PH101-Unit 4physics.iitm.ac.in/~labs/amp/PH1010 Unit 3 Motion in a Central Field.… · designing...

PCD_STiCM Unit 3 PH1010 IIT-Tirupati

1

PH1010: STiCM

↑ Unit 3

Unit 3: Motion in a Central Force Field

(2 lectures)

Central potential and conservation of energy and angular

momentum; effective potential, Kepler's laws (planetary

motion), satellite orbits.

Key theme: Symmetry and conservation laws systems

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

2 PCD_STiCM Unit 3 PH1010 IIT-

Tirupati

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 3 PCD_STiCM Unit

3 PH1010 IIT-

Tirupati

4

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 Unit 3 PH1010 IIT-

Tirupati

5

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 Unit 3 PH1010 IIT-

Tirupati

6

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 Unit 3 PH1010 IIT-

Tirupati

7

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?


Tirupati

8

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?


Tirupati

9

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 ?


Tirupati

10

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 ?


Tirupati

11

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 Unit 3 PH1010 IIT-

Tirupati

12

Galileo Galilei 1564 - 1642

Isaac Newton (1642-1727)

Causality, Determinism, Equation of Motion

‘Dynamics’ came well AFTER KEPLER!

Johannes Kepler

1571- 1630


Tirupati

http://en.wikipedia.org/wiki/Image:Galileo.arp.300pix.jpg

13

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


Tirupati

CMR

14

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


Tirupati

1 1 CMR r R

2 2 CMr R R

15

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 2L T PCD_STiCM Unit 3 PH1010 IIT-

Tirupati

16

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


17

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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18

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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19

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


Tirupati

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. 20 PCD_STiCM Unit 3 PH1010 IIT-

Tirupati

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.

21

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 Unit 3 PH1010 IIT-Tirupati

22

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


Tirupati

23

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)


24

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


Tirupati

25

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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26

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


Tirupati

27

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

2

3v - 0H r r rrr

r


Tirupati

28

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

2

3v - 0H r r rrr

r


Tirupati

29

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 Unit 3 PH1010 IIT-

Tirupati

30

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


Tirupati

31

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’:


Tirupati

32

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

Unit 3 PH1010

2

H p

,A r

X

F


Tirupati

1 cos

pr

What is ? A

34

vr

H Ar

2 2 22v vH H r A

r

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 A

r

, vr


35

2 2 2 2 2 2 22v v sinH r A

r

2 2 2 2 2 2 2

2

2v v sinH r A

rH

2 2 2 22vH A

r

2 2 22H E A

2 2

2 2

21

H E A

2

2

21

A EH

v H r


, vr

36

2 2

2

1 cos

2 & 1

pr

with

H A EHp


Tirupati

37

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 Unit 3 PH1010 IIT-

Tirupati

38

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


Tirupati

2 2

2

2 & 1

with

H A EHp

39

b

a

e=0 e=0.5 e=0.75 e=0.95

:e eccentricity

Focus

b a


Increasing eccentricity Note that as eccentricity increases, the foci

move closer to the apogee and the perigee

since they are closest to the foci than other

points on the ellipse.

2

2

semi-major axis

semi-minor axis

1

a

b

be

a

minr

40

X

min min when 0; cos 1; 1

pr r r

maxr perigee

apogee

; 1 cos

pr

2 2

2

2 & 1

H A EHp

perigee

apogee max max when ; cos 1; 1

pr r r

0 1


41

2 2

21 co

2;

s ; 1

Hpr

EHp

rr

perigee 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

21

1 cos

ar

2 2 2 for circular orbit 0

2

1 1

2 2 2E

H p a

apogee


Tirupati


42

c: focal length

perigee apogee rmin

rmax

43

24 GPS satellites ---- Wikimedia Commons

http://en.wikipedia.org/wiki/File:ConstellationGPS.gif PCD_STiCM Unit 3 PH1010 IIT-

Tirupati

44

1e1

ˆe

X

Y

O

“Unit Circle”

1

e

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


Tirupati

45

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

e

A

v H

e

S

1r

ˆvA H e


Tirupati

v

l

H

e

v H

e

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


Tirupati

47

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


Tirupati

48

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


Tirupati

49

2v

2E

r

2per unit m :

vˆThe for ace ss

de

dt

vˆH

dA de

dt dt

2

2 ˆ ˆˆz

dAe e

te

d

0dA

dt

ˆ ˆ ˆ, , :

ˆ ˆ ˆ

z

z

e e e right handed basis set

e e e


Tirupati

50

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.


Tirupati

51

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


Tirupati

52

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 Unit 3 PH1010 IIT-

Tirupati

53

“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)


Tirupati

54

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.


Tirupati

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

55

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 Unit 3 PH1010 IIT-

Tirupati


Tirupati

56

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’

57

Continuous Symmetries - Translation, Rotation

Dynamical Symmetries - LRL, Fock Symmetry

SO(4)

Discrete Symmetries

- P : Parity

- C : Charge Conjugation

- T : Time Reversal


Tirupati

58

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.


Tirupati

59

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 Unit 3 PH1010 IIT-

Tirupati

60

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


Tirupati

61

‘every symmetry in nature yields

a conservation law

and conversely,

every conservation law reveals

an underlying symmetry’.

Noether’s theorem


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62

“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


r

63

2 2

21 co

2;

s ; 1

Hpr

EHp

r

perigee 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

21

1 cos

ar

2 2 2 for circular orbit 0

2

1 1

2 2 2E

H p a

apogee


Tirupati


64

2 2

21 co

2;

s ; 1

Hpr

EHp

2 2

21 co

2;s ; 1

Hp

r

EHp

Eq.8.41, page

300 in

Thornton &

Marion’s

Classical

Dynamics

2

21

b

a

2 2

2 2

21 1

b EH

a

Eq.8.14, page

291 in

Thornton &

Marion’s

Classical

Dynamics PCD_STiCM Unit 3 PH1010 IIT-Tirupati 65

2( )

UF

2 ˆ ˆ=( )e (2 )ea

ˆ

ˆ ˆv

r e

e e

2 2 2 2v

2

2 22 4 2 2 2 2 2 2

2 2

ˆ ˆ ˆ ˆH v

HH ; ;

zr e e e e

2 2 2

2 22 2

2 2 2

( ); where ( )2

1 1( ) ( )

2 2 2

E KE PE U U

U U


66

22

2

1 1( )

2 2E U

22

2

1 1( )

2 2E U

2

2

2 1( )

2

dE U

dt

2

2 2

2( )

dE U

dt


67

2

2 2

2( )

dE U

dt

2

2 2

2

2

20 when ( ) 0

1. . when ( ) 0

2

dE U

dt

i e E U

Quadratic equation in

'

perigee 2a

apogee

Turning

points

are at the

apogee and

the perigee

0 at the

turning points


68

2

2

1( ) 0

2E U

2

2

1( ) 0

2E U

2

2

1( )

2

EffectiveU U

physical

potential centrifugal

potential;

associated with l

At the turning

points,

which

are at the

apogee and

the perigee


69

2

2

1

2

( )U min

max

min max

2

2

1( )

2E U KE


Tirupati

70

Unit 4: The Equation of

Continuity

The concepts of flux and

divergence of a force field;

Gauss' theorem, equation of

continuity and its physical

meaning; Local description of

conservation of mass and charge. Key theme: Physical significance of the equation of continuity

and its universality.

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