Lecture 5 69/6/2013

Functional minimization

0 :conditionNecessary

.,),( extremizes which )(function theFind

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

points end with the,)( functions offamily aConsider

L

xdx

dyxyLxy

xdx

dyxyLyxyyxy

xy

ffii

1,1

0,0

22

:Example

dydxL

PHY 711 Fall 2013 -- Lecture 5 79/6/2013

1

0

2

1,1

0,0

22

1

:Example

dxdx

dy

dydxL

4789.141 )(

4142.1211 )(

4789.14

11 )(

:functions Sample

1

0

222

1

0

2

1

0

1

dxxL xxy

dxL xxy

dxx

L xxy

PHY 711 Fall 2013 -- Lecture 5 89/6/2013

Calculus of variation example for a pure integral functions

0 :conditionNecessary

.,),(,),( where

,),( extremizes which )(function theFind

L

dxxdx

dyxyfx

dx

dyxyL

xdx

dyxyLxy

f

i

x

x

./

:Formally

)()(

)(

)()()(let ,any At

,,

dxdx

dy

dxdy

fy

y

fL

dx

xdy

dx

xdy

dx

xdy

xyxyxyx

f

i

x

x yxdx

dyx

PHY 711 Fall 2013 -- Lecture 5 99/6/2013

After some derivations, we find

fi

yxdx

dyx

fi

x

x yxdx

dyx

x

x yxdx

dyx

xxxdxdy

f

dx

d

y

f

xxxdxydxdy

f

dx

d

y

f

dxdx

dy

dxdy

fy

y

fL

f

i

f

i

allfor 0 /

allfor 0/

/

,,

,,

,,

PHY 711 Fall 2013 -- Lecture 5 109/6/2013

0

/1

/

0 /

1,),( 1

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

2

,,

21

0

2

dxdy

dxdy

dx

d

dxdy

f

dx

d

y

f

dx

dyx

dx

dyxyfdx

dx

dyL

yy

yxdx

dyx

xxy

K

KK

dx

dy K

dxdy

dxdy

)(

1'

/1

/

:Solution

22

PHY 711 Fall 2013 -- Lecture 5 119/6/2013

y

x

xi yi

xf yf

0

/1

/

0 /

1,),( 12

:Example

2

,,

22

dxdy

dxxdy

dx

d

dxdy

f

dx

d

y

f

dx

dyxx

dx

dyxyfdx

dx

dyxA

yxdx

dyx

x

x

f

i

PHY 711 Fall 2013 -- Lecture 5 129/6/2013

21

212

2

1

12

2

ln)(

1

1

/1

/

0/1

/

KxxKKxy

Kxdx

dy

Kdxdy

dxxdy

dxdy

dxxdy

dx

d

PHY 711 Fall 2013 -- Lecture 5 139/6/2013

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

a

xaxy

aydx

yd

adxaydx

dyI

yyxy

sin

sin)(

0

:equation Lagrange-Euler

0constant for

:integral theminimizethat

11 and 00 with curves allConsider

2

2

1

0

22

PHY 711 Fall 2013 -- Lecture 5 149/6/2013

0 /

: extremize tocondition necessary a

,,),(for :Review

,,

yxdx

dyx

x

x

dxdy

f

dx

d

y

f

dx,xdx

dyy(x),f

xdx

dyxyf

f

i

x

f

dx

dy

dxdy

ff

dx

d

x

f

dx

dy

dx

d

dxdy

f

dx

dy

dxdy

f

dx

d

x

f

dx

dy

dx

d

dxdy

f

dx

dy

y

f

dx

df

xdx

dyxyf

/

//

/

,,),(for that Note

PHY 711 Fall 2013 -- Lecture 5 159/6/2013

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(p)=-2 ?

PHY 711 Fall 2013 -- Lecture 5 169/6/2013

0

1

1

0/

1,),(

because 2

1

2

2

221

2

dxdy

ydx

d

dx

dy

dxdy

ff

dx

d

ydxdy

xdx

dyxyf

mgymvdxgy

dxdy

v

dsT

f

i

ff

ii

x

x

yx

yx

0

1

1

2

1

:dcomplicate more isequation aldifferenti

0, /

:equation Lagrange-Euler

of form original for the that Note

23

2

,,

dxdy

y

dxdy

dx

d

ydxdy

dxdy

f

dx

d

y

f

yxdx

dyx