9/11/2013 PHY 711 Fall 2013 -- Lecture 7 1 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 7: Continue reading Chapter 3 1.Lagrange’s equations 2.D’Alembert’s principle
9/11/2013PHY 711 Fall 2013 -- Lecture 71 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 7: Continue reading.
Jan 16, 2016
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PHY 711 Fall 2013 -- Lecture 7 19/11/2013
PHY 711 Classical Mechanics and Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 7:
Continue reading Chapter 3
1. Lagrange’s equations
2. D’Alembert’s principle
PHY 711 Fall 2013 -- Lecture 7 29/11/2013
PHY 711 Fall 2013 -- Lecture 7 39/11/2013
PHY 711 Fall 2013 -- Lecture 7 49/11/2013
PHY 711 Fall 2013 -- Lecture 7 59/11/2013
Summary of results from the calculus of variation
x
f
dx
dy
dxdy
ff
dx
d
dxdy
f
dx
d
y
f
dxxdx
dyxyfI
f
i
x
x
/
0 /
:equations Lagrange-Euler theiscondition necessary A
,),(
:form integralan on
ionextremizatan perform toneed we whereproblems of class For the
PHY 711 Fall 2013 -- Lecture 7 69/11/2013
Application to particle dynamics
Simple example: vertical trajectory of particle of mass m subject to constant downward acceleration a=-g.
221
2
2
)( gttvyty
mgdt
ydm
ii
PHY 711 Fall 2013 -- Lecture 7 79/11/2013
UTtdt
dytyL
,),(
:be todefined Lagrangian heconsider t Now
Kinetic energy
Potential energy
:)( physicalfor minimized is 2
1
:states principle sHamilton'
2
1,),(
:exampleour In
2
2
tydtmgydt
dymS
mgydt
dymUTt
dt
dytyL
f
i
t
t
PHY 711 Fall 2013 -- Lecture 7 99/11/2013
http://rjlipton.wordpress.com
PHY 711 Fall 2013 -- Lecture 7 109/11/2013
gdt
dy
dt
d
ymdt
dmg
y
L
dt
d
y
L
dtmgydt
dymS
f
i
t
t
0
0
:relations Lagrange-Euler
2
1
:action theminimizingfor Condition 2
221)( gttvyty ii
PHY 711 Fall 2013 -- Lecture 7 119/11/2013
323
322
321
321332
21
3
221222
21
2
212
21
1
221
2
150.0
167.0
125.0
:says Maple
//1
/1
/1)( :ies trajectorTrial
0 , ; ,0 Assume
2
1
:Check
TmgS
TmgS
TmgS
TgtHTtgTty
gtHTtgTty
gTtHTtgTty
yTtgTHyt
dtmgydt
dymS
ffii
t
t
f
i
PHY 711 Fall 2013 -- Lecture 7 129/11/2013
Jean d’Alembert 1717-1783
PHY 711 Fall 2013 -- Lecture 7 139/11/2013
sd ixq
:scoordinate dGeneralize
Uq
q
x
x
Ud
x
UF
xFd
d-m-m
i
i
i
ii
i
ii
sF
sF
saFaF
:force veconservati aFor
0 0
:laws sNewton'
D’Alembert’s principle:
PHY 711 Fall 2013 -- Lecture 7 149/11/2013
xm
q
xm
dt
ddm
q
x
dt
dx
x
dt
d
q
x
q
x
x
dt
dxm
q
xxm
dt
d
xxmdm
d-m-m
i
ii
iiiii
i
ii
ii
i
ii
2212
21
and :Claim
0 0
:laws sNewton'
sa
sa
saFaF
sd ixq
:scoordinate dGeneralize
PHY 711 Fall 2013 -- Lecture 7 159/11/2013
T
q
T
dt
ddm
xmT
xm
q
xm
dt
ddm
ii
i
ii
sa
sa
2
21
2212
21
:energy kinetic -- Define
sd ixq
:scoordinate dGeneralize
0
:Recall
T
q
T
dt
dq
q
Ud-m
Uq
q
x
x
Ud
i
i
i
saF
sF
PHY 711 Fall 2013 -- Lecture 7 169/11/2013
sd ixq
:scoordinate dGeneralize
0
0
UT
q
UT
dt
d
T
q
T
dt
dq
q
Ud-m
saF
0
if only true is This :Note
q
U
UTtqqL ;,
PHY 711 Fall 2013 -- Lecture 7 179/11/2013
sd ixq
:scoordinate dGeneralize
f
i
t
t
dttqqLS
L
q
L
dt
dd-m
tqqLL
UTL
,, :integralon Minimizati
0
,,
:Lagrangian -- Define
saF
PHY 711 Fall 2013 -- Lecture 7 189/11/2013
0
,, :equations LagrangeEuler
q
L
q
L
dt
d
UTtqqLL
Example:
q
d
sin
0sin 0
cos,
2
2
2
2221
d
g
dt
d
mgdmddt
d
q
L
q
L
dt
d
ddmgmdLL
PHY 711 Fall 2013 -- Lecture 7 199/11/2013
coscossin),,,,,(
0
,, :exampleAnother
232
122212
1 MgdIILL
q
L
q
L
dt
d
UTtqqLL
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