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(cid:132) Shift-register
IN
DQ
Q0
Q1
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OUT
CLK
IN
Q0
Q1
CLK
all measurements are made
from the clocking event that is,
the rising edge of the clock
100
L4: 6.111 Spring 2006
Introductory Digital Systems Laboratory
25 | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/60ded5de196ed8fa1ff1b65a29d27976_lec4.pdf |
System Architecture
Creativity
Thomas H. Speller, Jr.
Thomas H. Speller, Jr.
Engineering Systems Division
Engineering Systems Division
Doctoral Candidate
Doctoral Candidate
January 12, 2007
January 12, 2007
ESD.34
Professor Ed Crawley
1
Today’s Topics on Creativity
• Introduction
• Creativity
– Nature
– Design Rules a... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
of project vehicles removed due to copyright restrictions.
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
6
System Architecture Solution Space
within the Creative Space
Definition
SA Sol’n Space:= SA’s satisfying the given
specification
DCI Variant
DCI Variant... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
i
c
s
f
o
r
e
b
m
u
N
1000
900
800
700
600
500
400
300
200
100
0
1900
1920
1950
1980
2000
Worldwide Publication of Scientific Articles
Figure by MIT OCW.
Stephen Hawking, The
Universe in a Nutshell,
2001, pp. 156-168.
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technol... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
– Limited time and budget
• Dominance of paradigms, subjective personalities,
political positions and financial influencers (The Structure
of Scientific Revolutions, Thomas Kuhn, 1970)
– Individuals
– Teams
– Enterprises
Insufficient interaction of concept design and selection
with stakeholders to elicit their true ... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
u
F
F
Concept
Concept
Aggregation
(combining of F-F)
Evaluation with respect
to fitness function and
“ilities”
Conflict resolution may follow each of the above
steps. Iteration accompanies all these steps
What is your creative process at each step?
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), ... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
Value Engineering and the Semantic
Web tool
• Radiant Thinking, Mind Mapping tool
• Appendix: Technological change: from its
creation to economic growth and societal
welfare
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
20
Nature’s Processes as Creativity
... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
7, Engineering Systems Division (ESD), Massachusetts Institute of Technology
24
Evolutionary Computing:= {GA, EP, ES, GP}
Evolutionary Computing approach
1. EC consists of at least 4 sub-approaches
• Genetic Algorithms (GA)
• Evolutionary programming (EP)
• Evolutionary strategies (ES)
• Genetic programming (GP)
2. A ... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
27
Example of Evolutionary Computing:
Genetic Programming
Courtesy of John Koza. Used with permission.
Genetic and Evolutionary Conference (2005).
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Inst... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
The basic assembly module ,
developed from an initial
primitive,
wt.
wt.
wt.
90
90
moment
direction
• T module derived from the
physics that the least energy
structure requires only 50%
brick support to maintain
structural equilibrium.
• The T acting as a column
• How many ways can you
sequentially assemble
t... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
0
14000
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Bins
But, of these 19,880 solutions only 20 are uniqu... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
6103
3580
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18000
16000
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10000
8000
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© Thomas H... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
-dimensional cellular
automaton
– Transcribe the shapes into symbols, then
– Compute generatively the system architectures by computing
with the symbolically represented shapes
• Translate back to shape and provide graphical visual
output
1Described in T. Speller, D. Whitney, E. Crawley, “Use of shape grammar to der... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
$3,000,000
$2,000,000
$1,000,000
$0
($1,000,000)
Baseline Yrly Net-Cash
Option Yrly Net-Cash
Option NPV
Baseline NPV
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
46
Arising out of observations from my
doctoral study is a definition of creativity
• Creativity... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
individual
• formal or informal
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
Adapted from mini lecture
by Prof. E. Crawley
50
THOUGHTS
• Creativity is the making of the new, or the
remaking of the old, in a new way1
• Inspired people create1 Motivated peopl... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
Personal / Group Stimulants
5 senses1, challenges2, provocations2, random inputs2, “motion”2
• Models
Edison, F.L. Wright, Fuller, Welch ...
any Nobel Prize winner
1 Vance 2 de Bono
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
getting out of a pattern1
54
SIX... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
pattern1
• Don’t look back - keep trying to move ahead
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
57
MOVEMENT STIMULATED BY RANDOM INPUT &
PROVOCATION
• Random input - pick a word at random -- let the brain connect to idea at
hand
• PO -- Provocative Oper... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
In what circumstances is there direct value?
PO -- drinking glasses have round bottoms
(cid:198) bars sell more drinks
•
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
59
Scenarios as a creative practice
Scenarios are another creative practice increasingly bein... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
consider testing it for robustness
in this outside usage or ‘protecting’ the product from being misused. Once these
scenarios are developed and discussed, their impact must be discounted to the present in
your SA design as much as you judge is appropriate. In this viewpoint one might consider
the selected present S... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
Solving
Many thanks to Dan Frey, Don
Clausing and Victor Fey for
materials and advice
See http://www.triz-journal.com/
62
Axiomatic Design, Prof. Nam Suh, MIT
• Independence of Functions
• Minimum information
• A recommended course and reading:
http://www.amazon.com/gp/product/0195134664/ref=nosim/002-5285815-518323... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
Ideal Technological System
System
System
Function
Object
System
Function
Object
An ideal system does not exist as a physical entity,
but its function is fully performed
http://www.trizgroup.com/
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
67
Ways to Achieve... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
amber
Acid
Specimen
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
72
Ideality: Practice
• A delivery company is shipping meat by
aircraft, but the refrigeration system is heavy
and is reducing the useful payload.
• A Swiss company manufactures confections.
... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
78
Solution
• Make the front half of the armor plate hard, and
the back half tough
– US Patent 3,475,812
– Huge improvement in ballistics
• Basic invention principle: separate the
requirements in space
• Current project in MIT’s soldier n... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
7, Engineering Systems Division (ESD), Massachusetts Institute of Technology
83
Law/Line of Evolution
Domination of Higher-Level Systems
• Evolutionary trends of a system determine
directions of evolution of its components.
• Also, a problem at the system’s level may
be a symptom of a problem at the higher
hierarch... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
from its
creation to economic growth and societal
welfare
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
88
The Invention Machine
The Invention Machine
Computational adaptation of TRIZ,
Computational adaptation of TRIZ,
Value Engineering and
Value Engineer... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
's
Hierarchy of
Needs
Unlimited
Wants
Creative
Environment
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
© Speller 2007
94
Human Talent Applied
Maslow's
Hierarchy of
Needs
Unlimited
Wants
Creative
Environment
Human
Talent
© Thomas H. Speller, Jr. 2007, Enginee... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
--
Technological
Change
Technology
98
Social Welfare
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
© Speller 2007
99
Combining System Dynamics
notation with Object Process
Methodology
Stock ª Object
Flow ª Process
Maslow's
Hierarchy of
Needs
Unlimited
Wants... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
ESD.33 -- Systems Engineering
Session #9
Critical Parameter Management &
Error Budgeting
Dan Frey
Plan for the Session
• Follow up on session #8
• Critical Parameter Management
• Probability Preliminaries
• Error Budgeting
– Tolerance
– Process Capability
– Building and using error budgets
• Next steps
S - Curves
Ati... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
2
RB211-524D4
JT9D-7R4D (B)
JT8D-17A
JT8D-219
TAY
CFM56-3
RB211-535C
RB211-524D4 UP
JT9D-7R4H1 CFM56-5B
CF6-80A/A1 CFM56-5A1
RB211-535E4
2037
CF6-80C2
V2500
4460
RB211-524D4D
4056
4156
CFM56-5C2
4380
4168
4083
P&W
deHavilland
RR
GE
5
4
9
1
0
5
9
1
5
5
9
1
0
6
9
1
5
6
9
1
0
7
9
1
5
7
9
1
0
8
9
1
5
8
9
1
0
9
9
1
5
9
9
1
... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
Session
• Follow up on session #8
• Critical Parameter Management
• Probability Preliminaries
• Error Budgeting
– Tolerance
– Process Capability
– Building and using error budgets
• Next steps
Probability Definitions
• Sample space – a list of all possible
outcomes of an experiment
– Finest grained
– Mutually exclusi... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
Arithmetic average
• Median
• Mode
n
1
∑
n 1
i
=
ix
fx(x)
x
Measures of Dispersion
• Variance
VAR
x
)(
2
=σ
=
xExE
−
((
(
2
))
)
• Standard deviation
=σ
2xExE
))
−
((
(
)
• Sample variance
2
S
=
1
−
1
n
n
∑
i
1
−
(
x
i
2
−
x
)
• nth central moment
• Covariance
nxExE
))
−
((
(
)
xExE
−
((
(
))(
yEy
−
(
)))
Sums of Ran... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
y, and z?
{
aP
x
≤<
}
b
b
∫=
a
f
x
d)(
x
x
∞
∫
∞−
f x
x
d)(
x
=
1
0 ≤
f x
)(
x
for
all
x
Simulation Can Quickly Answer the
Question
trials=10000;nbins=trials/1000;
x= random('Uniform',0,1,trials,1);
y= random('Uniform',0,2,trials,1);
z=x+y;
subplot(3,1,1); hist(x,nbins); xlim([0 3]);
subplot(3,1,2); hist(y,nbins)... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
8.3%
99.7%
1-2ppb
Joint Normal Distribution
p
x
( )
=
1
m
)
2
π
(
exp
−
⎧
⎨
⎩
1
2
K
(
x
−
µ
)
T
1
−
K x
(
−
µ
)
⎫
⎬
⎭
• The lines of constant probability density are
ellipsoids
• If the matrix K is diagonal, then the variables are
uncorrelated and independent
correlated
uncorrelated
Independence
• Random variables ... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
tolerance zone
THIS ON A DRAWING
MEANS THIS
GD&T Symbols
Geometric Characteristic Symbols
Type of
Tolerance
Characteristic
Symbol
For Individual
Features
Form
For Individual
or Related
Features
Profile
Straightness
Flatness
Circularity (Roundness)
Cylindricity
Profile of a Line
Profile of a Surface
Angularity
Ori... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
3 2
2
C
p
(
1
+
k
)
⎞
⎟
⎠
⎤
⎥
⎥
⎦
This function to maps Cp and k to yield
39
Cp and k Determine Quality Loss
L(q)
Ao
L
U L+
2
U
q
Taguchi's quality loss function
ANSI's implied quality loss function
Quality
Loss
=
Ao
[
LU
(
−
2/)
⎛
⎜
⎝
2
]
d
−
2
LU
+
2
⎞
⎟
⎠
E(Quality
Loss)
=
⎛
⎜
kA
o
⎜
⎝
2
+
1
C
9
p
2
⎞
⎟
⎟
⎠
Cran... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
o
r
r
e
e
d
i
s
o
t
e
d
i
S
r
o
r
r
e
e
d
i
s
o
t
e
d
i
S
20
40
60
80
100
120
Lead number
Side #2
r
o
r
r
e
e
d
i
s
o
t
e
d
i
S
Side #1
20
40
Lead number
20
40
60
80
100
120
Lead number
6
Plan for the Session
• Follow up on session #8
• Critical Parameter Management
• Probability Preliminaries
... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
0 rad
σ
0.0001 rad
0.0001 rad
Drive error of joint #3
Z ·0.0001
0.01mm
Pitch of joint #3
Yaw of joint #3
Parallelism of joint 2 in the x direction
0 rad
0 rad
0.0002
rad
0.00005
rad
0.00005
rad
0.0001 rad
εz1
εz2
δz3
εx3
εy3
xp2
A Model of a Robot
• The matrices describe the intended
motions and the errors
NOTE: T... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
00)
01
0
0
0
⋅
⎡
⎢
⎢
⎢
⎢
⎣
mm
mm
mm
1
⎤
⎥
⎥
⎥
⎥
⎦
⋅
⎡
⎢
⎢
⎢
⎢
⎣
001
400
mm
010
100
000
0
0
mm
mm
1
⎤
⎥
⎥
⎥
⎥
⎦
⋅
⎡
⎢
⎢
⎢
⎢
⎣
1
0
0
1
−
y
εε
x
3
3
0
0
−
ε
y
3
ε
x
1
3
0
0
0
Z
−
mm
mm
δ
z
3
−
1
⎤
⎥
⎥
⎥
⎥
⎦
• Can be applied to any point on the end
effector
x
y
z
'
'
'
p
p
p
1
⎧
⎪
⎪
⎨
⎪
⎪
⎩
⎫
⎪
⎪
⎬
⎪
⎪
⎭
=
0 T
3
⎧
⎪
⎪
⎨
⎪... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
7.3.6 Homogeneous and Inhomogeneous Broadening
Laser media are also distinguished by the line broadening mechanisms in-
volved. Very often it is the case that the linewidth observed in the absorption
or emission spectrum is not only due to dephasing process that are acting on
310
CHAPTER 7. LASERS
Laser Medium
Wave-
l... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/6135a85d75be96fb5a556b3f879b0d8c_homo_nonhomo_br.pdf |
16
20
20
20
14
18
12
Upper-St.
Lifetime
τ L (μs)
1,200
87
450
50
350
10,000
1,000
3
67
170
88
0.7
0.07
2,900,000
0.0033
0.002
∼
Linewidth
∆fF W HM =
2
(THz)
T2
0.210
1.2
0.390
0.300
3
4
0.06
100
80
65
80
0.0015
0.0035
0.000060
5
25
Refr.
index
n
1.82
1.47 (ne)
1.82 (ne)
2.19 (ne)
1.5
1.46
1.76
1.76
1.4
1.4
1.4
1
∼
1
∼
... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/6135a85d75be96fb5a556b3f879b0d8c_homo_nonhomo_br.pdf |
an atomic ensemble there
are groups of atoms with a different center frequency of the atomic tran-
sistion. The overall ensemble therefore may eventually show a very broad
linewidth but it is not related to actual dephasing mechanism that acts upon
each atom in the ensemble. This is partially the case in Nd:silicate gla... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/6135a85d75be96fb5a556b3f879b0d8c_homo_nonhomo_br.pdf |
. The velocity distribution of an
ideal gas with atoms or molecules of mass m in thermal equilibrium is given
by the Maxwell-Boltzman distribution
³
´
(7.5)
p(v) =
m
2πkT
exp
mv2
2kT
.
¶
−
µ
r
(7.6)
312
CHAPTER 7. LASERS
This means that p(v)dv is equal to the brobability that the atom or molecule
has a velocity in the... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/6135a85d75be96fb5a556b3f879b0d8c_homo_nonhomo_br.pdf |
4. LASER DYNAMICS (SINGLE MODE)
313
Figure 7.19: Possible cavity configurations. (a) Schematic of a linear cavity
laser. (b) Schematic of a ring laser.
The laser resonators can be modelled as Fabry Perots as discussed in
section . Typically the techniques are used to avoid lasing of transverse modes
and only the longitu... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/6135a85d75be96fb5a556b3f879b0d8c_homo_nonhomo_br.pdf |
the laser
is homogenously broadened the laser gain will satured to the loss level and
only the mode at the maximum of the gain will lase.
If the gain is not
homogenously broadend and in the absence of a filter many modes will lase.
For the following we assume a homogenously broadend laser medium and
only one cavity mode... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/6135a85d75be96fb5a556b3f879b0d8c_homo_nonhomo_br.pdf |
vgnL is the photon flux, σ is the stimulated emission cross section,
τ L = γ21 the upper state lifetime and Rp is the pumping rate into the upper
laser level. A similar rate equation can be derived for the photon density
d
dt
nL =
nL
τ p
−
+ 2∗
σvg
V
N2
nL +
µ
1
V
.
¶
(7.18)
Here, τ p is the photon lifetime in the cavit... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/6135a85d75be96fb5a556b3f879b0d8c_homo_nonhomo_br.pdf |
the round-trip amplitude gain g = 2∗
the circulating intracavity power P = I
Aef f
·
σvg
2V N2TR experienced by the light and
d
dt
d
dt
g =
P =
gP
Esat
g
g0
−
τ L −
2g
TR
P +
−
−
1
τ p
(P + Pvac) ,
with
Esat =
hfL
2∗σ
1
2∗
Aef f =
IsAef f τ L
Psat = Es/τ L
Pvac = hfLvg/2∗L = hfL/TR
Rp
2Aef f
g0 = 2∗
στ L,
(7.21)
(7.22)... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/6135a85d75be96fb5a556b3f879b0d8c_homo_nonhomo_br.pdf |
2 (g0
−
ln
l)
Aef f TR
στ L
.
(7.29)
318
CHAPTER 7. LASERS
Figure 7.23: Built-up of laser power from spontaneous emission noise.
Some time after the built-up phase the laser reaches steady state, with
the saturated gain and steady state power resulting from Eqs.(7.21-7.22),
neglecting in the following the spontaneous ... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/6135a85d75be96fb5a556b3f879b0d8c_homo_nonhomo_br.pdf |
d∆P
dt
d∆g
dt
1 + Ps
P sat
1
τ stim
= 1
τ L
where
is the inverse stimulated lifetime. The stimulated
lifetime is the lifetime of the upper laser state in the presence of the optical
field. The perturbations decay or grow like
¢
¡
µ
which leads to the system of equations (using gs = l)
µ
¶
¶
∆P
∆g
=
∆P0
∆g0
est.
(7.36)
∆... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/6135a85d75be96fb5a556b3f879b0d8c_homo_nonhomo_br.pdf |
) for g0 < l and (Ps, gs) for g0 > l are
si
always stable, i.e. Re
{
< 0.
}
(ii): For lasers pumped above threshold, r > 1, and long upper state
lifetimes, i.e.
,
r
4τ L
< 1
τ p
the relaxation rate becomes complex, i.e. there are relaxation oscilla-
tions
1
s1/2 =
−
2τ stim ±
jωR.
(7.42)
with a frequency ωR approximate... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/6135a85d75be96fb5a556b3f879b0d8c_homo_nonhomo_br.pdf |
7.25 shows the typically observed fluctuations of the output of a solid-
Figure 7.25: Relaxation oscillations in the time and frequency domain.
state laser in the time and frequency domain. Note, that this laser has a long
upperstate lifetime of several 100 μs
One can also define a quality factor for the relaxation oscil... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/6135a85d75be96fb5a556b3f879b0d8c_homo_nonhomo_br.pdf |
the output coupling mirror. The internal losses can be a
significant fraction of the total losses. The output power of the laser is
Pout = T
Psat
·
µ
2g0
2lint + T −
1
¶
The pump power of a laser is minimized given
Pp = RphfP ,
(7.48)
(7.49)
where hfP is the energy of the pump photons. In discussing the efficiency of
a la... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/6135a85d75be96fb5a556b3f879b0d8c_homo_nonhomo_br.pdf |
2)
d
dt
d
dt
g =
P =
gP
Esat
g
g0
−
τ L −
2g
TR
P +
−
−
1
τ p
(P + Pvac) ,
(7.54)
(7.55)
where Pvac is the power of a single photon in the mode. The steady state
conditions are
gs =
g0
(1 + Ps/Psat)
,
0 = (2gs
−
2l) P + 2gsPvac.
(7.56)
(7.57)
324
CHAPTER 7. LASERS
Substitution of the saturated gain condition (7.56) in... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/6135a85d75be96fb5a556b3f879b0d8c_homo_nonhomo_br.pdf |
3
p v=10
0.5
1.0
1.5
2.0
Pump parameter r
1
-1
10
-2
10
-3
p v =10
10
-10
0.5
1.0
1.5
2.0
Pump parameter r
p
,
r
e
w
o
p
y
t
i
v
a
c
a
r
t
n
I
p
,
r
e
w
o
p
y
t
i
v
a
c
a
r
t
n
I
Figure 7.26: Intracavity power as a function of pump parameter r on a linear
scale (a) and a logarithmic scale (b) for various values of the ... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/6135a85d75be96fb5a556b3f879b0d8c_homo_nonhomo_br.pdf |
7.91 / 20.490 / 6.874 / HST.506
7.36 / 20.390 / 6.802
C. Burge Lecture #10
March 11, 2014
Markov & Hidden Markov Models of
Genomic & Protein Features
1
Modeling & Discovery of Sequence Motifs
• Motif Discovery with Gibbs Sampling Algorithm
• Information Content of a Motif
• Parameter Es... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/613e7bd5b9e83a805d32cde2ce585e76_MIT7_91JS14_Lecture10.pdf |
0.0
0.0
0.1
0.1
0.1
0.2
0.2
0.1
0.2
0.1
0.8 1.0
0.1 0.0
0.0
1.0
0.4
0.1
0.1
0.1
0.8
0.0
0.2
0.5
S = S1 S2 S3 S4 S5 S6 S7 S8 S9
Ex: TAGGTCAGT
P(S|+) = P-3(S1)P-2(S2)P-1(S3) ••• P5(S8)P6(S9)
‘Inhomogeneous’, assumes independence between positions
What if this is not true?
4
... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/613e7bd5b9e83a805d32cde2ce585e76_MIT7_91JS14_Lecture10.pdf |
Decoy 5’ss
Markov
True
5’ss
True
5’ss
I1M 5’ss Score
Markov models also improve modeling of transcriptional motifs - Zhou & Liu Bioinformatics 2004
© sources unknown. All rights reserved. This content is excluded from our Creative
Commons license. or more information, see http://ocw.mit.edu/help/faq-fair-use/.
... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/613e7bd5b9e83a805d32cde2ce585e76_MIT7_91JS14_Lecture10.pdf |
. Bayes est.
9
A
2
C
2
G
1
T
14
8 + 1
1 + 1
1 + 1
0 + 1
0.64
0.14
0.14
0.07
0.80
0.10
0.10
1.00
0.00
1.00
10
ML = maximum likelihood (of generating the observed data)
Bayes est. = Bayesian posterior relative to Dirichlet prior
Good treatment of this in appendix of:
Biological Sequence Ana... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/613e7bd5b9e83a805d32cde2ce585e76_MIT7_91JS14_Lecture10.pdf |
, e.g., Bart’s genotype is
conditionally independent of Grandpa Simpson’s
genotype given Homer’s genotype:
P(Bart = a/a | Grandpa = A/a & Homer = a/a)
= P(Bart = a/a | Homer = a/a)
Future
Bart
Images of The Simpsons © FOX. All rights reserved. This content is excluded from our Creative
Commons license. For more i... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/613e7bd5b9e83a805d32cde2ce585e76_MIT7_91JS14_Lecture10.pdf |
of
CpG dinucleotides (normally rare) which are unmethylated
• Associated with promoters of many human genes (~ 1/2)
16
CpG Island Hidden Markov Model
Hidden
Pig
Pgg
Pii
Genome
Pgi
Island
…
A
C
T
C
G
A
G
T
A
Observable
17
“Initiation
probabilities” πj
Rabiner notation... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/613e7bd5b9e83a805d32cde2ce585e76_MIT7_91JS14_Lecture10.pdf |
ation for HMM Calculations
Random vector of hidden states
Random vector of observable
data (DNA bases)
P( H = 1h , ...,
nh , O = 1o ,..., no )
Specific hidden state values
Specific sequence of bases
Another specific set of hidden state values
P( H = 1ʹ′ h , ...,
nʹ′ h
, O = 1o , ...,
no )
21
Reversing the H... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/613e7bd5b9e83a805d32cde2ce585e76_MIT7_91JS14_Lecture10.pdf |
i ending in state h
(h)
Solve recursively, i.e. determine R2
in terms of R1 , etc.
(h)
23
© source unknown. All rights reserved. This content is
excluded from our Creative Commons license For more
information, see http://ocw.mit.edu/help/faq-f... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/613e7bd5b9e83a805d32cde2ce585e76_MIT7_91JS14_Lecture10.pdf |
is the optimal parse of the sequence for the CpG
island HMM defined previously?
• (ACGT)10000
• A1000C80T1000C20A1000G60T1000
Powers of 1.5:
N = 20
40
60
80
(1.5)N = 3x103
1x107 3x1010 1x1014
28
Run time for k-state HMM on
sequence of length L?
O(k2L)
The computational efficiency of the Viter... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/613e7bd5b9e83a805d32cde2ce585e76_MIT7_91JS14_Lecture10.pdf |
BWT
T Feb 25 DG L6 Genome assembly
R Feb 27 DG L7 ChIP-Seq analysis (DNA-protein interactions)
T Mar 04 DG L8 RNA-seq analysis (expression, isoforms)
R Mar 06 CB L9 Modeling & Discovery of Sequence Motifs
T Mar 11 CB L10 Markov & Hidden Markov Models (+HMM content on 3/13)
Exam may have some overlap with topics f... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/613e7bd5b9e83a805d32cde2ce585e76_MIT7_91JS14_Lecture10.pdf |
18.354 Nonlinear Dynamics II: Continuum
Systems
Spring 2015 Lecture Notes
Instructor: J¨orn Dunkel
Authors of Lecture Notes: Michael Brenner, Tom Peacock, Roman Stocker,
Pedro Reis, J¨orn Dunkel
1
Contents
1 Math basics
1.1 Derivatives and differential equations . . . . . . . . . . . . . . . . . . . . . .
. .... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
. . . . . . . . . .
3 Dimensionless groups
3.1 The pendulum
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Taylor’s blast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 The drag on a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Kepler’s... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
diffusion
5.1 Derivation of the diffusion equation using particle fluxes . . . . . . . . . . .
5.2 Derivation using probabilities . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Suggestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Solving the diffusion equation
6.1 Fourier meth... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
. . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Two species in one space dimension
7.4.2 Lotka-Volterra model
6
6
7
7
8
9
9
9
10
10
11
12
12
13
14
15
15
15
16
18
18
19
20
20
22
23
24
25
25
28
30
31
31
32
33
36
37
37
2
8 Variational Calculus
8.1 ... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
. . . . . . . . . . . . . . . . . . . . . . . . .
10 Some basic differential geometry
. . . . . . . . . . . . . . . . . . . . . . . . .
10.1 Differential geometry of curves
10.2 Two-dimensional surfaces
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Minimal surfaces . . . . . . . . . . . . . . . . . .... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
. . . . . . . . . . . . . .
12.1.1 The continuity equation . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.2 Momentum equations
. . . . . . . . . . . . . . .
12.2 From Newton’s laws to hydrodynamic equations
13 The Navier-Stokes Equations
13.1 Viscosity ... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.5 Force dipoles
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.6 Boundary effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.6.1 Reminder: Cartesian vs. cylindrical corodinates . . . . . . . . . . . .
14... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
. . . . . . . . . . . .
16.3 An elementary differential equation . . . . . . . . . . . . . . . . . . . . . . .
17 Towards airplane flight
17.1 High-Reynolds number limit . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2 Kelvin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18 Eu... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
. . . . . . . . . . . . . . . . . . . . . .
19.3 Simple conformal maps
20 Classical aerofoil theory
20.1 An elliptical wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.2 Flow past an aerofoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.3 Blasius’ lemma . . . . . . . . ... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
. . . . . . . . . . . . . . . 100
. . . . . . . . . . . . . . . . . . . . 100
21.5 More on the Taylor-Proudman theorem
22 The Ekman layer
101
22.1 A small deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
History
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
24.2 Korteweg-de Vries (KdV) equation . . . . . . . . . . . . . . . . . . . . . . . 111
5
1 Math basics
1.1 Derivatives and differential equations
In this course, we will mostly deal with ordinary differential equations (ODEs) an... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
ned with respect to some global
orthonormal frame Σ, spanned by the basis vectors (e1, e2, e3), the gradient-operator ∇ is
defined by
∇ = ∂xe1 + ∂ye2 + ∂ze3 =
3
(cid:88)
i=1
∂iei ≡ ∂iei,
(3)
where we have introduced the Einstein summation convention on the rhs. Applying ∇ to a
scalar function f gives a vector
∇f = (∂1f,... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
2f (t, x) = 0.
(9)
Functions f satisfying this equation are called harmonic.
Later on, we will often try to approximate nonlinear PDEs through linear PDEs.
1.3 Complex numbers and functions
Although we will mostly deal with real fields in this course, it is sometimes helpful to rewrite
equations in terms of complex quan... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
)
7
By differentiating again, we find
∂2
xu = ∂x∂yv = −∂2
y v = ∂y∂xu = −∂2
∂2
y u = 0
xv = 0;
this means that analytic functions f = (u, v) are harmonic
∇2f = 0.
(15b)
(15c)
(15d)
(15e)
An analytic function that we will frequently encounter is the exponential function
Euler’s formula
exp(z) =
∞ zn
(cid:88)
k!
k=0
= 1 +... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
∞
dt eiωt
δ(t) =
1
√
2π
,
δ(t) =
1 (cid:90) ∞
2π −∞
dω e−iωt
(20a)
(20b)
(21)
(22)
These definitions and properties extend directly to higher dimensions.
A main advantage of Fourier transformations is that they translate differential equations
into simpler algebraic equations.
8
2 Dimensional analysis
Before moving on t... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
, however, we will rarely find useful exact solutions to the Navier-Stokes equations,
and so dimensional analysis will often give us insight before diving into the mathematics
or numerical simulations. Before formalising our approach, let us consider a few examples
where simple dimensional arguments intuitively lead to ... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
smaller triangles are a2f (φ) and b2f (φ)
(where elementary geometry shows that the acute angle φ is the same for the two little
triangles as the big triangle). Moreover, since these are all right triangles, the function f is
the same for each. Therefore, since the area of the big triangle is just the sum of the areas
... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
.4 The oscillation of a droplet
What happens if instead of considering a large body of fluid, such as a star, we consider a
smaller body of fluid, such as a raindrop. Well, in this case we argue that surface tension γ
10
provides the relevant restoring force and we can neglect gravity. γ has dimensions of en-
ergy/area,... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
1. We can construct a quantity with
the dimensions of T −1 through the relation
(cid:112)ω = c
gk.
(32)
We see that the frequency of water waves is proportional to the square root of the wavenum-
ber, in contrast to light waves for which the frequency is proportional to the wavenumber.
As with a droplet, we might worry... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
grouped into
n−d independent dimensionless ratios, or dimensionless groups Πi, expressible in functional
form by
Π1 = G(Π2, Π3, ..., Πn−d),
(36a)
or, equivalently,
0 = F (Π1, Π2, ..., Πn−d),
where d is the number of independent dimensions (mass, length, time...). The formal
proof can be found in the book Scaling, Self ... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
�1 is just the constant of proportionality c from above. Thus we have
(cid:112)c = τ
g/l
where c is a constant to be determined from an experiment.
3.2 Taylor’s blast
(39)
(40)
This is a famous example, of some historical and fluid mechanical importance. The story
goes something like this. In the early 1940’s there appe... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
=
Et2
ρR5
.
R = c
(cid:19) 1
5
(cid:18) E
ρ
2
5 .
t
13
(43)
(44)
The relation shows that if one measures the radius of the shock wave at various instants in
time, the slope of the line on a log-log plot should be 2/5. The intercept of the graph would
provide information about the energy E released in the explosion, if... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
the
exponents of mass length and time gives, α = −1, β = −2 and γ = −2. Thus
Π2 =
D
ρU 2R2
,
(48)
and this is called the dimensionless drag force. Buckingham’s Pi theorem tells us that we
must have the functional relationship
or alternatively
Π2 = G(Π1)
D
ρU 2R2
= G(Re).
(49)
(50)
The functional dependence is determine... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
mathematics we do, came from Newton’s desire to understand the motion
of the planets, which were known to obey Kepler’s laws.
4.1 Kepler’s laws of planetary motion
In the early seventeenth century (1609-1619) Kepler proposed three laws of planetary motion
(i) The orbits of the planets are ellipses, with the Sun’s c... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
r ∧ f = r ∧ f (r)rˆ = 0,
(54)
where f (r)rˆ is the central force, depending only on the distance r = |r| and pointing in the
direction rˆ = r/r. It can therefore be seen that the angular momentum of a particle moving
under a central force is constant, a consequence of this being that motion takes place in a
plane. The ... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
2 ˙θ|.
r2 ˙θ = l,
(56)
(57a)
(57b)
(58a)
(58b)
(59)
(60)
where l = L/m is the angular momentum per unit mass. Given a radial force f (r), equa-
tions (58a) and (58b) can now be solved to obtain r and θ as functions of t. A more practical
result is to solve for r(θ), however, and this requires the definition of a new var... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
replacing u by the original radial coordinate r = 1/u,
(cid:18)
r = Acosθ +
(cid:19)−1
k
ml2
(66)
which is the equation of a conic section with the origin at the focus. This can be rewritten
in standard form
r = r0
1 + (cid:15)
1 + (cid:15)cosθ
,
where
(cid:15) =
Aml2
k
,
r0 =
ml2
k(1 + (cid:15))
.
(cid:15) is called t... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
ity
Using this expression to substitute b in (71)
(cid:112)
=
1 − (cid:15)2.
b
a
The length of the major axis is
τ =
2πa2
l
(cid:112)
1 − (cid:15)2
2a = r0 + r1 =
2ml2
k(1 − (cid:15)2)
.
Squaring (71) and replacing gives
confirming Kepler’s 3rd law.
τ 2
=
4π2m
k
a3,
(71)
(72)
(73)
(74)
(75)
4.2 Hamiltonian dynamics of m... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
m
n
p˙ n = mnx¨ n = −∇x
nU.
(78)
An important observation is that many physical systems obey certain conservation laws.
For instance, the Hamiltonian (76a) itself remains conserved under the time-evolution (77)
d
dt
H =
=
(cid:88) (cid:2)
(∇pnH) · p˙ n + (∇x
nH) · x˙ n
(cid:3)
n
(cid:88)
n
(cid:2)
(∇pnH) · (−∇xnH) + (∇... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
there is no general analytical solution. Lagrange showed that there are some solutions
to this problem if we restrict the planets to move in the same plane, and assume that the
mass of one of them is so small as to be negligible. In the absence of an explicit solution
to the ‘three body problem’ one must use ideas from... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
The mathematics of how to solve ‘macroscopic equations’, which are nonlinear partial
differential equations, is non-trivial. We will need to introduce many new ideas.
In tackling these problems we will spend a lot of time doing fluid mechanics, the reason being
that it is by far the most developed field for the study o... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
(i) Each particle steps to the right or the left once every τ seconds, moving a distance
dxi = ±δ.
20
(ii) The probability of going to the right at each step is 1/2, and the probability of going
to the left is 1/2, independently of the previous history.
(iii) Each particle moves independently of all the other par... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
:105) + δ2.
i(cid:105) + δ
Repeating this procedure n times and recalling that xi(0) = 0, we find
(cid:104)xi(n)2(cid:105) = (cid:104)xi(0)2(cid:105) + nδ2 = nδ2
and, hence, for the mean square displacement of the cloud’s mean value
(cid:104)X(n)2
(cid:105) =
1
N 2
N
(cid:88)
i=1
nδ2 =
δ2
N
n.
(84)
(85)
(86)
If we write... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
left. The net number of particles crossing to the right
is therefore
− [N (x + δ, t) − N (x, t)] .
(88)
1
2
To obtain the net flux, we divide by the area normal to the x-axis, A, and by the time
interval τ ,
Multiplying by δ2/δ2 gives
which can be rewritten
Jx = −
[N (x + δ, t) − N (x, t)]
2Aτ
.
Jx = −
δ2
2τ
1
δ
(cid:20... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
τ ) − n(x, t)
τ
= −
[Jx(x + δ/2, t) − Jx(x − δ/2, t)]
δ
.
In the limit τ → 0 and δ → 0, this becomes
∂n
∂t
= −
∂Jx
∂x
= D
∂2n
∂x2
,
(93)
(94)
which is Fick’s law. This is commonly known as the diffusion equation. It tells us how
a cloud of particles will redistribute itself in time. If we know the initial distribution a... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
rewrite (96) in terms of the associated probability density
p(x, t + τ ) = [p(x + δ, t) + p(x − δ, t)] .
1
2
Performing a Taylor expansion about the position x and time t in the limit δ, τ → 0,
p(x, t) +
∂p
∂t
τ ≈
1 (cid:20)
2
p(x, t) +
∂p
∂x
δ +
∂2p
∂x2
δ2
2
+ p(x, t) −
∂p
∂x
δ +
∂2p
∂x2
δ2 (cid:21)
2
.
which simplifie... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
is
,
P
L4
P
L2 ∼ O
which is typically very small. This is an example of a scaling argument, which in this case
implies that we are justified in neglecting the extra terms provided L is much greater than δ.
You must be careful however, as this estimate is not correct everywhere. For example, in
the tails of the distribut... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf |
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