STiCM

Select / Special Topics in Classical Mechanics

P. C. Deshmukh

Department of Physics

Indian Institute of Technology Madras

Chennai 600036

pcd@physics.iitm.ac.in

STiCM Lecture 13: Unit 4

Symmetry and Conservation Laws

Dynamical Symmetry in the Kepler Problem

1 PCD_STiCM

2

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)

PCD_STiCM

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

3 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 4 PCD_STiCM

5

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

PCD_STiCM

6

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

7

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

8

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?

PCD_STiCM

9

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?

PCD_STiCM

10

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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11

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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12

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

13

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

14

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

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

PCD_STiCM

16

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

17

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

PCD_STiCM

18

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

PCD_STiCM

19

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

20

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

PCD_STiCM

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

22

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

PCD_STiCM

23

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

PCD_STiCM

( )

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

24 PCD_STiCM

25

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

PCD_STiCM

26

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.

PCD_STiCM

27

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

28

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

PCD_STiCM

29

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

PCD_STiCM

30

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

PCD_STiCM

31

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

32

We will take a Break… …… Any questions ?

pcd@physics.iitm.ac.in

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)

PCD_STiCM

STiCM

Select / Special Topics in Classical Mechanics

P. C. Deshmukh

Department of Physics

Indian Institute of Technology Madras

Chennai 600036

pcd@physics.iitm.ac.in

STiCM Lecture 14: Unit 4

Symmetry and Conservation Laws

Dynamical Symmetry in the Kepler Problem

33 PCD_STiCM

34

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

PCD_STiCM

35

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

36

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

37 PCD_STiCM

38

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

39

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

40

2 2

2

1 cos

2 & 1

pr

with

H A EHp

PCD_STiCM

41

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

42

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

43

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

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

PCD_STiCM

45

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

46

24 GPS satellites ---- Wikimedia Commons

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

47

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

48

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

49 PCD_STiCM

50

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

PCD_STiCM

51

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

PCD_STiCM

52

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

PCD_STiCM

53

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.

PCD_STiCM

54

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

PCD_STiCM

55

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

56

“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)

PCD_STiCM

57

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

58

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 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’

60

Continuous Symmetries - Translation, Rotation

Dynamical Symmetries - LRL, Fock Symmetry

SO(4)

Discrete Symmetries

- P : Parity

- C : Charge Conjugation

- T : Time Reversal

PCD_STiCM

61

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.

PCD_STiCM

62

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

63

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

PCD_STiCM

64

‘every symmetry in nature yields

a conservation law

and conversely,

every conservation law reveals

an underlying symmetry’.

Noether’s theorem

PCD_STiCM

65

“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

PCD_STiCM

66

We will take a Break… …… Any questions ?

pcd@physics.iitm.ac.in

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