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STiCM
Select / Special Topics in Classical Mechanics
P. C. Deshmukh
Department of Physics
Indian Institute of Technology Madras
Chennai 600036
STiCM Lecture 13: Unit 4
Symmetry and Conservation Laws
Dynamical Symmetry in the Kepler Problem
1 PCD_STiCM
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Unit 1 : Equations of Motion (Newton / Lagrange / Hamilton )
Unit 2: Oscillators (Free/damped/driven), Resonances, Waves
Unit 3: Polar Coordinate Systems
Unit 4 : Dynamical Symmetry in the Kepler Problem
Symmetry Conservation laws (Noether)
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Unit 4: Learning goals: Recapitulate:
Conservation of energy that is well-known in the Kepler-Bohr
problem stems from the symmetry with regard to temporal
translations (displacements on the time axis).
Conservation of angular momentum, likewise, stems from the
central field symmetry in the Kepler problem.
Neither of these accounts for the fact that the Kepler ellipse
remains fixed; that the ellipse does not undergo a ‘rosette’
motion.
GETTING CONSERVATION LAWS FROM THE EQUATION OF MOTION
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This unit will discover the ‘dynamical’
symmetry of the Kepler problem and
its relation with the constancy
(conservation) of the LRL vector,
which keeps the orbit ‘fixed’.
Motivation: Connection between symmetry &
conservation laws has important
consequences on issues at the very frontiers
of physics and technology.
Mechanics of
Flights into Space 4 PCD_STiCM
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Mechanics of Flights into Space
Konstantin Tsiolkovsky
(1857-1935)
Robert H. Goddard
(1882-1945)
Hermann
Oberth
(1894-1989)
Dr. Vikram
Sarabhai
Father of India’s
Space Program
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Indian Space Research Organisation (ISRO)
[1] Indian National Satellites (INSAT)
- for communication services
[2] Indian Remote Sensing (IRS) Satellites
- for management of natural resources
[3] Polar Satellite Launch Vehicle (PSLV)
- for launching IRS type of satellites
[4] Geostationary Satellite Launch Vehicle (GSLV)
- for launching INSAT type of satellites. PCD_STiCM
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Gravity plays the most important role in
designing satellite trajectories,
of course,
and hence we study the Kepler TWO-BODY
problem
We must then adapt the formalism to
understand the models, methods
and applications of satellite orbits,
etc. PCD_STiCM
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Other than ‘energy’ and ‘angular momentum’, what else
is conserved, and what is the associated symmetry?
For given physical laws of nature, what
quantities are conserved?
Rather, if you can observe what physical quantities
are conserved, can you discover the physical laws
of nature?
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How did Kepler deduce that planetary orbits are
ellipses around the sun ?
Kepler had no knowledge of :
(a) differential equations
(b) inverse square force
(gravity).
Johannes Kepler
1571- 1630
How would you solve this
problem?
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Kepler got his elliptic orbits not
by solving differential
equations for gravitational
force, but by doing clever
curve fitting of Tycho Brahe’s
experimental data.
How did Kepler deduce that planetary
orbits are ellipses around the sun ?
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Tycho Brahe (1546-1601):
Danish astronomer appointed as the imperial astronomer under
Rudolf II in Prague, which then came under the Roman empire.
Brahe had his own “Tychonian” model of planetary motion:
Brahe had planets revolve around the sun (like the Copernican
system), and the sun and the moon going around the earth (like
the system of Ptolemy).
What was Brahe’s nose was made of ?
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Tycho Brahe (1546-1601):
He discovered the supernova in 1572, in
“Cassiopeia’.
Kepler and Brahe never could collaborate
successfully.
They quarreled, and Tycho did not provide Kepler
any access to the high precision observational data
he (Brahe) had complied.
It was only on his deathbed, saying
“……let me not seem to have lived in vain…….”
that Brahe handed over his observational data to
Kepler [Ref.:Sagan]. PCD_STiCM
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Galileo Galilei 1564 - 1642
Isaac Newton (1642-1727)
Causality, Determinism, Equation of Motion
‘Dynamics’ came well AFTER KEPLER!
Johannes Kepler
1571- 1630
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http://en.wikipedia.org/wiki/Image:Galileo.arp.300pix.jpg
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Two-body problem
1R
2R
2 1 r R Rˆ
ru
r
_1_ _ 2
1 22 2 2
2 1
ˆ
by onF
m mm R G u
R R
1m
2m
_ 2_ _1
1 21 1 2
2 1
ˆ
by onF
m mm R G u
R R
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CMR
15
Two-body problem: Centre of Mass
1R
2R
1m
2m
1 1 2 2
1 2
CM
m R m RR
m m1 1 CMr R R
2 2 CMr R R
1 1 2 2 1 1 2 2
1 1 2 2 1 2
( ) ( )
0.
CM CM
CM
m r m r m R R m R R
m R m R m m R
2 1 2 1 r R R r r
1 2m m
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2 1 1 2 3
3 1 1 22 1...where ....for
rr R R G m m
G m m Gmr
m
r
rm
_1_ _ 2
1 22 2 2
2 1
ˆ
by onF
m mm R G u
R R
_ 2_ _1
1 21 1 2
2 1
ˆ
by onF
m mm R G u
R R
30.
rr
r
This equation of motion describes the
‘relative motion’ of the smaller mass
relative to the larger mass, assuming that
the difference in the masses is huge.
2 1 2 1 r R R r r
ˆ r
ur
3 2LT PCD_STiCM
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30.
rr
r
30r r r r
r
ˆ ˆˆru rur r rrru v vv vrr
3vv 0rr
r
0
20
limv
v limt
t
r
t
t r
We now take the dot product of the
velocity with the ‘Eq. of motion’:
ˆ ˆ ˆd d
r r ru ru rudt dt
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30.
rr
r
0
20
limv
v limt
t
r
t
t r
2
2
v
2
v
2
Er
Er
Total ‘SPECIFIC’
(i.e. per unit mass)
MECHANICAL
ENERGY: Constant
of integral /
Constant of Motion.
INVARIANCE,
SYMMETRY.
We took the dot product of the ‘Eq. of
motion’ with velocity:
Integration
w.r.t. time
Differentiation
w.r.t. time
2 2
3 2
DIMENSIONS
E L T
L T
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30.
rr
r
30r r r r
r
Equation of Motion
We took the dot product of the ‘Eq. of motion’ with velocity:
2v
2E
r
…….. and discovered a CONSERVED QUANTITY!
E: constant
symmetry with respect to translations in time.
(E,t): canonically conjugate pair of variables PCD_STiCM
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We now take the cross product of the position vector with the
‘Eq. of motion’:
30.
rr r r
r
: 0 Therefore r r
, ( ) 0d
Now r r r r r r r rdt
= v is also a constant of motion.
SPECIFIC ANGULAR MOMENTUM
Therefore H r r r
INVARIANCE, SYMMETRY
Force: RADIAL
central force symmetry
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Homogeneity of time
Homogeneity of space
Translational Symmetry
Conservation of energy
Conservation of linear
momentum
Isotropy of space
Rotational Symmetry Conservation of angular
momentum
Emmy Noether
1882 to 1935
SYMMETRY CONSERVATION
LAWS
Her entry to the Senate of the University of Gottingen,
Germany, was resisted.
David Hilbert argued in favor of admitting her to the University
Senate. 21 PCD_STiCM
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Time is homogeneous:
Lagrangian of a closed system
does not depend explicitly on time.
Hamiltonian / Hamilton’s Principal Function
q - is a CONSTANT.
LL
q
HamiltonianConservation of Energy is
thus connected with the
symmetry principle
regarding invariance with
respect to temporal
translations.
“ENERGY”
0
L
t
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since 0, this means . . is conserved.
. ., is independent of time, is a constant of motion
L d L Li e p
q dt q q
i e
Space is homogenous and isotropic
Law of conservation of momentum,
arises from the homogeneity of space.
the condition for homogeneity of space : ( , , ) 0
. ., 0
which implies 0 where , ,
L x y z
L L Li e L x y z
x y z
Lq x y z
q
Symmetry Conservation Laws
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( )
1
( )
1
Displacement is arbitrary, hence, 0;
. . 0 =
Thus P is conserved in the absence of external forces.
Ni
kk
Ni
kk
F
d di e p P
dt dt
The conservation of momentum was secured in ‘Unit 1’ on
the basis of ‘translational invariance in homegenous
space’, and not on the basis of Newton’s III law.
( )
1
0 N
i
kk
F S
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Given the symmetry related to
translation in homogenous
space, could you have
discovered Newton’s 3rd law?
2 1
12 21
0
d P
dt
d p d p
dt dt
F F
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Are the conservation principles consequences of the laws of
nature? Or, are the laws of nature the consequences of the
symmetry principles that govern them?
Until Einstein's special theory of relativity,
it was believed that
conservation principles are the result of the laws of nature.
Since Einstein's work, however, physicists began to analyze
the conservation principles as consequences of certain
underlying symmetry considerations,
enabling the laws of nature to be revealed from this analysis.
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Instead of introducing Newton’s III law as a
fundamental principle,
we deduced it (in Unit 1) from symmetry / invariance.
This approach places SYMMETRY ahead of LAWS OF NATURE.
It is this approach that is of greatest value to contemporary
physics. This approach has its origins in the works of
Albert Einstein, Emmily Noether and Eugene Wigner.
(1882 – 1935) (1902 – 1995) (1879 – 1955) PCD_STiCM
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30.
rr
r Equation of Motion for
Kepler’s two-body problem
We shall get yet another constant of motion, a
conserved quantity, by taking the cross product of
the ‘SPECIFIC ANGULAR MOMENTUM with
the equation of motion:
30.
rH r H
r
H
= v,
the SPECIFIC ANGULAR MOMENTUM
H r r r
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3
3
v 0
v - v 0
H r r rr
H r r r r rr
30.
rH r H
r
ˆ ˆ: vd dr
Use r r re r e rrdt dt
23 v - 0H r r rrrr
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since, v .. 0d d
Now H H r H r Hdt dt
2 2
3
2
ˆv v - 0
vv 0
dH r r re
dt r
d rH r
dt r r
v 0
v 0
d d rH
dt dt r
d rH
dt r
23 v - 0H r r rrrr
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v 0
ˆv ,
constant
d rH
dt r
H e A
LAPLACE – RUNGE – LENZ
VECTOR
Observe how a
constant of
motion has
emerged –
- yet again
-– by playing with
the equation of
motion!
ˆv , constantA H e
3 2
1 2 1 3 2v
L T
H LT L T L T
Physical
Dimensions PCD_STiCM
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We will take a Break… …… Any questions ?
Next Lecture: Dynamical Symmetry
of the Kepler Problem
References:
[1] Oliver Montenbruck and Eberhard Gill
‘Satellite orbits – Models, Methods, Applications’.
(Springer, Berlin, 2000)
[2] Francis J. Hale
‘Introduction to Space Flight’
(Prentice Hall, Englewood Cliffs, 1994)
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STiCM
Select / Special Topics in Classical Mechanics
P. C. Deshmukh
Department of Physics
Indian Institute of Technology Madras
Chennai 600036
STiCM Lecture 14: Unit 4
Symmetry and Conservation Laws
Dynamical Symmetry in the Kepler Problem
33 PCD_STiCM
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Take dot product with :
Equation to the orbit / trajectory
Without solving the differential equation of motion!
r
LAPLACE – RUNGE – LENZ VECTOR
ˆv , constantH e A
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Take dot product with : r
2
,
cosH r A r Ar
A r
ˆv H e A where vH r
: vsign reversal H r r A r
vH r r A r
v H r r A r
Interchange ‘dot’ and ‘cross’:
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2 cos
,
H r Ar
A r
,A r
X
2
2
2
cos
cos 1 cos1 cos
r A H
HH p
rAA
What is ?A PCD_STiCM
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,A r
X
F
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vr
H Ar
2 2 22
v vH H r Ar
0 0v , 90 & , v 90
ˆv v
H r H r H
H H u
2v v v v v - vH r r r
22 2 2
2 2 2
v v v - v v cos v
v sin
H r r r r r r r
r
2 2 2 2 2 2 22v v sinH r Ar
, vr
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2 2 2 2 2 2 22v v sinH r Ar
2 2 2 2 2 2 2
2
2v v sinH r A
rH
2 2 2 22vH Ar
2 2 22H E A
2 22 2
21
H E A
2
2
21
A EH
v H r
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2 2
2
1 cos
2 & 1
pr
with
H A EHp
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1 cos
1:
1:
0 1:
0 :
pr
Hyperbola (open trajectory)
Parabola (open trajectory)
Ellipse (closed trajectory)
Circle degenerate ellipse
For satellite
and ballistic
missile
trajectories,
ellipses
(inclusive of
the circle) are
of primary
interest. PCD_STiCM
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1 cos
1:
1:
0 1:
0 :
pr
Hyperbola (open trajectory)
Parabola (open trajectory)
Ellipse (closed trajectory)
Circle degenerate ellipse
1 1
Deep space probes
leave earth’s gravity
on hyperbolic orbits
Earth-orbiting
satellites are in
elliptic motion.
0 1
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b
a
Increasing eccentricity
e=0 e=0.5 e=0.75 e=0.95
2
2
semi-major axis
semi-minor axis
1
a
b
be
a:e eccentricity
Focus
b a
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minr
44
X
min min when 0 : 1
pr r r
maxr perigee
apogee
; 1 cos
pr
2 2
2
2 & 1
H A EHp
perigee
apogee max max when : 1
pr r r
0 1
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2 2
21 co
2;
s ; 1
Hpr
EHp
rrperigee
2a
2b At apogee/perigee: 2 , constantr r a
2
2
22
1 1 1
1
perigee apogee
p p pa r r
p a
211 cos
ar
2 2 2 for circular orbit2
1 1
2 2 2E
H p a
apogee
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24 GPS satellites ---- Wikimedia Commons
http://en.wikipedia.org/wiki/File:ConstellationGPS.gif PCD_STiCM
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1̂e1
ˆe
X
Y
O
“Unit Circle”
1
ê
ˆ ˆv e e
2
v
ˆ ˆ ˆ
ˆz
H r
e e e
e
Plane/Cylindrical Polar Coordinate System
ˆ ˆ ˆ, , ze e e ˆvA H e
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The
(specific)
angular
momentum
vector is
out of the
plane of
this figure,
toward us.
Laplace Runge Lenz Vector
is constant for a strict
potential.
v
H
ê
A
v H
ê
S
1r
ˆvA H e PCD_STiCM
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v
l
H
êv H
ê
S
ˆ - - : ,
ˆv
p lThe Laplace Runge Lenz vector A e
or alternatively defined in terms
of the specific angular momentum H as
A H e
49 PCD_STiCM
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ˆvA H e
ˆvv
dedA d dHH
dt dt dt dt
ˆv dedA dH
dt dt dt
ˆ ˆˆ
de eBut e
dt
vˆ
dA dH e
dt dt
Central Field Symmetry
Angular Momentum
is Conserved
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ˆvA H e
vˆ
dA dH e
dt dt
We must know the form of the interaction
vWhat is the form of the force : ?
per unit ma
!
s
sd
dt
vdF ma m
dt
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2v
2E
r
2per unit m :
vˆThe for ace ss -
de
dt
vˆH
dA de
dt dt
22- ˆˆ ˆzdA
dte e e
0dA
dt
ˆ ˆ ˆ, , : ˆ ˆ ˆ
z
z
e e e right handed basis set
e e e
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For this potential and for the
associated field:
no precession of orbit.
The constancy of the orbit
suggests a conserved quantity
and one must look for an
associated symmetry.
Angular momentum is
conserved,
but major-axis not fixed.
Rosette motion
Two-body central field Kepler-Bohr
problem,
Attractive force:
inverse-square-law.
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Conclusion: 0d A
dt
ˆvA H e
0A H LRL vector is in the plane of the orbit
perigee
A
Direction of the LRL vector is: focus to perigee.
: Must remain constant
– no matter where the planet is! A
This is precisely what FIXES the orbit!
Find the direction of at the perigee. A
H
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Given the constancy related to the conservation of
the LRL vector, could you have discovered the law of
gravity?
If you did that, wouldn’t you
have discovered a law of nature?
2
vˆForce (per unit mass):
Newton told us (but first, to Halley)!
de
dt
Newton
Halley PCD_STiCM
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“It is now natural for us to try to derive the laws of nature
and to test their validity by means of the laws of invariance,
rather than to derive the laws of invariance from what we
believe to be the laws of nature.” - Eugene Wigner
Symmetry & Conservation Principles!
Emmily
Noether
(1882 – 1935)
Eugene
Paul
Wigner
(1902-1995)
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Pierre-Simon Laplace
1749 - 1827
Carl David Tolmé
Runge
1856 - 1927
Wilhelm
Lenz
1888 -1957
Symmetry of the H atom: ‘old’
quantum theory. En ~ n-2
ˆv
Laplace Runge Lenz Vector :
1constant for a strict potential.
A H e
r
Conservation law associated
with ‘dynamical / accidental ’
symmetry.
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http://en.wikipedia.org/wiki/Image:CarleRunge.jpg
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Further reading: • Symmetry & Conservation laws play an
important role in understanding the very frontiers
of Physics.
• The implications go as far as testing the
‘standard’ model of physics, and exploring if
there is any physics beyond the standards
model. Do visit:
Feynman's Messenger Lectures Online
AKA Project Tuva
http://www.fotuva.org/news/project_tuva.html PCD_STiCM
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PCD_STiCM 59
P. C. Deshmukh and Shyamala Venkataraman
Obtaining Conservation Principles from Laws of Nature -- and the
other way around!
Bulletin of the Indian Association of Physics Teachers, Vol. 3, 143-
148 (2011)
P. C. Deshmukh and J. Libby
(a) Symmetry Principles and Conservation Laws in Atomic
and Subatomic Physics -1
Resonance, 15, 832 (2010)
b) Symmetry Principles and Conservation Laws in Atomic and
Subatomic Physics -2
Resonance, 15, 926 (2010)
Useful references on ‘Symmetry & Conservation Laws’
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Continuous Symmetries - Translation, Rotation
Dynamical Symmetries - LRL, Fock Symmetry
SO(4)
Discrete Symmetries
- P : Parity
- C : Charge Conjugation
- T : Time Reversal
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Lorentz symmetry: associated with the PCT symmetry.
PCT theorem (Wolfgang Pauli)
No experiment has revealed any violation of PCT symmetry.
This is predicted by the Standard Model of particle physics.
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The ‘standard model’ unifies all the
fundamental building blocks of matter,
and three of the four fundamental forces.
The Standard Model today
To complete the Model a new particle is needed – the Higgs
Boson – that the physics community hopes to find in the new
built accelerator LHC at CERN in Geneva. PCD_STiCM
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Yoichiro Nambu
½ prize
Enrico Fermi Institute,
Chicago, USA
Makoto Kobayashi
¼ prize
High Energy
Accelerator
Research Organization,
Tsukuba, Japan
Toshihide Maskawa
¼ prize
Kyoto Sangyo Univ,
KyotoJapan
"for the discovery of the
mechanism of spontaneous
broken symmetry in
subatomic physics"
"for the discovery of the origin of
the broken symmetry which
predicts the existence of at least
three families of quarks in nature"
2008 Nobel Prize in Physics
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‘every symmetry in nature yields
a conservation law
and conversely,
every conservation law reveals
an underlying symmetry’.
Noether’s theorem
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“In the judgment of the most competent living
mathematicians, Fräulein Noether was the most significant
creative mathematical genius thus far produced since the
higher education of women began.”
“In the realm of algebra, … , she discovered methods which
have proved of enormous importance …. ”
“………. Her unselfish, significant work over a period of many
years was rewarded by the new rulers of Germany with a
dismissal, which cost her the means of maintaining her
simple life and the opportunity to carry on her mathematical
studies……”
ALBERT EINSTEIN. Princeton University, May 1, 1935.
New York Times May 5, 1935, Excerpts http://www-history.mcs.st-and.ac.uk/history/Obits2/Noether_Emmy_Einstein.html
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We will take a Break… …… Any questions ?
Next: Unit 5: Inertial and non-inertial reference frames.
Moving coordinate systems. Pseudo forces.
Inertial and non-inertial reference frames.
‘Deterministic’ cause-effect relations in inertial frame,
and their modifications in a non-inertial frame. PCD_STiCM