physics 319 classical mechanics - jefferson lab · undergraduate classical mechanics spring 2017...
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![Page 1: Physics 319 Classical Mechanics - Jefferson Lab · Undergraduate Classical Mechanics Spring 2017 Ignorable or Cyclic Coordinates • Lagrangian for particle acted on by gravity •](https://reader036.vdocuments.site/reader036/viewer/2022062402/5f02248a7e708231d402c877/html5/thumbnails/1.jpg)
Undergraduate Classical Mechanics Spring 2017
Physics 319
Classical Mechanics
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 14
![Page 2: Physics 319 Classical Mechanics - Jefferson Lab · Undergraduate Classical Mechanics Spring 2017 Ignorable or Cyclic Coordinates • Lagrangian for particle acted on by gravity •](https://reader036.vdocuments.site/reader036/viewer/2022062402/5f02248a7e708231d402c877/html5/thumbnails/2.jpg)
Undergraduate Classical Mechanics Spring 2017
Lagrangian Method
1. Choose coordinates to incorporate constraint
automatically.
2. Write down the Euler-Lagrange Equation.
3. Solve the Euler-Lagrange Equation.
• Double Pendulum example
• Analyzed in detail in Chapter 11
(0,0)
1 1 1 1 1 1
2 2 1 1 2 2 2 2
2 2 2 2 2 21 2 21 2 1 2 1 1 1 1 2 1 2 1 2 1 2 1 2 2 2
1 1 1 2 1 1 2 2 2
, cos , sin
, , cos , sin
, , , sin sin cos cos2 2 2
cos cos cos
x y l l
x y x y l l
m m ml l m l l l
m gl m gl m gl
L
l1
l2
1
2
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Undergraduate Classical Mechanics Spring 2017
Ignorable or Cyclic Coordinates
• Lagrangian for particle acted on by gravity
• Does not depend on x or y. Euler-Lagrange imply
• When the Lagrangian is independent of a generalized
coordinate that coordinate is said to by ignorable or cyclic.
Automatically, generalized momentum is conserved.
Usually this degree of freedom can be integrated completely
using the conservation of generalized momentum
2 2 2
2
mx y z mgz L
0 0x y
d dmx p C my p C
dt dt
0 0ii
i i
dpdp C
q dt q dt
L L
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Undergraduate Classical Mechanics Spring 2017
Conservation Laws/Nöther’s Theorem
• Suppose Langrangian is translationally invariant
This invariance in Lagrangian implies momentum conserved
• Example: N interacting bodies
, ,
0 0i i ii i i
q q q q
dC
q dt q q
L L
L L L L
12
int
1 1 12
, ,
N N ii
i i j
i i j
j j j j
mv U r r
v r v r
L
L L
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Undergraduate Classical Mechanics Spring 2017
Momentum Conservation
• Total momentum conserved
• When Lagrangian invariant to (infinitesimal) rotation
1 1
int,
1 1 1 1
0N N i i
i x i i i
i i j j
Udm v x x
dt x
, ,
, , ,
, , ,
,
ˆ ˆ, ,
0 0
i i i i
i i jk j i k i i i jk j i k i i i
i jk i k i jk i k i jk i k
i i ij i i i i i i
r r r r
r r x r r x r r
dr r r
r r dt r
L
L L
L L L L
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Undergraduate Classical Mechanics Spring 2017
Angular Momentum Conservation
• Total angular momentum conserved
• When Lagrangian independent of time
, , 0i i i i jk i k
i
j i i ij
i
dm r r
dt
L r m r C
1 1 1 1
1 1
1 1 1
, , , , , , , , , , , ,
0
n n n n
n n
i i
i ii i
n n n
i i
i i ii i i
q q q q t q q q q t
d dq q
t dt q dt q
d d d dq q
dt q dt dt q dt q
L L
L L L L L
L L L LL
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Undergraduate Classical Mechanics Spring 2017
Hamiltonian Function
• The negative of the preceding combination is called the
Hamiltonian function
• The Hamiltonian is conserved
• Taylor shows for time-independent generalized coordinate
transformations, leading to quadratic dependence of the
Lagrangian on the
• Energy conserved!
1
n
i
i i
L
H L
2T T U T U H
iq
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Undergraduate Classical Mechanics Spring 2017
Lagrangian for Electromagnetic Force
• Electromagnetic Lorentz force follows from the
Lagrangian
• Canonical momentum
2 2 2
2x y z
mx y z q q xA yA zA L
mr qA Px
L
yx x x x x z
d A Amr qA mr q r
dt t x xx
AA A A A A Amx q qx qy qz q qx qy qz
t x y z x x x x
L L
P
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Undergraduate Classical Mechanics Spring 2017
y y y y yx z
yxz z z z z
y x x zx
A A A A AA Amy q qx qy qz q qx qy qz
t x y z y y y y
AAA A A A Amz q qx qy qz q qx qy qz
t x y z z z z z
A A A Amx qE qy qz
x y z x
my q
y y xzy
yx z zz
A A AAE qz qx
y z x y
AA A Amz qE qx qy
z x y z
ma q E v B
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Undergraduate Classical Mechanics Spring 2017
Lagrange Multipliers
• Sometimes constraints are not easily accounted for only by
coordinate choice. If the constraint can be put in the form
of a time-integral along the path, the problem can be
solved using Lagrange multipliers
• Find stationary condition using Euler-Lagrange for
• Works because the constrained variation of the added λf is
zero (it integrates to a constant) and does not change the
fact that the constrained solution is still has stationary
Lagrangian
1 1
1 1
, , , , , ,
, , , , , ,
n n
n n
S q q q q t dt
C f q q q q t dt
L
S f dt L
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Undergraduate Classical Mechanics Spring 2017
Snell’s Law 3!
• In terms of two angles
1 1 2 2
1 2
1 1 2 2 2
1 1 2 21 1 2 2
1 2
2
2
1 1 2 2
cos cos
tan tan
tan tan 0cos cos
sinsec 0 sin
cos
sin sin
tot
i
i i ii i i i
i
n h n hL
h h x
n h n hh h
n hh n
n n