PCD_STiCM PHY101 IISER-Tirupati

1

PHY101: THE WORLD of PHYSICS I - MECHANICS

Unit 1

↑ Unit 1

Unit 2

↑ Unit 3 ↑ Unit 4 Unit 1 10 classes

2 5

3 5

4 5

We shall reorganize the

curriculum in 4 units

STiCM:

PCD’s

NPTEL

course

PCD_STiCM 2

NOTE!

This file is being uploaded for IISER students at

Tirupati.

This is the first draft version of Unit 2 and consists

of the material that will be covered in the first few

classes of this unit.

Some more material will be added as integral part

of Unit 2 and this file will be updated, and then

uploaded again, possibly after a few weeks…..

may be sooner?

PCD_STiCM PHY101 IISER-Tirupati

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PHY101: THE WORLD of PHYSICS I - MECHANICS

Unit 2

STiCM:

PCD’s

NPTEL

course

Done: Unit 1↑ ↑Introduced↑

in Unit 1

• Central field / spherically symmetric potentials

• Planetary Motion

• Rotational Motion of a Rigid Body

• Multiparticle Systems

• Conservation Laws – further insights

Unit 2: 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

4 PCD_STiCM

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 5 PCD_STiCM

6

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

8

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

9

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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12

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

PCD_STiCM

CMR

16

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 2CMR 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

PCD_STiCM

1 1 CMR r R

2 2 CMr R R

17

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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19

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

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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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30.

rr

r Equation of Motion

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. 22 PCD_STiCM

Consider a system of N particles in a medium that is

homogenous.

A displacement of the entire N-particle system through in

this medium would result in a new configuration that would find

itself in an environment that is completely indistinguishable

from the previous one.

23

S

Begin with a symmetry principle:

translational invariance in homogenous space.

PCD-L03

The connection between ‘symmetry’ and ‘conservation law’ is

so intimate, that we can actually derive Newton’s III law using

‘symmetry’.

This invariance of the environment of the entire N

particle system following a translational

displacement is a result of translational symmetry

in homogenous space. PCD_STiCM PHY101 IISER-Tirupati

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FOUR essential considerations:

1. Each particle of the system is under the influence only of

all the remaining particles;

the system is isolated: no external forces act on any of its

particles.

2 The entire N-particle system is deemed to have

undergone simultaneous identical displacement;

all inter-particle separations and relative orientations

remain invariant.

3 Entire medium: essentially homogeneous;

- spanning the entire system both

before and after the displacement.

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4. The displacement the entire system under

consideration is deemed to take place at a certain instant

of time. PCD_STiCM PHY101 IISER-

Tirupati

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The implication of these FOUR considerations:

“Displacement considered is only a virtual

displacement.”

- only a mental thought process;

PCD-L03

PCD_STiCM PHY101 IISER-

Tirupati

- real physical displacements would require a certain

time interval over which the displacements would

occur

- virtual displacements can be thought of to occur at

an instant of time and subject to the specific four

features mentioned.

26

PCD-L03

Displacement being

virtual,

it is redundant to ask

“what agency has caused

it”

PCD_STiCM PHY101 IISER-

Tirupati

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Now, the internal forces do no work in this virtual

displacement.

Therefore ‘work done’ by the internal forces in the

‘virtual displacement’ must be zero.

This work done (rather not done)

is called ‘virtual work’.

1 1 1,

th

0 ,

where : force on particle by the ,

and

force on the particle due to the

-1 particles.

N N N

k ik

k k i i k

thik

thk

remai

W F s F s

F k i

F k

ng Nni

PCD-L03 PCD_STiCM PHY101 IISER-

Tirupati

Under what conditions can the above relation

hold for an ARBITRARY displacement ?

28

s

The mathematical techniques: Jean le Rond

d'Alembert (1717 – 1783)

1 1 1

0N N N

kk k

k k k

dp d dPF p

dt dt dt

PCD-L03

1 1 1,

0N N N

k ik

k k i i k

W F s F s

Newton’s I,II laws used; not the III. PCD_STiCM PHY101 IISER-

Tirupati

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is obtained from the properties of translational

symmetry in homogenous space.

2 1

12 21

0,

. ., ,

which gives ,

the .

d P

dt

d p d pi e

dt dt

F F

III law of Newton

Amazing!

- since it suggests a path to

discover the laws of

physics by exploiting the

connection between

symmetry and

conservation laws!

PCD-10

For just two particles:

1 1 1

0N N N

kk k

k k k

dp d dPF p

dt dt dt

The relation

PCD_STiCM PHY101 IISER-

Tirupati

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Newton’s III law need not be introduced as a

fundamental principle/law;

we deduced it from symmetry / invariance.

SYMMETRY placed ahead of LAWS OF NATURE.

Albert Einstein, Emmily Noether and Eugene Wigner.

(1882 – 1935) (1879 – 1955) PCD-L03 (1902 – 1995)

PCD_STiCM PHY101 IISER-Tirupati

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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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32

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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33

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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34

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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35

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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36

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

37

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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38

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’:

PCD_STiCM

39

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

,A r

X

F

40 PCD_STiCM

41

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

PCD_STiCM

42

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

PCD_STiCM

, vr

43

2 2

2

1 cos

2 & 1

pr

with

H A EHp

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44

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

45

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

PCD_STiCM

46

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

PCD_STiCM

minr

47

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

PCD_STiCM

48

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

PCD_STiCM

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24 GPS satellites ---- Wikimedia Commons

http://en.wikipedia.org/wiki/File:ConstellationGPS.gif PCD_STiCM

50

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

PCD_STiCM

51

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

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

52 PCD_STiCM

53

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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54

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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55

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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56

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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57

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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58

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

59

“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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60

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.

PCD_STiCM

http://en.wikipedia.org/wiki/Image:CarleRunge.jpg

61

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

PCD_STiCM 62

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’

63

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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65

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 – FOUND in the experiments at the accelerator LHC at

CERN in Geneva. PCD_STiCM

66

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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PCD_STiCM 69

Multi-particle systems

Fluid Mechanics

Rotation of a Rigid Body