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PHY 7 11 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 6: Start reading Chapter 3 – F irst focusing on the “calculus of variation”. Comment on HW #3. - PowerPoint PPT Presentation
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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