PHY 711 Fall 2014 -- Lecture 6 19/08/2014

PHY 711 Classical Mechanics and Mathematical Methods

10-10:50 AM MWF Olin 103

Plan for Lecture 6:

Continue reading Chapter 3

Further development of the “calculus of variation”

PHY 711 Fall 2014 -- Lecture 6 29/08/2014

PHY 711 Fall 2014 -- Lecture 6 39/08/2014

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 2014 -- Lecture 6 49/08/2014

ydxdy

xdx

dyxyf

mgymvdxgy

dxdy

v

dsT

f

i

ff

ii

x

x

yx

yx

2

221

2

1,),(

because 2

1

Alternative relationships for extremization:

0 /

f d f

y dx dy dx

/

d f dy ff

dx dy dx dx x

PHY 711 Fall 2014 -- Lecture 6 59/08/2014

aKdx

dyy

dxdy

ydx

d

x

f

dx

dy

dxdy

ff

dx

d

ydxdy

xdx

dyxyf

21 0

1

1

/

1,),(

2

2

2

PHY 711 Fall 2014 -- Lecture 6 6

Parametric equations for Brachistochrone:

1cos

sin

ay

ax

9/08/2014

dx

ya

dy

y

a

dx

dy

aKdx

dyy

12

12

212

0

22

22

22

sin''cos1

1sin22

cossin2

12

1cossin2Let

adax

dx

aa

da

ya

dy

aay

PHY 711 Fall 2014 -- Lecture 6 79/08/2014

Brachistochrone problem -- summary

Parametric equations;

1cos

sin

ay

ax

12

212

y

a

dx

dy

aKdx

dyy

2

2

2

2

sin

cos

1cos

1cos

21

2

sin

cos

cos1

sin

/

/

:Check

y

ay

y

a

a

a

ddx

ddy

dx

dy

0

PHY 711 Fall 2014 -- Lecture 6 89/08/2014

Brachistochrone problem -- summary

adxy

dxdy

gT

xy

a

21

2

:)( optimalFor :Check

0

2

a

agT

xxy

21.185

212

2)( :Check

221

1

a

agT

a

xaxy

22.378

212

2sin2)( :Check

222

2

PHY 711 Fall 2014 -- Lecture 6 99/08/2014

Shape of a rope of length L and mass density r hanging between two points

x1 y1

x2 y2

PHY 711 Fall 2014 -- Lecture 6 109/08/2014

LEW

dxdx

dyL

dxdx

dyygE

x

x

x

x

:minimize ofunction t composite a Define

1

:rope ofLength

1

:rope hanging ofenergy Potential

2

1

2

1

2

2

Lagrange multiplier

PHY 711 Fall 2014 -- Lecture 6 119/08/2014

K

dxdy

dxdy

dx

dygy

x

f

dx

dy

dxdy

ff

dx

d

dx

dygy

dx

dyyf

dxdx

dygyW

x

x

2

2

2

2

2

1

1

/

1,

12

1

PHY 711 Fall 2014 -- Lecture 6 129/08/2014

gK

axK

gxy

K

dxdy

gy

K

dxdy

dxdy

dx

dygy

/cosh

1)(

1

1

1

1

2

2

2

2

PHY 711 Fall 2014 -- Lecture 6 139/08/2014

Ldxdx

dy

yxy

yxy

aK

gK

axK

gxy

x

x

2

1

2

22

11

1

)(

)( :sConstraint

,, :constantsn Integratio

/cosh

1)(

PHY 711 Fall 2014 -- Lecture 6 149/08/2014

Summary of results

x

f

dx

dy

dxdy

ff

dx

d

dxdy

f

dx

d

y

f

δIdxxdx

dyxyfI

f

i

x

x

/

0 /

:equations Lagrange-Euler theiscondition necessary A

0 ,),(

:form integralan on

ionextremizatan perform toneed we whereproblems of class For the

PHY 711 Fall 2014 -- Lecture 6 159/08/2014

Application to particle dynamics

2

1

;,

:Denote

(action)

n)(Lagrangia

)coordinate ed(generaliz

(time)

t

t

dttqqLA

dt

dqq

AI

Lf

qy

tx