Post on 07-Jun-2020
c-v c +vv c
u=4c/5
√(c-v)(c +v)=c√(1-v2/c2)=c sech ρ=c cos σ
v=c tanh ρ= c sin σσ
c sin σ
Multiply segments by cosh ρ = sec σ=1/√(1-v2/c2)to recover dimensions in (ck, ω) plot
Space-Time Geometry
2012 INBRE workshop
Weapons of Math Instruction:Geometry in Elementary and Advanced Physics
2012 INBRE workshop
Weapons of Math Instruction:Geometry in Elementary and Advanced Physics
Bill Harter Tyle Reimer Alvason LiBill Harter Tyle Reimer Alvason Li1
c-v c +vv c
u=4c/5
√(c-v)(c +v)=c√(1-v2/c2)=c sech ρ=c cos σ
v=c tanh ρ= c sin σσ
c sin σ
Multiply segments by cosh ρ = sec σ=1/√(1-v2/c2)to recover dimensions in (ck, ω) plot
Space-Time Geometry
2012 INBRE workshop
Weapons of Math Instruction:Geometry in Elementary and Advanced Physics
2012 INBRE workshop
Weapons of Math Instruction:Geometry in Elementary and Advanced Physics
Bill Harter Tyle Reimer Alvason LiBill Harter Tyle Reimer Alvason LiLogistics and love: Carole Harter
2
SOME PHYSICS YOU CAN DO BETTER WITH GEOMETRY
•Superball collision problems (Discover momentum-energy rules and Lagrange, Hamilton, and Poincare classical mechanics)
•Rutherford-Coulomb scattering orbits and caustics •Runga-Lenz-Lagrange scattering orbits and caustics
•Space-time wave fractal (“quantum carpet”) gives a lesson in fractions that is quite appealing
•Einstein-Lorentz-Minkowskii relativity (Discover relativistic quantum mechanics)
•Accurate pocket sundial that tells time and predicts sunrise, sunset, civil and nautical twilight, and sunburn hazard.
SOME PHYSICS YOU CAN DO BETTER WITH GEOMETRY
3
SOME PHYSICS YOU CAN DO BETTER WITH GEOMETRY
•Superball collision problems (Discover momentum-energy rules and Lagrange, Hamilton, and Poincare classical mechanics)
•Rutherford-Coulomb scattering orbits and caustics •Runga-Lenz-Lagrange scattering orbits and caustics
•Space-time wave fractal (“quantum carpet”) gives a lesson in fractions that is quite appealing
•Einstein-Lorentz-Minkowskii relativity (Discover relativistic quantum mechanics)
•Accurate pocket sundial that tells time and predicts sunrise, sunset, civil and nautical twilight, and sunburn hazard.
SOME PHYSICS YOU CAN DO BETTER WITH GEOMETRY
4
Ka-bong!Ka-bong! Ka-runch!Ka-runch!
Totallyinelastic
case
Perfectlyelasticcase
??????????
1060
????????
1060
0-0.2-0.4-0.6-0.8-1-6 sec.-12 sec.
-24 sec.
-36 sec.
-48 sec.
1 mile
1minute(60sec.) (b) Collision!
After collision...what velocities?Before collision.....A problem in ssppaaccee--ttiimmee : (6600mph Cell-faxing 4ton SUV rear-ends 1100mph 1ton VW)
6600mph1100mph
5
Ka-bong!Ka-bong! Ka-runch!Ka-runch!
Totallyinelastic
case
Perfectlyelasticcase
??????????
1060
????????
1060
0-0.2-0.4-0.6-0.8-1-6 sec.-12 sec.
-24 sec.
-36 sec.
-48 sec.
1 mile
1minute(60sec.) (b) Collision!
After collision...what velocities?Before collision.....A problem in ssppaaccee--ttiimmee : (6600mph Cell-faxing 4ton SUV rear-ends 1100mph 1ton VW)
6600mph1100mph
Conventional solution:Get out formulas: Σ mV(before)= Σ mV(after) [momentum conservation]
Σ mV2(before)= Σ mV2(after) [energy conservation]
6
Ka-bong!Ka-bong! Ka-runch!Ka-runch!
Totallyinelasticcase
Perfectlyelasticcase
??????????
1060
????????
1060
0-0.2-0.4-0.6-0.8-1-6 sec.-12 sec.
-24 sec.
-36 sec.
-48 sec.
1 mile
1minute(60sec.) (b) Collision!
After collision...what velocities?Before collision.....A problem in ssppaaccee--ttiimmee : (6600mph Cell-faxing 4ton SUV rear-ends 1100mph 1ton VW)
6600mph
Conventional solution:Get out formulas:ΣmV(before)=ΣmV(before) [momentum conservation]ΣmV2(before)=ΣmV2(before)[energy conservation]etc.
But an UNconventional way is quicker and slicker.......... (Just have to draw 2 lines! ... (and a circle...)
7
Ka-bong!Ka-bong! Ka-runch!Ka-runch!
Totallyinelasticcase
Perfectlyelasticcase
??????????
1060
????????
1060
0-0.2-0.4-0.6-0.8-1-6 sec.-12 sec.
-24 sec.
-36 sec.
-48 sec.
1 mile
1minute(60sec.) (b) Collision!
After collision...what velocities?Before collision.....A problem in ssppaaccee--ttiimmee : (6600mph Cell-faxing 4ton SUV rear-ends 1100mph 1ton VW)
6600mph
Conventional solution:Get out formulas:ΣmV(before)=ΣmV(before) [momentum conservation]ΣmV2(before)=ΣmV2(before)[energy conservation]etc.
But an UNconventional way is quicker and slicker.......... (Just have to draw 2 lines! ... (and a circle...)
..... and most importantly, DERIVE LOGICAL STRUCTURE!
8
Ka-bong!Ka-bong! Ka-runch!Ka-runch!
Totallyinelasticcase
Perfectlyelasticcase
??????????
1060
????????
1060
0-0.2-0.4-0.6-0.8-1-6 sec.-12 sec.
-24 sec.
-36 sec.
-48 sec.
1 mile
1minute(60sec.) (b) Collision!
After collision...what velocities?Before collision.....A problem in ssppaaccee--ttiimmee : (6600mph Cell-faxing 4ton SUV rear-ends 1100mph 1ton VW)
50mph20 40 60 90
20
40
60
80
50mph
100mph
100mph
VVVW
0
INITIAL
10 30 70 80
90
10
30
70
I
VVSUV
6600mph1100mph
VVeelloocciittyy-vveelloocciittyy PPlloott
6600mph
1100mph
Conventional solution:Get out formulas:ΣmV(before)=ΣmV(before) [momentum conservation]ΣmV2(before)=ΣmV2(before)[energy conservation]etc.
9
Ka-bong!Ka-bong! Ka-runch!Ka-runch!
Totallyinelasticcase
Perfectlyelasticcase
??????????
1060
????????
1060
0-0.2-0.4-0.6-0.8-1-6 sec.-12 sec.
-24 sec.
-36 sec.
-48 sec.
1 mile
1minute(60sec.) (b) Collision!
After collision...what velocities?Before collision.....A problem in ssppaaccee--ttiimmee : (6600mph Cell-faxing 4ton SUV rear-ends 1100mph 1ton VW)
50mph20 40 60 90
20
40
60
80
50mph
100mph
100mph
VVVW
0
INITIAL
10 30 70 80
90
10
30
70
I
VVSUV
6600mph1100mph
VVeelloocciittyy-vveelloocciittyy PPlloott
6600mph
1100mph
Conventional solution:Get out formulas:ΣmV(before)=ΣmV(before) [momentum conservation]ΣmV2(before)=ΣmV2(before)[energy conservation]etc.
50mph20 40 60 90
20
40
60
80
50mph
100mph
100mph
VVVW
0
INITIAL
10 30 70 80
90
10
30
70
I
VVSUV6600mph
1100mph
MSUVVVSUV+MVWVVVW =constant is AAxxiioomm ##11
4
-1
10
Ka-bong!Ka-bong! Ka-runch!Ka-runch!
Totallyinelasticcase
Perfectlyelasticcase
??????????
1060
????????
1060
0-0.2-0.4-0.6-0.8-1-6 sec.-12 sec.
-24 sec.
-36 sec.
-48 sec.
1 mile
1minute(60sec.) (b) Collision!
After collision...what velocities?Before collision.....A problem in ssppaaccee--ttiimmee : (6600mph Cell-faxing 4ton SUV rear-ends 1100mph 1ton VW)
50mph20 40 60 90
20
40
60
80
50mph
100mph
100mph
VVVW
0
INITIAL
10 30 70 80
90
10
30
70
I
VVSUV
4
-1
slope =-4
6600mph1100mph
VVeelloocciittyy-vveelloocciittyy PPlloott
6600mph
1100mph
Conventional solution:Get out formulas:ΣmV(before)=ΣmV(before) [momentum conservation]ΣmV2(before)=ΣmV2(before)[energy conservation]etc.
50mph20 40 60 90
20
40
60
80
50mph
100mph
100mph
VVVW
0
INITIAL
10 30 70 80
90
10
30
70
I
VVSUV6600mph
1100mph
= ΔVSUV
ΔVVWMSUV-mVW
4-1 =
MSUVVVSUV+MVWVVVW =constant is AAxxiioomm ##11
11
Ka-bong!Ka-bong! Ka-runch!Ka-runch!
Totally
inelastic
case
Perfectly
elastic
case
??????????
1060
????????
1060
0-0.2-0.4-0.6-0.8-1
-6 sec.
-12 sec.
-24 sec.
-36 sec.
-48 sec.
1 mile
1minute(60sec.) (b) Collision!
After collision...what velocities?Before collision.....
A problem in ssppaaccee--ttiimmee : (6600mph Cell-faxing 4ton SUV rear-ends 1100mph 1ton VW)
50mph20 40 60 90
20
40
60
80
50mph
100mph
100mph
VVVW
0
INITIAL
10 30 70 80
90
10
30
70
I
VVSUV
“Ka-Runch!”
V VW
=V SUV
line
FINAL
Ka-runch!
4
-1
slope =-4
6600mph1100mph
VVeelloocciittyy-vveelloocciittyy PPlloott
6600mph
1100mph
Conventional solution:
Get out formulas:
ΣmV(before)=ΣmV(before) [momentum conservation]ΣmV2(before)=ΣmV2(before)[energy conservation]etc.
50mph20 40 60 90
20
40
60
80
50mph
100mph
100mph
VVVW
0
INITIAL
10 30 70 80
90
10
30
70
F
45°
45°
I
VVSUV
6600mph
1100mph
= ΔVSUV
ΔVVW
MSUV
-mVW
4
-1 =
MSUVVVSUV
+MVWVVVW
=constant is AAxxiioomm ##11
BINGO!slope=+1
12
Ka-bong!Ka-bong! Ka-runch!Ka-runch!
Totally
inelastic
case
Perfectly
elastic
case
??????????
1060
????????
1060
0-0.2-0.4-0.6-0.8-1
-6 sec.
-12 sec.
-24 sec.
-36 sec.
-48 sec.
1 mile
1minute(60sec.) (b) Collision!
After collision...what velocities?Before collision.....
A problem in ssppaaccee--ttiimmee : (6600mph Cell-faxing 4ton SUV rear-ends 1100mph 1ton VW)
50mph20 40 60 90
20
40
60
80
50mph
100mph
100mph
VVVW
0
INITIAL
10 30 70 80
90
10
30
70
I
VVSUV
“Ka-Runch!”
V VW
=V SUV
line
FINAL
Ka-runch!
4
-1
slope =-4
6600mph1100mph
VVeelloocciittyy-vveelloocciittyy PPlloott
6600mph
1100mph
Conventional solution:
Get out formulas:
ΣmV(before)=ΣmV(before) [momentum conservation]ΣmV2(before)=ΣmV2(before)[energy conservation]etc.
50mph20 40 60 90
20
40
60
80
50mph
100mph
100mph
VVVW
0
INITIAL
10 30 70 80
90
10
30
70
F
45°
45°
I
VVSUV
6600mph
1100mph
= ΔVSUV
ΔVVW
MSUV
-mVW
4
-1 =
FINAL
Ka-bong!
MSUVVVSUV
+MVWVVVW
=constant is AAxxiioomm ##11
BINGO!
DOUBLE
BINGO!
“!hnuR-aK”
13
Ka-bong!Ka-bong! Ka-runch!Ka-runch!
Totally
inelastic
case
Perfectly
elastic
case
??????????
1060
????????
1060
0-0.2-0.4-0.6-0.8-1
-6 sec.
-12 sec.
-24 sec.
-36 sec.
-48 sec.
1 mile
1minute(60sec.) (b) Collision!
After collision...what velocities?Before collision.....
A problem in ssppaaccee--ttiimmee : (6600mph Cell-faxing 4ton SUV rear-ends 1100mph 1ton VW)
50mph20 40 60 90
20
40
60
80
50mph
100mph
100mph
VVVW
0
INITIAL
10 30 70 80
90
10
30
70
I
VVSUV
“Ka-Runch!”
V VW
=V SUV
line
FINAL
Ka-runch!
4
-1
slope =-4
6600mph1100mph
VVeelloocciittyy-vveelloocciittyy PPlloott
6600mph
1100mph
Conventional solution:
Get out formulas:
ΣmV(before)=ΣmV(before) [momentum conservation]ΣmV2(before)=ΣmV2(before)[energy conservation]etc.
50mph20 40 60 90
20
40
60
80
50mph
100mph
100mph
VVVW
0
INITIAL
10 30 70 80
90
10
30
70
F
45°
45°
I
VVSUV
6600mph
1100mph
= ΔVSUV
ΔVVW
MSUV
-mVW
4
-1 =
FINAL
Ka-bong!
MSUVVVSUV
+MVWVVVW
=constant is AAxxiioomm ##11
BINGO!
DOUBLE
BINGO!
“!hnuR-aK”
Superball Collision Simulatorhttp://www.uark.edu/rso/modphys/animations/BounceItWeb.html
Superball Collision Simulator
14
A problem in ssppaaccee--ttiimmee : (6600mph Cell-faxing 4ton SUV rear-ends 1100mph 1ton VW)Geometry of Galilean translation (A ssyymmmmeettrryy ttrraannssffoorrmmaattiioonn)
5020 40 60 90
10010 30 70 80
FEarth
10
20
30
40
50
60
70
80
90
100
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10-100-90-80-70-60
-50-40-30-20-10
FCOM
ICOM
IEarth
VSUV
VVW
subtracting
(50,50)
(a) Galileo transforms to COM frame
Fig. 2.5ain Unit 1
If you increase your velocity by 50 mph,...
...the rest of the world appears to be 50 mph sslloowweerr
15
M1=70gmM1=70gm
M0=10kgM0=10kgbounceplatebounceplate
RRrrd
Superballpenetrationdepthr22Rd=
SuperballSuperball
ballpointpen
M2=10gm
ballpointpen
M2=10gm
The X-2pen-
launcher
The X-2pen-
launcher
The X-2 Pen launcher and Superball Collision Simulator*
*SimulatorWebsite:http://www.uark.edu/rso/modphys/animations/BounceItWeb.html16
M1=70gmM1=70gm
M0=10kgM0=10kg
bounce
plate
bounce
plate
RRrr
d
Superball
penetration
depth
r2
2Rd=
SuperballSuperball
ballpoint
pen
M2=10gm
ballpoint
pen
M2=10gm
The X-2pen-
launcher
The X-2pen-
launcherFig. 4.1 and Fig. 4.3
in Unit 1
M1
m2BANG!M1
(BiggerBANG!)
(StillBiggerBANG!)
m2
M1M1
m2
Bang1!
Bang2!M1
m2
(a) Super-elastic 2nd-body bounce (b) 2-Bang Model (c) n-BodySupernovaSuperballs
m2
-1.0
1.0
1.0
0.5
2.0
m2 Velocity axis
Vym2
m1 Velocity axis
Vym1
(0,0)
Bang-1(01)
INIT point at
(-1.0,-1.0)
(a)Bang-1(01)
Fig. 4.4a-bin Unit 1
Bang-1(01)
FINAL point
(+1.0,-1.0)
This 1st bang is a floor-bounce of
M1off very massive plate/Earth M
0
2.02.01st bang:
mass (M0)vs. mass (M
1)
-7
+1
1.0
1.0
0.5
2.0
m1 Velocity axis
Vym1
(0,0)
COM-point at
(0.75,0.75))
-1.0
Bang-2(12)
INIT point at
(+1.0,-1.0)
Bang-2(12)
FINAL point
(0.5,2.5)
(b)Bang-2(12)
2nd bang:
(M1)vs.(M
2)
17
M1=70gmM1=70gm
M0=10kgM0=10kgbounceplatebounceplate
RRrrd
Superballpenetrationdepthr22Rd=
SuperballSuperball
ballpointpen
M2=10gm
ballpointpen
M2=10gm
The X-2pen-
launcher
The X-2pen-
launcherFig. 4.1 and Fig. 4.3
in Unit 1
M1
m2BANG!M1
(BiggerBANG!)
(StillBiggerBANG!)
m2
M1M1
m2
Bang1!
Bang2!M1
m2
(a) Super-elastic 2nd-body bounce (b) 2-Bang Model (c) n-BodySupernovaSuperballs
m2
Fig. 4.5ain Unit 1Start at
(1.0,-1.0)
1.0
1.0
-1.0
0.5
2.0 C
L
P
L is 15::1
P is 7::1C is 4::1
Line CPLis elastic collisionfinal pt. locus fordifferentmomentumslopesormassratiosM1::M2
M2 Velocity axisV2
M1 Velocity axis V1(0,0)
3.0
Bang-2(12)FINALpoints
(a)
18
M1=70gmM1=70gm
M0=10kgM0=10kgbounceplatebounceplate
RRrrd
Superballpenetrationdepthr22Rd=
SuperballSuperball
ballpointpen
M2=10gm
ballpointpen
M2=10gm
The X-2pen-
launcher
The X-2pen-
launcherFig. 4.1 and Fig. 4.3
in Unit 1
M1
m2BANG!M1
(BiggerBANG!)
(StillBiggerBANG!)
m2
M1M1
m2
Bang1!
Bang2!M1
m2
(a) Super-elastic 2nd-body bounce (b) 2-Bang Model (c) n-BodySupernovaSuperballs
m2
Fig. 4.5a-bin Unit 1Start at
(1.0,-1.0)
1.0
1.0
-1.0
0.5
2.0 C
L
P
L is 15::1
P is 7::1C is 4::1
Line CPLis elastic collisionfinal pt. locus fordifferentmomentumslopesormassratiosM1::M2
1.0
1.0
-1.0
0.5
2.0 C
L
P
L is 15::1
P is 7::1C is 4::1
II iiss 11::::00oorr ∞∞::::11II
MM iiss 33::::11
MM
3.0M2 Velocity axisV2
M2 Velocity axisV2
M1 Velocity axis V1 M1 Velocity axis V1(0,0) (0,0)
3.0
D is 2::1
Uis 1::1
Bang-2(12)FINALpoints
(a) (b)
-1.0
U2
U1
19
Geometric “Integration” (Converting Velocity data to Spacetime)
Start at
(-1.0,-1.0)
1.0
1.0
-1.0
0.5
2.0
Bang-2(12)
Bang-3(20)
Bang-1(01)
Floor at y=0
Bang-1(01)
Bang-2(12)
Bang-3(20)
M2
slope
+2.5
M1
slope
+0.5
Height y
y=3
y=1
Ceiling at y=7
Bang-4(12)
??
Vy1
Vy2
??
(a)
Time t
(b)
M2
slope
-2.5
20
Geometric “Integration” (Converting Velocity data to Spacetime)
Start at
(-1.0,-1.0)
1.0
1.0
-1.0
0.5
2.0
Bang-2(12)
Bang-3(20)
Bang-1(01)
Floor at y=0
Bang-1(01)
Bang-2(12)
Bang-3(20)
M2
slope
+2.5
M1
slope
+0.5
Height y
y=3
y=1
Ceiling at y=7
Bang-4(12)
??
Vy1
Vy2
??
(a)
Time t Bang-1(01)
Bang-2(12)
Bang-3(20)
Bang-4(12)
Bang-5(20)
Bang-6(12)
Bang-7(01)
Bang-8(20)
Bang-9(12)
Time t
Floor at y=0
Ceiling at y=7
y
Start at
(-1.0,-1.0)
1.0
1.0
-1.0
0.5
2.0
Bang-2(12)
Bang-3(20)
Bang-4(12)
Bang-1(01)
Bang-5(20)
Bang-6(12)
Bang-7(01)
Vy1
Vy2
(b)
(c)
(d)
M2
slope
-2.5
Fig. 4.7a-din Unit 1
Kinetic Energy Ellipse
KE = 12
M1V12 + 1
2M 2V2
2 =12
+ 72= 4
1 = V12
2KE /M1
+V2
2
2KE /M 2
=x1
2
a12 +
x22
a22
21
Geometric “Integration” (Converting Velocity data to Spacetime)
Start at
(-1.0,-1.0)
1.0
1.0
-1.0
0.5
2.0
Bang-2(12)
Bang-3(20)
Bang-1(01)
Floor at y=0
Bang-1(01)
Bang-2(12)
Bang-3(20)
M2
slope
+2.5
M1
slope
+0.5
Height y
y=3
y=1
Ceiling at y=7
Bang-4(12)
??
Vy1
Vy2
??
(a)
Time t Bang-1(01)
Bang-2(12)
Bang-3(20)
Bang-4(12)
Bang-5(20)
Bang-6(12)
Bang-7(01)
Bang-8(20)
Bang-9(12)
Time t
Floor at y=0
Ceiling at y=7
y
Start at
(-1.0,-1.0)
1.0
1.0
-1.0
0.5
2.0
Bang-2(12)
Bang-3(20)
Bang-4(12)
Bang-1(01)
Bang-5(20)
Bang-6(12)
Bang-7(01)
Vy1
Vy2
(b)
(c)
(d)
M2
slope
-2.5
Fig. 4.7a-din Unit 1
Kinetic Energy Ellipse
KE = 12
M1V12 + 1
2M 2V2
2 = 72
+12= 4
1= V12
2KE /M1
+ V22
2KE /M 2
= x12
a12 +
x22
a22
Ellipse radius 1
a1 = 2KE /M1
= 2KE /7
= 8/7 = 1.07
Ellipse radius 2
a2 = 2KE /M1
= 2KE /1
= 8/1 = 2.83
22
Difficult to see high mass-ratio-skinny-ellipse improved byScale transformation M1v1 -> √M1 v1
M1 =49m2 =1
23
Ellipse rescaling geometry and reflection symmetry analysis Convert to rescaled velocity: symmetrize: V1 = v1 ⋅ m1 , V2 = v2 ⋅ m1 , KE =2
1m1v12+2
1m2v22 =2
1 V12 +21 V22Then collisions become reflections and double-collisions become rotations
where:
cosθ sinθsinθ −cosθ
⎛
⎝⎜⎞
⎠⎟
cosθ ≡ m1 − m2
m1 + m2
⎛⎝⎜
⎞⎠⎟
and: sinθ ≡2 m1m2
m1 + m2
⎛
⎝⎜
⎞
⎠⎟ with: m1 − m2
m1 + m2
⎛⎝⎜
⎞⎠⎟
2
+2 m1m2
m1 + m2
⎛
⎝⎜
⎞
⎠⎟
2
= 1
θ=16.26°1
m1 - m2m1+m2 2√m1 m2
m1+m2
4850=
1450=
slope :
−m2
m1
= −17
slope :
+m2
m1
= +17
11
33
77
1177
1199
11331155
55
991111
2211
θ=16.26°V1
V2
[11]-tangent slope-√m1/√m2= -7
22
44
11
33
77
1177
1199
11331155
55
991111
2211
v1
v2
slope1/1
slope-1/1
[11]-tangentslope
-m1/m2= -49
22
44
24
(a) Lagrangian L = L(v1,v2)
v1
v2(b) Estrangian E = E(V1,V2)
V1=√m1v1
(c) Hamiltonian H = H(p1,p2)
p1=m1v1
p2=m2v2
V2=√m2v2
COM Bisectorslope = 1/1
Collision line andCOM tangent slope= -m1/m2 =-16
Collision line andCOM tangent slope=-√m1/√m2=-4
COM Bisector slope= √m2/√m1 =1/4
Collision line andCOM tangent slope
= -1/1
COM Bisector slope= m2/m1 =1/16
slope√m1√m2
=4
slope=1
25
p2=m2v2
p1=m1v1
Hamiltonian plotH(p)=const.=p•M-1•p/2(b)Lagrangian plot
L(v)=const.=v•M•v/2
v2=p2 /m2
L=const = E
v1=p1 /m1
(a)
v v = ∇∇pH=M-1•p
p = ∇∇vL=M•v
p
Lagrangian tangent at velocity vis normal to momentum p
Hamiltonian tangent at momentum pis normal to velocity v
(c) Overlapping plotsv
p
v
p
p
v (d) Less mass
(e) More mass
H=const = E
L=const = E
H=const = E
26
-1
-1
+2
0 +1
+1
p-slope=v
dH/dp=+1.0
-1 0 +1
-1
+1
+2
v-slope=p
dL/dv =+1.6
1.6
1
1.0
1
Lagrangian plot
L(v)= v•p - H(p)(a) Hamiltonian plot
H(p)=p•v-L(v)(b)
v-slope
intercept
-H = -0.6 p-slope
intercept
-L = -1.0
Lagrangian
L(v) = 1.0 Hamiltonian
H(p) = 0.6
momentum
pvelocity
v
velocity
v = 1.0
momentum
p = 1.6
L(v) H(p)
v-slope is momentum p
velocity v is p-slope
Lagrangian L(v) is -p-slope intercept
-v-slope intercept is Hamiltonian H(p)
Unit 1Fig. 12.3
27
“Monster Mash”classical segue to Heisenberg action relationsExample of very very large M1 ball-walls crushing a poor little m2
How m2 keeps its action An interesting wave analogy: The “Tiny-Big-Bang” [Harter, J. Mol. Spec. 210, 166-182 (2001)],[Harter, Li IMSS (2012)]
A lesson in geometry of fractions: Ford Circles and Farey Sums [Lester. R. Ford, Am. Math. Monthly 45,586(1938)] [John Farey, Phil. Mag.(1816)]
28
2
-2
4
-46
-6
Δx=1.5
Δx=0.75Δx=0.5
Δx=3.0
Δx=0.375
Δx=1.5
Δx=0.75Δx=0.5
Δx=3.0
Δx=0.375
11//11
11//22
11//33
11//4411//5511//6611//77
v2=0
v2=2
v1=1
Time
Space
The Wall
Time
Space
(a) Big ball moves in and traps small ball between it and The Wall
(b) Trajectory geometry exposed
Y Y
Vy2
Vy2
(a) Big space
Low speed
Bang (1)12
Bang (2)20
(b) Decreasing space
Increasing speed
Y
Vy2(c) Small space
High speed
Bang (n)12
Bang (n+1)20
VyY = const.
Y
Vy2
Y
Vy2
Y
Vy2
Unit 1Fig. 6.4
The Classical“Monster Mash”
Classical introduction to
Heisenberg “Uncertainty” Relations
v2 = const.Y
or: Y ⋅v2 = const.
is analogous to: Δx ⋅ Δp = N ⋅
29
-3
-57
-7 8-8
Δx=2.0
Δx=1.5Δx=1.2
Δx=3.0
Δx=1.0
v2=3
Time
v2=-1
5
2
-2
-44
6-6
11//11
11//22
11//33
11//4411//5511//6611//77
9-9
v2=+1
v2=0 v2=0
V1=+1 V1=-1
v2=-0.2
v2=+2.2
Double “Monster Mash”
30
-3
3
The Wall-1
5
3
-1
5
-3
0
4
-26
2
-2
-4
4
6-6
The Wall(a) Galilean shift by V=1 (b)
−vINvFIN=vIN+2V
1
+vIN
V1=1V1=1
V1=1V1=1
V1=1V1=1
V1=1V1=1
V1
V1=1V1=1
V1=1V1=1
B− B0 B+
A−A0 A+
C
O
V1=1V1=1
V1=1V1=1
(a) (b)
(c)
V1V1
vFIN2
+4
+3
+2
+1
0
-1
-2
-3
+5B− B0 B+
Unit 1Fig. 6.6
andFig. 6.7
31
Δm = 9
2Δx = 4 %
-15 -10 -5 0 5 10 15 = m
1/1
0/11/1
2Δx = 4 % 0/1Time t (units of fundamental period ) τ1
Coordinate φ (units of 2π )
0/1
1/10
1/5
3/10
2/5
1/2
3/5
7/10
4/5
9/10
1/1
3/4
1/4
1/21/40-1/4-1/2
1/21/4
0-1/4
-1/2
(Imagine "wrap-around" φ-coordinate) +π /2
−π /2
+π−π
3/4
1/4
0/1
1/2
1/1
Time t
[Harter, J. Mol. Spec. 210, 166-182 (2001)]
32
Δm = 9
2Δx = 4 %
-15 -10 -5 0 5 10 15 = m
1/1
0/1
1/2
1/4
1/61/7
1/3
1/5
2/5
2/7
3/7
1/8
Wave packet starts hereZeros start here Zeros start here
Time t(units of τ1)
Coordinate φ(units of 2π)
1/1
0/10 1/4 1/2-1/2 -1/4
N-level-system and revival-beat wave dynamics (9 or10-levels (0, ±1, ±2, ±3, ±4,..., ±9, ±10, ±11...) excited) Zeros (clearly) and “particle-packets” (faintly) have paths
labeled by fraction sequences like: 07, 17, 27, 37, 47, 57, 67,11
17
27
37
47
57
67
47
57
67
11
01
[Harter, J. Mol. Spec. 210, 166-182 (2001)]
33
Farey Sum algebra of revival-beat wave dynamicsLabel by numerators N and denominators D of rational fractions N/D
Time t(units of τ1)
0 1/4 1/2
1/d1
1/d2
Coordinate φ(units of 2π)
1/1
0/1-1/2 -1/4
1/d2
2/d2
3/d2
n2/d2
14/d113/d112/d1
n1/d1(n2-1)/d2
(n1+1)/d1
•••
•••
1/d2
2/d2
n2/d2 path slope is 1/d2
n1/d1 path slope is -1/d1
•••
•••
n1/d1 and n2/d2 pathfractions
numerator/denominator
34
n1+n2d1+d2
tx=
Time t(units of τ1)
0 1/4 1/2
1/d1
1/d2
n2/d2- t
1/2 -φ= 1/d2
Coordinate φ(units of 2π)
1/1
0/1-1/2 -1/4
n1/d1 - t1/2 -φ = -1/d1
(φx ,tx)
1/d2
2/d2
3/d2
n2/d2
14/d113/d112/d1
n1/d1(n2-1)/d2
(n1+1)/d1
•••
•••
1/d2
2/d2
x
n1/d1 and n2/d2 pathintersection time
(Farey-Sum)
•••
•••
d1n2-n1d2d1+d2
φx=
n1/d1 and n2/d2 pathintersection point
(Ford-Cross)
n2/d2 path slope is 1/d2
n1/d1 path slope is -1/d1
Farey Sum algebra of revival-beat wave dynamicsLabel by numerators N and denominators D of rational fractions N/D
[Lester. R. Ford, Am. Math. Monthly 45,586(1938)] [John Farey, Phil. Mag.(1816)] 35
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
20 3 4 51 7 8 9 106 12 13 14 1511 17 18 19 2016-1-2-3
11
v1=(1,1)v0=(0,1)
(a) 01
Numerator Axis N
DenominatorAxisD
Unit Real Interval Farey Sum related to
vector sumand
Ford Circles1/1-circle has
diameter 1
36
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
20 3 4 51 7 8 9 106 12 13 14 1511 17 18 19 2016-1-2-3
12
11
v1=(1,1)v0=(0,1)
(b) 01
v1=(1,2)
Numerator Axis N
DenominatorAxisD
Unit Real Interval
12
11
01
Farey-Sum Tree
Farey Sum related to
vector sumand
Ford Circles
1/2-circle hasdiameter 1/22=1/4
1/1-circle hasdiameter 1
37
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
20 3 4 51 7 8 9 106 12 13 14 1511 17 18 19 2016-1-2-3
12
13
11
v1=(1,1)v0=(0,1)
(c) 01
v1/2=(1,2)v1/3=(1,3)
Numerator Axis N
DenominatorAxisD
Unit Real Interval
12
11
01
13
23
Farey-Sum Tree
Farey Sum related to
vector sumand
Ford Circles
1/2-circle hasdiameter 1/22=1/4
1/3-circles havediameter 1/32=1/9
38
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
20 3 4 51 7 8 9 106 12 13 14 1511 17 18 19 2016-1-2-3
122
513
14
15
Numerator Axis N
DenominatorAxisD
Unit Real Interval 11
v1=(1,1)v0=(0,1)
(d) 01
v1/2=(1,2)v1/3=(1,3)
v2/5=(2,5)
16
17
27
37
18
38
12
11
01
13
23
14
34
15
25
35
45
16
56
17
27
37
47
57
67
18
38
58
78
Farey-Sum Tree
Farey Sum related to
vector sumand
Ford Circles
1/2-circle hasdiameter 1/22=1/4
1/3-circles havediameter 1/32=1/9
n/d-circles havediameter 1/d2
39
U of A Physics Day 2000(Quantum computer simulation)
That makes an ∞-ly deep “3D-Magic-Eye” picture
40
•Einstein-Lorentz-Minkowskii relativity (Discover relativistic quantum mechanics)
41
42
43
43
44
45
Wavevector ck
Frequency ω
1 300THz
2 600THz
2 900THz
4 1200THz
1 300THz-1 300THz 2-2 1-1 3 4
3
46
Wavevector ck
Frequency ω
1 300THz
2 600THz
2 900THz
4 1200THz
1 300THz-1 300THz 2-2 1-1 3
3
4
Phase
Group
47
Wavevector ck
Frequency ω
1 300THz
2 600THz
2 900THz
4 1200THz
1 300THz-1 300THz 2-2 1-1 3 4
3
Phase
Group
Doppler blue-shiftUP by factor e+ρ=2
Doppler red-shiftDN by factor e-ρ=1/2
48
Wavevector ck
Frequency ω
1 300THz
2 600THz
2 900THz
4 1200THz
1 300THz-1 300THz 2-2 1-1 3 4
3
Phase
Group
Doppler blue-shiftUP by factor e+ρ=2
Doppler red-shiftDN by factor e-ρ=1/2
49
Wavevector ck
Frequency ω
1 300THz
2 600THz
2 900THz
4 1200THz
1 300THz-1 300THz 2-2 1-1 3 4
3
Phase
Group
Doppler blue-shiftUP by factor e+ρ=2
Doppler red-shiftDN by factor e-ρ=1/2
50
Wavevector ck
Frequency ω
1 300THz
2 600THz
2 900THz
4 1200THz
1 300THz-1 300THz 2-2 1-1 3 4
3
Phase
Group
Doppler blue-shiftUP by factor e+ρ=2
Doppler red-shiftDN by factor e-ρ=1/2
Group
Phase
Group vector shiftedfrom ϖ(1,0) to ϖ(cosh ρ ,sinh ρ)
Phase vector shiftedfrom ϖ(0,1) to ϖ(sinh ρ,cosh ρ)ϖ=2
51
52
Wavevector ck
Frequency ω
1 300THz
2 600THz
2 900THz
4 1200THz
1 300THz-1 300THz 2-2 1-1 3 4
3
Phase
Group
Doppler blue-shiftUP by factor e+ρ=2
Doppler red-shiftDN by factor e-ρ=1/2
Group
Phase
Group vector shiftedfrom ϖ(1,0) to ϖ(cosh ρ ,sinh ρ)
Phase vector shiftedfrom ϖ(0,1) to ϖ(sinh ρ,cosh ρ)ϖ=2
53
Wavevector ck
Frequency ω
1 300THz
2 600THz
2 900THz
4 1200THz
1 300THz-1 300THz 2-2 1-1 3 4
3
Phase
Group
Doppler blue-shiftUP by factor e+ρ=2
Doppler red-shiftDN by factor e-ρ=1/2
Group
Phase
Group vector shiftedfrom ϖ(1,0) to ϖ(cosh ρ ,sinh ρ)
Phase vector shiftedfrom ϖ(0,1) to ϖ(sinh ρ,cosh ρ)ϖ=2
54
Wavevector ck
Frequency ω
1 300THz
2 600THz
2 900THz
4 1200THz
1 300THz-1 300THz 2-2 1-1 3 4
3
Phase
Group
Doppler blue-shiftUP by factor e+ρ=2
Doppler red-shiftDN by factor e-ρ=1/2
Group
Phase
Group vector shiftedfrom ϖ(1,0) to ϖ(cosh ρ ,sinh ρ)
Phase vector shiftedfrom ϖ(0,1) to ϖ(sinh ρ,cosh ρ)ϖ=2
55
56
57
58
The Circular Functions “Urban elite” The Hyperbolic Functions “Country-cousins”They’re related by Legendre contact transformation L = p•v-H
tan σ = sinh ρ
Circular Geometry of Lagrangian Functions versus Hyperbolic Geometry of Hamiltonian Functions
59
The Circular Functions “Urban elite” The Hyperbolic Functions “Country-cousins”They’re related by Legendre contact transformation L = p•v-H
tan σ = sinh ρ sin σ = tanh ρ
Circular Geometry of Lagrangian Functions versus Hyperbolic Geometry of Hamiltonian Functions
60
The Circular Functions “Urban elite” The Hyperbolic Functions “Country-cousins”They’re related by Legendre contact transformation L = p•v-H
tan σ = sinh ρ sin σ = tanh ρcos σ = sech ρ
Circular Geometry of Lagrangian Functions versus Hyperbolic Geometry of Hamiltonian Functions
61
The Circular Functions “Urban elite” The Hyperbolic Functions “Country-cousins”They’re related by Legendre contact transformation L = p•v-H
tan σ = sinh ρ sin σ = tanh ρcos σ = sech ρ sec σ = cosh ρ
Circular Geometry of Lagrangian Functions versus Hyperbolic Geometry of Hamiltonian Functions
62
The Circular Functions “Urban elite” The Hyperbolic Functions “Country-cousins”
Circular Geometry of Lagrangian Functions versus Hyperbolic Geometry of Hamiltonian Functions
In spacetime: asimultaneity factor In per-spacetime: momentum velocity u/c group velocity Lorentz contraction -Lagrangian Einstein time dilation Hamiltonian
They’re related by Legendre contact transformation L = p•v-Htan σ = sinh ρ sin σ = tanh ρcos σ = sech ρ sec σ = cosh ρcot σ = csch ρcsc σ = coth ρ
sin σ = tanh ρ = u/csec σ = cosh ρ =1/√1-u2/c2
Old-fashionednotation:
63
64
65
66
Rest energy
Mc2=1
u/c=4/5
Frequency ωorTotal energy
Hamiltonian
H=ω=5/3=coshρ=secσ
Momentum cp or Wavevector ck
cp=ck=4/3=sinhρ=tanσ
-Lagrangian
-L=3/5=sechρ=cosσ
Momentum or Wavevector ck
cp=ck=4/3=sinhρ=tanσPhase velocity
c/u=5/4=cothρ=cscσ
DeBroglie
wavelength λ=3/4=cschρ=cotσ
u/c slope angle υ=atan(u/c)
stellar aberation angle σ=asin(u/c)
DeBroglie
wavelength
λ=3/4=cschρ=cotσ
Phase velocity
c/u=5/4=cothρ=cscσ
Doppler
blue-shift
b=3=e+ρ
Doppler
red-shift
r=1/3=e−ρ
Group velocity
u/c=4/5
=tanhρ=sinσ
Kinetic energy
KE=2/3=coshρ -1=secσ -1
ω
ck
ω´
ck´
67
-2-3 1 2 3 4 5 6 7 8 9
1
2
3
5
6
7
8
9
-1
Phasevector
Groupvector
BHamiltonian
H=Bcosh ρAreaB·ρ/2
per-Space-Time Geometry(frequency vs wavevector)
68
VVW100110120
-120
(60,10)Initial-point
Final“Ka-Bong”-point
a=√=60.21
2·KEMSUV
b=√=120.42
2·KEmVW
ElasticKineticEnergyellipse(KE=7,250)
MomentumPTotal=250line
(50,50)
(40,90)
MSUV=4MSUV=4
mVW=1mVW=1
VSUV
Fig. 3.1 ain Unit 1
69
VVW
100
110
120
-120
(60,10)
Initial-point
Final“Ka-Bong”-point
a=√=60.21
2·KE
MSUV
b=√=120.42
2·KEmVW
Elastic
Kinetic
Energy
ellipse
(KE=7,250)
Momentum
PTotal=250
line
(50,50)
(40,90)
MSUV=4M
SUV=4
mVW=1m
VW=1
VSUV
Fig. 3.1 ain Unit 1
a=√=55.9
2·IE
MSUV
b=√=111.8
2·IEmVW
COM
Energy
ellipse
(ECOM=1,000)
elastic
FIN
IN
inlastic
FIN is
VCOM
point
Inelastic
Kinetic
Energy
ellipse
(IE=6,250)
Fig. 3.1 bin Unit 1
70