ClassicalD

ynamics

P2

NE

WTO

NIA

NM

EC

HA

NIC

S

•N

ewtonian

Mechanics

is:

-N

on-relativistici.e.

velocitiesv

c(speed

oflight=3×10

8m

s−

1.)

-C

lassicali.e.E

th

(Planck’s

constant=1.05×

10−

34

Js.)

•A

ssumptions:

-m

assindependentofvelocity,tim

eorfram

eofreference;

-m

easurements

oflengthand

time

areindependentofthe

frame

ofreference;

-allparam

eterscan

beknow

nprecisely.

•M

echanics:

=S

tatics(absence

ofmotion);

+K

inematics

(descriptionofm

otion,usingvectors

forpositionand

velocity);

+D

ynamics

(predictionofm

otion,andinvolves

forcesand/orenergy).

ClassicalD

ynamics

P1

CLA

SS

ICA

LD

YN

AM

ICS

20Lectures

Prof.

S.F.G

ull

•H

AN

DO

UT

—com

prehensivesetofnotes

containingallrelevantderivations.

Please

reportallerrorsand

typos.

•N

OTE

S—

Provisionalhardcopy

availablein

advance.

Definitive

copiesofoverheads

availableon

web.

Please

reportallerrorsand

typos.

•S

UM

MA

RY

SH

EE

TS—

1page

summ

aryofeach

lecture.

•E

XA

MP

LES

—2

example

sheets—

2exam

plesperlecture.

•W

OR

KE

DE

XA

MP

LES

—W

illbeavailable

onthe

web

later.

•W

EB

PAG

E—

Forfeedback,additionalpictures,movies

etc.

http://ww

w.m

rao.cam.ac.uk/∼

steve/part1bdyn/

Thereis

alink

toitfrom

theC

avendishteaching

pages.

ClassicalD

ynamics

P4

•The

simple

harmonic

oscillator(SH

O)occurs

many

times

duringthe

course.

SIM

PLE

HA

RM

ON

ICO

SC

ILLATOR

•M

assm

moving

inone

dimension

with

coordinatex

ona

springw

ithrestoring

force

F=

−kx

.The

constantkis

known

asthe

springconstant.

•N

ewtonian

equationofm

otion:mx

=−

kx

,where

xdenotes

dxdt

etc.

•G

eneralsolution:x=

Acos

ωt+

Bsin

ωt

where

ω2

=k/m

.

•C

analso

write

solutionas

x=

<(A

eiω

t),where

Ais

complex.

•W

ecan

integratethe

equationofm

otionto

getaconserved

quantity—

theenergy.

•M

ultiplyingthe

equationofm

otionby

x(a

goodgeneraltrick)w

eget

mxx

+kxx

=0⇒

12m

x2

+12kx

2=

E=

constant

•H

erethe

quantityT

≡12m

x2

isthe

kineticenergy

ofthem

assand

V≡

12kx

2is

the

potentialenergystored

inthe

spring.

•Form

anydynam

icalsystems

(suchas

theS

HO

)thetim

et

doesnotappearexplicitly

inthe

equationsofm

otionand

thetotalenergy

E=

T+

Vis

conserved.This

conserved

quantityis

alsoknow

nas

theH

amiltonian.

ClassicalD

ynamics

P3

•This

coursecontains

many

applicationsofN

ewton’s

Second

Law.

BA

SIC

PR

INC

IPLE

SO

FN

EW

TON

IAN

DY

NA

MIC

S

•M

assesaccelerate

ifaforce

isapplied...

•The

rateofchange

ofmom

entum(m

ass×velocity)is

equaltothe

appliedforce.

•Vectorially

(p=

mv

):dpdt

=d(m

v)

dt

=F

•U

suallym

isa

constantsom

dvdt

=F

•G

eneralcasem

dvdt

+dmdt

v=

Fenables

youto

dorocketscience.

v

u

mdm

0

•R

ocketofmass

m(t)

moving

with

velocityv(t)

expelsa

mass

dm

ofexhaustgasesbackw

ardsatvelocity−

u0

relativeto

therocket.

•In

theabsence

ofgravityorotherexternalforces

dm

u0

+m

dv

=0.

•Integrating,w

efind

v=

u0log

(mi /m

f ),

where

mi,f

arethe

initialandfinalm

asses.

•Fora

rocketacceleratingupw

ardsagainstgravity

mdvdt

+dmdt

u0

+m

g=

0.

ClassicalD

ynamics

P6

TH

EE

NE

RG

YM

ETH

OD:

EX

AM

PLE

•Ladderleaning

againstasm

oothw

all,restingon

asm

oothfloor

(notrecomm

endedpractice).

•N

ewtonian

method

needsreaction

forcesN

andR

.

Takem

oments

togetthe

angularacceleration.Try

it...

•The

energym

ethodis

easier:

(1)Potentialenergy.This

iseasy:

V=

12m

glcos

θ.

(2)Kinetic

energy.W

ritethis

asthe

sumofthe

kineticenergy

ofthecentre

ofmass

plusthe

energyofrotation

about

thecentre

ofmass:

T=

12m

(x2

+y2)

+12Iθ2,

where

I=

112m

l2

isthe

mom

entofinertiaofa

uniformrod

aboutitscentre.

Coordinates

ofcentreofm

ass:x

=12lsin

θ;

y=

12lcos

θ

Work

outvelocities:x

=12lcos

θθ

;y

=−

12lsin

θθ

T=

12m

(x2

+y2)

+12Iθ2

=18m

l2θ

2+

124m

l2θ

2=

16m

l2θ

2

(3)Energy

method:

d(T

+V

)

dt

=0⇒

θ(

13θm

l2−

12m

glsin

θ)

=0.

(4)Equation

ofmotion:

θ=

3g2lsin

θ.

ClassicalD

ynamics

P5

TH

EE

NE

RG

YM

ETH

OD

•Ifw

eknow

fromphysicalgrounds

thattheenergy

isconserved,w

ecan

always

derivethe

equationsofm

otionofsystem

sthatonly

haveone

degreeoffreedom

(suchas

theS

imple

Harm

onicO

scillator):12m

x2

+12kx

2=

E⇒

x(m

x+

kx)

=0⇒

mx

=−

kx

.

•This

works

becausex

isnotalw

ayszero.

We

willcallthis

theenergy

method.

•W

ecan

sometim

esderive

theequations

ofmotion

ofmuch

more

complicated

systems

with

ndegrees

offreedomis

asim

ilarway.

It’scertainly

notrigorous,butworks

formostofthe

systems

studiedin

thiscourse.

Thetheoretically

more

advancedm

ethodsofLagrangian

andH

amiltonian

mechanics

derivethe

equationsofm

otionfrom

avariationalprinciple.

They

arerigorous,butstilluse

thekinetic

energyT

andpotentialenergy

V(actually

inthe

combinationL

=T−

V).

•W

ew

illseelaterin

thecourse

theuse

oftheenergy

method

toderive

theequation

ofmotion

ofaparticle

atradiusr

ina

centralforce:12m

r2

+V

eff(r)

=E

.

Differentiating

with

respecttotim

ew

eget:

r

(

mr

+dV

eff

dr

)

=0

•W

estrike

outther

toobtain

theequation

ofmotion.

ClassicalD

ynamics

P8

RE

VIS

ION:

VE

CTO

RC

ALC

ULU

SII

(See

Section

1.2ofH

andout)

Gradientoperator

∇(Vectordifferentialoperator).

•grad(Φ

);∇

ΦVector

gradientofascalar

fieldΦ

.

•div(E

);∇·E

Scalar

divergenceofa

vectorfield

E.

•curl(E

);∇

××× ××E

.Vector

curlofavector

fieldE

.

Divergence

Theorem(G

auss’Theorem)

d=

d| |

Sn

S

•R

elatesintegralofflux

vectorE

throughclosed

surfaceS

(noutw

ards)to

volume

integralof∇·E

.

∮

dS·E

=

∫

dV

∇·E

Stokes’Theorem

•R

elatesline

integralvectorE

aroundclosed

loopl

to

surfaceintegralof

∇××× ××E

.∮

dl·E

=

∫

dS·∇

××× ××E

Furtheridentities:

∇××× ××(∇

Φ)

=0;

∇·(

∇××× ××E

)=

0;∇

××× ××(∇

××× ××E

)=

∇(∇·E

)−∇

2E

ClassicalD

ynamics

P7

RE

VIS

ION:

VE

CTO

RC

ALC

ULU

S

(See

Section

1.2ofH

andout)

•In

dynamics

we

usevectors

todescribe

thepositions,velocities

andaccelerations

of

particlesand

otherbodies,asw

ellasthe

forcesand

couplesthatacton

them

•W

eneed

torevise

vectors,vectorfunctions,vectoridentitiesand

integraltheorems.

•R

evisescalar

producta·b

andvector

producta××× ××b

(preferableto

a∧b

).

•There

areonly

acouple

ofvectoridentities,butyouM

US

TLE

AR

NTH

EM

.

(1)Scalar

tripleproduct

a·(b××× ××c)

=(a

××× ××b)·c

(Interchangeofdotand

cross)

a·(b××× ××c)

=b·c

××× ××a

=−

b·a××× ××c

(Permutations

changesign)

(2)Vectortriple

product

Thisis

them

ostimportantidentity:

a××× ××(b

××× ××c)=

a·cb−

a·bc

Rule:

Vectoroutsidebracketappears

inboth

scalarproducts.

Rule:

Outerpairtakes

theplus

sign.(a

××× ××b)××× ××c

=a·c

b−b·c

a

ClassicalD

ynamics

P10

OV

ER

ALL

MO

TION

(See

Section

3.1ofH

andout)

•D

evelopmentfrom

New

ton’sLaw

s:m

ar

a=

Fa ,

where

a=

1,Nforthe

ath

ofNparticles.

Arigid

bodyis

aspecialcase.

•O

verallmotion

∑

a

mar

a=

∑

a

Fa

=∑

a

Fa0

+∑

a

∑

b

Fab ,

where

Fa0

isthe

externalforceon

particlea

andF

ab

isthe

forceon

adue

tob.

Since

Fab

=−

Fba

byN

ewton’s

3rdLaw

,the∑

a

∑

b

termabove

sums

tozero.

•D

efineM

≡∑

a

ma

andM

R≡

∑

a

mar

a .R

isthe

positionofthe

Centre

ofMass.

•W

iththese

definitionsM

R=

∑

a

Fa0 ≡

F0

•The

Centre

ofMass

moves

asifitw

erea

particleofm

assM

actedupon

bythe

total

externalforceF

0 .

•In

terms

ofmom

entump

a=

Fa ;

P=

F0 ,w

hereP

isthe

totalmom

entum.

ClassicalD

ynamics

P9

ME

CH

AN

ICS

—R

EV

IEW

OF

PA

RT

IA

Revision

ofdynamics

ofmany-particle

system.(D

etailedderivations

inS

ection3

ofHandout.)

•S

ystemofN

particles.The

ath

particleofm

assm

ais

atpositionr

aand

velocityv

a .

Acted

onby

externalforceF

a0

andinternalforces

Fab

fromotherparticles.

•Im

portantdefinition:C

entreofm

assR

.D

efineM

≡∑

a

ma

andM

R≡

∑

a

mar

a .

•O

therconcepts:

Totalmom

entumP

.Totalangularm

omentum

J.

TotalexternalforceF

0

andcouple

G0 .

Kinetic

energyT

,potentialenergyU

,totalenergyE

.

•Totalm

omentum

actsas

ifitwere

actedupon

bythe

totalexternalforce.

•Totalangular

mom

entumacts

asifitw

ereacted

uponby

thetotalexternalcouple.

•Intrinsic

angularm

omentum

:J

′inthe

frame

S′in

which

P′=

0(zero-m

omentum

,or

Centre

ofMass

(CoM

)frame).

Theintrinsic

angularmom

entumis

independentoforigin.

•The

Centre

ofMass

frame

isthus

special,andshould

beused

whereverpossible.

•G

alileantransform

ationfrom

CoM

frame

S′to

frame

Sm

ovingatvelocity

V

(CoM

atR′=

Vt).

Mom

entum:

P=

P′+

MV

Angular

Mom

entum:J

=J

′+M

R′××× ××V

.K

ineticenergy:T

=T

′+12M

V2

ClassicalD

ynamics

P12

CH

OIC

EO

FO

RIG

IN

(See

Section

3.3ofH

andout)

•S

upposew

em

ovethe

originby

aconstanta

,givingnew

coordinatesr′w

ithr

=r′+

a.

•Then

r=

r′and

theoverallm

otionis

unaffected.

•W

hatabouttheangularm

omentum

J?

Foroneparticle

Ja

=J

′a+

a××× ××p

a ,orforthe

system

J=

J′+

∑

a

a××× ××p

a=

J′+

a××× ××P

i.e.Jdepends

onthe

choiceoforigin

unlessP

=0.

•Intrinsic

angularm

omentum

:J

inthe

frame

inw

hichP

=0

(zero-mom

entum,orC

entre

ofMass

frame).

TheIntrinsic

angularmom

entumis

independentoforigin.

•The

Centre

ofMass

frame

isthus

special,andshould

beused

whereverpossible.

•S

imilarly

G=

G′+

a××× ××F

.

ClassicalD

ynamics

P11

MO

ME

NTS

(See

Section

3.2ofH

andout)

•C

ouple,torque:G

≡r××× ××F

.A

ngularm

omentum

:J

≡r××× ××p

.

•S

incep

a=

Fa ,w

ehave

∑

a

ra××× ××p

a=

∑

a

ra××× ××F

a

•E

xpandR

HS

:R

HS

=∑

a

ra××× ××F

a0

+∑

a

∑

b

ra××× ××F

ab

︸︷︷

︸∑

b

∑a<

b

(ra −

rb )

××× ××F

ab

︸︷︷

︸

=0

•The

lattertermis

zerosince

Fab

isassum

edto

bealong

theline

between

aand

b.

We

haveagain

usedN

ewton’s

ThirdLaw

.

•The

LHS

foroneparticle

isJ

a=

ddt (r

a××× ××p

a )=

ra××× ××p

a︸

︷︷

︸+

ra××× ××p

a

zero,sincem

r=

p•

Forthesystem

ofparticles

J≡

∑

a

Ja

=∑

a

ra××× ××p

a=

RH

S=

∑

a

ra××× ××F

a0 ≡

G0

•G

0is

theresultantcouple

Gfrom

allexternalforces.

ClassicalD

ynamics

P14

PO

TEN

TIAL

AN

DT

OTA

LE

NE

RG

Y

(See

Section

3.4ofH

andout)

•Potentialenergy:

Uis

definedas

dU

=∑

∑a<

b Fabd|r

a −r

b |

•N

otethe

zeroofU

=

∫

dU

isundefined.

Itisoften

takenw

ithU

=0

with

particlesat

infiniteseparation,giving

negativeU

forasystem

ofparticlesw

ithattractive

forces.

•Fora

rigidbody

dU

=0

since|ra −

rb |is

fixed.

•TotalE

nergy:E

=T

+U

.

•A

sdefined

above

dE

=dT

+dU

=∑

a

Fa0 ·d

ra

•The

RH

Sterm

isthe

work

doneby

externalforces;itcanbe

incorporatedinto

Uifdesired.

ClassicalD

ynamics

P13

KIN

ETIC

EN

ER

GY

(See

Section

3.4ofH

andout)

•W

orkdone:

force ×distance

moved‖

force=

changein

energy.

•Fora

singleparticle

F·d

r=

mr·d

r=

mr·r︸︷︷︸dt

ddt (

12r·r

)

or

F·d

r=

d(12m

v2)

•K

ineticenergy:

T≡

12m

v2.

•W

orkdone

onparticle

=change

inkinetic

energy.

•Fora

systemofparticles

dT

=∑

a

dT

a=

∑

a

Fa ·d

ra

=∑

a

Fa0 ·d

ra

+∑

∑a<

b

Fab ·(

dr

a −dr

b )

where

we

haveused

Fab

=−

Fba .

We

canw

ritethe

ab-term

as−F

abd|r

a −r

b |,where

Fab

hasm

agnitude=

|Fab |

andis

positiveifforce

isattractive,negative

ifrepulsive.

ClassicalD

ynamics

P16

GA

LILEA

NT

RA

NS

FOR

MATIO

N—

EN

ER

GY

(See

Section

3.5.3ofH

andout)

•E

nergy

T=

∑

a

12m

a v2a

=12m

a (v′a

+V

)·(v′a

+V

)

=T

′+∑

a

mav′a

︸︷︷

︸ ·V+

12M

V2.

(=0,

ifS′

=zero-m

omentum

frame)

or

T=

KE

inzero-m

omentum

frame

+12M

V2

ClassicalD

ynamics

P15

GA

LILEA

NT

RA

NS

FOR

MATIO

N

(See

Section

3.5ofH

andout)

•G

ofrom

frame

S′to

Sw

ithr

=r′+

Vt;V

steady;t=

t′.

•M

omentum

p=

p′+

mV

;P

=P

′+M

V

i.e.Pin

Sand

P′in

S′change

together(orremain

steadytogetherifthere

ifnoexternal

force).IfP

′=

0,thenS′is

thezero-m

omentum

orCentre

ofMass

frame.

•A

ngularm

omentum

J=

∑

a

(r′a

+V

t)××× ××(p

′a+

maV

)

•There

are4

terms.

The4th

isV

××× ××V

=0.

Theothers

give

J=

J′+

Vt××× ××P

′+∑

a

r′a××× ××m

aV

︸︷︷

︸∑

a

(mar′a )

××× ××V

=M

R′××× ××V

•Thus

ifS′is

thezero-m

omentum

frame,P

′=

0and

J=

J′

+M

R′××× ××V

︸︷︷

︸.

inS

intrinsicm

otionofC

ofMin

S

ClassicalD

ynamics

P18

DY

NA

MIC

SIN

CY

LIND

RIC

AL

PO

LAR

S

(See

Section

2.1ofH

andout)

•In

Cartesians,the

equationofm

otionofa

particleis

mr

=F

;or

mx

=F

xetc,

where

we

denotetim

ederivatives

drdt≡

r,

d2r

dt2≡

retc.

eρ

eφ

•C

onsidercylindricalpolars;ignorez

-motion

forthem

oment

r=

ρe

ρw

heree

ρ ,e

φand

ez

areunitvectors

inthe

directionsofincreasing

ρ,φ

,z.

•N

otethatthe

directionofthe

vectorse

ρand

eφ

changeas

theparticle

moves:

r=

ρe

ρ+

ρ˙eρ

•A

sthe

particlem

ovesfrom

sayP

toP

′indt,

eρ

ande

φrotate

bydφ

.

•E

lementary

geometry

givesde

ρ=

dφ

eφ

or˙eρ

=φe

φand

similarly

˙eφ

=−

φe

ρ

givingr

=ρe

ρ︸︷︷︸

+ρφe

φ︸︷︷︸

radialtransverse

•In

cylindricalpolarcoordinatesthe

radialvelocityis

ρand

thetransverse

velocityis

ρφ

.

ClassicalD

ynamics

P17

CO

OR

DIN

ATES

YS

TEM

S

(See

Section

1.1ofH

andout)

•Position

vectorr

hasC

artesiancoordinates

(x,y

,z),

cylindricalpolarcoordinates(ρ

,φ,z

)

andsphericalpolarcoordinates

(r,θ,φ).

•R

elationbetw

eencoordinate

systems:

x=

ρcos

φ=

rsin

θcos

φ

y=

ρsin

φ=

rsin

θsin

φ

z=

z=

rcos

θ

ρ=

√

x2

+y2

r=

√

x2

+y2

+z2

•W

edefine

unitvectors(e

x,e

y ,e

z )along

x,y

,zaxes.

•S

imilarly

we

defineunitvectors

(e

ρ ,e

φ,e

θ )along

directionsofincreasing

(ρ,φ

,θ)

(unambiguous

becausethese

areorthogonalcoordinate

systems).

•Position

vectorr

=xe

x+

ye

y+

ze

z=

ρe

ρ+

ze

z=

re

r

ClassicalD

ynamics

P20

PO

LAR

SA

ND

THE

AR

GA

ND

DIA

GR

AM

eiφ

ieiφ

(See

HandoutS

ection2.1)

•The

complex

planez≡

x+

iy=

ρe

iφhas

the

same

structureas

thetw

o-dimensionalplane

r=

xe

x+

ye

y=

ρe

ρ .

•The

unitvectorscorrespond

tocom

plexnum

bers:

eρ

↔eiφ

eφ

↔ie

iφ

•W

ecan

thereforederive

theradialand

transversecom

ponentseasily

usingthe

Argand

diagram.

•Velocity:

ddt (ρ

eiφ

)=

ρe

iφ+

ρφ

ieiφ

.

•A

cceleration:d2

dt2(ρ

eiφ

)=

ρe

iφ+

2ρφ

ieiφ

+ρφ

ieiφ

−ρφ

2eiφ

=(ρ−

ρφ

2

︸︷︷

︸ )eiφ

+(ρ

φ+

2ρφ

︸︷︷

︸ )ie

iφ

radialtransverse

ClassicalD

ynamics

P19

AC

CE

LER

ATION

INP

OLA

RC

OO

RD

INATE

S

(See

Section

2.1ofH

andout)

•S

imilarly,w

ecan

work

outtherate

ofchangeofvelocity:

r=

ρe

ρ+

ρ˙eρ

︸︷︷︸+

ρφe

φ+

ρφe

φ+

ρφ

˙eφ︸︷︷︸

φe

φ−

φe

ρ

=(ρ−

ρφ

2)︸

︷︷

︸e

ρ+

(2ρφ

+ρφ)

︸︷︷

︸e

φ

radialtransverse

•The

z-m

otionis

independent:(r)

zis

justze

zsince

˙ez

=0.

•The

radialaccelerationis

ρ−ρφ

2,thesecond

termbeing

thecentripetalacceleration

requiredto

keepa

particlein

anorbitofconstantradius.

•The

transverseacceleration

is2ρ

φ+

ρφ

=1ρ

ddt

(

ρ2φ

)

andshow

sthatitis

relatedto

the

angularmom

entumperunitm

assρ2φ

.

•S

phericalpolarscan

betreated

byputting

r=

re

r ,andexpanding

retc.

with

˙er

expressedin

terms

ofe

r ,e

θand

eφ

.W

eshallnotneed

thishere,as

it’sslightly

complicated,butifyou

havecom

puteralgebraavailable

it’svery

useful...

ClassicalD

ynamics

P22

RO

TATING

FRA

ME

S

ω××× ××e

z

ez

ey

ex

(See

Section

2.3ofH

andout)•

Case

2:Fram

eS

rotatesw

ithangularvelocity

ω,so

that

theunitvectors

rotatew

ithrespectto

theinertialfram

eS

0 .

•The

rateofchange

isgiven

by˙ez

=ω

××× ××e

zetc.

Letthefram

escoincide

at t=

0:

r0

=xe

x+

ye

y+

ze

z=

r

r0

=xe

x+

x˙ex

+y

andz

terms

=v

+ω

××× ××r

,

v≡

xe

x+

ye

y+

ze

zis

theapparentvelocity

inS

.

•The

accelerationin

S0

isr

0=

xe

x+

2x˙ex+

x¨e

x+

yand

zterm

s

=xe

x+

2(ω××× ××e

x)x

+ω

××× ××(ω

××× ××e

x)x

+y

andz

terms

=a

+2ω

××× ××v

+ω

××× ××(ω

××× ××r)

where

a≡

xe

x+

ye

y+

ze

zis

theapparentacceleration

inS

.W

erew

ritethe

mom

entumequation

mr

0=

Fin

terms

oftheapparentquantities

r,v

anda

:

ma

=F

−2m

(ω××× ××v)−

mω

××× ××(ω

××× ××r)

•The

observerinS

addsC

oriolisand

Centrifugalforces

(inertialorfictitiousforces).

ClassicalD

ynamics

P21

FR

AM

ES

INR

ELATIV

EM

OTIO

N

(See

HandoutS

ection2.2)

•S

upposew

ehave

afram

eS

0in

which

mr

0=

F,w

ithF

ascribedto

known

physical

causes.W

hatisthe

apparentequationofm

otionin

am

ovingfram

eS

?

•C

ase1:

Suppose

r=

r0 −

R(t).

Suppose

theaxes

inS

0and

Srem

ainparalleland

t=

t0

(asalw

aysin

classicalphysics):r

=r0 −

R

•Forthe

specialcaseR

=0

(i.e.steadym

otionbetw

eenfram

es),mr

=m

r0

=F

,

i.e.thesam

eequation

ofmotion

(Galilean

transformation).

•ForgeneralR

(t)m

r=

mr0 −

mR

=F

−m

R

•The

apparentforcein

Sincludes

boththe

actualforcem

r0

anda

fictitiousforce−

mR

.

•Fictitious

forcesare:

(a)associatedw

ithaccelerated

frames;(b)proportionalto

mass.

•Q

uestion:Is

gravitya

fictitiousforce?

•A

nswer:

(accordingto

generalrelativity).Yes!

Gravity

isequivalentto

acceleration.

ClassicalD

ynamics

P24

CE

NTR

IFUG

AL

AN

DC

OR

IOLIS

FO

RC

ES

•C

entrifugalForce:−m

ω××× ××(ω

××× ××r).

−m

ω××× ××(ω

××× ××r)

=m

(ω2r

−r·ω

ω)

•C

entrifugalForcem

ω2ρ

outwards.

•C

oriolisForce:−

2m(ω

××× ××v)

appearsifa

bodyis

moving

with

respecttoa

rotatingfram

e.

•C

oriolisForce

isa

sideways

force,perpendicularbothto

therotation

axisand

tothe

velocity.

•P

roblems

involvingC

oriolisForce

canoften

bedone

byconsidering

angularmom

entum.

•A

dvice:D

onotm

eddlew

iththe

signsorthe

orderingofthe

terms.

Them

inussign

reminds

usthatthese

terms

came

fromthe

othersideofthe

equation,andω

××× ××(ω

××× ××r)

construction

reminds

usofthe

operatorrelation

[ddt

]

S0

=

[ddt

]

S

+ω

××× ××.

ClassicalD

ynamics

P23

RO

TATING

FR

AM

ES

(See

Section

2.3ofH

andout)

•There

isan

operatorapproachto

rotatingfram

esthatis

agood

aidto

mem

ory(and

isrigorous).

•Forany

vectorA

therates

ofchangein

frame

S0

andin

frame

S

arerelated

by

[dAdt

]

S0

=

[dAdt

]

S

+ω

××× ××A

•A

pplythis

operatorrelationtw

iceto

r(r

=r

0att=

0):[d

2r0

dt2

]

S0

=

([

ddt

]

S

+ω

××× ××

)(

[drdt

]

S

+ω

××× ××r

)

•E

xpandingand

setting

[drdt

]

S

=v

and

[dvdt

]

S

=a

we

recover

mr

0=

F=

ma

+2m

(ω××× ××v)+

mω

××× ××(ω

××× ××r)

•G

eneralcase:O

bservermoves

ona

pathR

(t)and

usesa

frame

rotatingatangular

velocityω

(t)w

hichis

alsochanging.

Fromprevious

resultsand,because

thetim

e

derivativenow

operateson

ωw

egetthe

generalformula:

ma

=F−

2m(ω

××× ××v)−

mω

××× ××(ω

××× ××r)−

mR−

mω

××× ××r

•The

mω

××× ××r

termis

calledthe

Eulerforce.

ClassicalD

ynamics

P26

OR

BITS

—C

EN

TRA

LF

OR

CE

FIE

LD

(See

Section

4.1ofH

andout)

F

O

P

r

v•

Particle

moving

incentralforce

field.PotentialU

(r)yields

radialforceF

=−

∇U

=−

dUdre

r .

•M

otionrem

ainsin

theplane

definedby

positionvector

r

andvelocity

v.

•N

ocouple

fromcentralforce ⇒

angularmom

entumis

conserved:

mr2φ

=J

=constant

(KeplerII)

•Totalenergy

isconserved:

E=

U(r)

+12m

(r2

+r2φ

2)=

12m

r2

+U

(r)+

J2

2mr2

•The

effectivepotentialU

eff(r)

hasa

contributionfrom

theangularvelocity.

Ueff(r)≡

U(r)

+J

2

2mr2

•The

effectivepotentialhas

acentrifugalrepulsive

term∝

1r2

.

ClassicalD

ynamics

P25

FIC

TITIOU

SF

OR

CE

S—

AP

PLIC

ATION

S

•C

entrifugalforcegives

riseto

theE

arth’sequatorialbulge:∼

Ω2Rg

≈1300

.

l

N

Yo

u a

re h

ere

W•

Coriolis

forcedue

tom

otionon

Earth’s

surface:F

=2m

Ωv

sinλ

Direction

issidew

aysand

tothe

rightinthe

Northern

Hem

isphere.

Independentofdirectionoftravel[N

SE

W].

F=

2m

vW

lco

s

zx

East

•C

oriolisforce

ona

fallingbody.

Starts

fromrestattim

et=

0.v

=gt

mx

=2m

gtΩ

cosλ⇒

x=

13gΩ

t3cos

λ

•Foucaultpendulum

.P

recessesatΩ

sinλ

.

Which

way?

•R

oundaboutsand

otherfairgroundrides.

Rollercoasters.

Low

Hig

h

Hig

h

Hig

h

N

w

Low

Hig

h

•W

eatherpatterns,tradew

inds,jetstreams,

tornados,bathtubs(??).

ClassicalD

ynamics

P28

OR

BITS

INP

OW

ER-L

AWF

OR

CE

(See

Section

4.2ofH

andout)

r

eff

U

r0

E0

rr

eff

U

0

r

eff

U

r0

U0

n≥

−1

−3

<n

<−

1

n<

−3

•The

potentialisqualitatively

differentfordifferentvaluesofn

:

n≥

−1:

Orbitatr

0stable.

Allorbits

bound.

−3

<n

<−

1:O

rbitatr0

stable.U

nboundorbits

forE>

0.

n<

−3:

Orbitatr

0unstable.

Willgo

tor

=0

orr=

∞

ClassicalD

ynamics

P27

OR

BITS

INP

OW

ER-L

AWF

OR

CE

(See

Section

4.2ofH

andout)

•W

ecan

gaina

lotofinsightintoorbits

bystudying

theforce

lawF

=−

Ar

nw

ithA

positive,

soforce

isattractive.

•The

effectivepotentialis

thenU

eff(r)

=A

rn+

1

n+

1+

J2

2mr2

theonly

exceptionbeing

n=

−1

(Ueff

thencontains

alog

rterm

).

•The

centrifugalpotentialisrepulsive

and ∝r−

2.A

plotofUeff(r)

shows

which

valuesof

theindex

nlead

tobound

orunboundorbits,and

which

leadto

stableorunstable

orbits.

•For n

≥−

1(including

thelog

rpotential),the

potentialincreasesas

r→

∞and

the

orbitsare

boundand

stable.

•For−

3<

n<

−1

thepotentialgoes

tozero

atr=

∞and

theorbits

caneitherbe

bound

orunbound.

•For n

<−

3the

attractionatr

→0

overcomes

thecentrifugalrepulsion

andthe

orbitsare

notstable(this

isthe

caseforthe

centralregionofblack

holesin

GR

).

ClassicalD

ynamics

P30

NE

AR

LYC

IRC

ULA

RO

RB

ITSII

•H

owdoes

ωp

oftheperturbation

compare

with

ωc

ofthecircularorbitatr

0 ?

ωc

=φ

=J/m

r20 .

Thereforeω

p=

√n

+3

ωc .

•The

simple

casesare

1.n

=1.

Forceproportionalto

r,i.e.simple

harmonic

motion.

ωp

=2ω

c ,givinga

centralellipse(Lissajous

figure).

2.n

=−

2.Inverse

squareforce.

ωp

=ω

c ,

givingan

ellipsew

itha

focusatr

=0

(planetaryorbit).

•G

eneralngives

non-comm

ensurateω

pand

ωc ,

with

non-repeatingorbits

(e.g.galactic

orbits).

Case

illustratedis

n=

−1.

ClassicalD

ynamics

P29

NE

AR

LYC

IRC

ULA

RO

RB

ITSIN

PO

WE

R-LAWF

OR

CE

r

eff

U

r0

U0

•LetF

=−

Ar

n,n

=index,w

ithcom

mon

cases

n=

+1

(2DS

HM

)andn

=−

2(gravity,electrostatics).

Ueff

=A

rn+

1

n+

1+

J2

2mr2

•N

earlycircular

orbitsare

oscillations/perturbationsaboutr

0 .

TaylorexpansionofU

eff

gives

Ueff

=U

0+

(r−r0 )

dU

eff

dr

∣∣∣∣r0

+12(r−

r0 )

2d

2Ueff

dr2

∣∣∣∣r0

+···

•A

tr=

r0

dU

eff

dr

iszero,giving

dU

eff

dr

=A

rn−

J2

mr3

=0

atr0 .

•The

secondderivative

ofUeff

isd

2Ueff

dr2

=nA

rn−

1+

3J2

mr4

=(n

+3)J

2

mr40

atr0 .

•U

singthe

energym

ethodddt

(12m

r2

+U

eff

)=

r

(

mr

+dU

eff

dr

)

=0

we

gettheS

HM

equationm

r+

(n+

3)J2

mr40

(r−r0 )

=0,

i.e.sim

pleharm

onicm

otionaboutr

0w

ithangularfrequency

ωp=√

n+

3J

mr20

.

ClassicalD

ynamics

P32

OR

BITS

ININ

VE

RS

ES

QU

AR

EL

AWF

OR

CE

•Inverse

squarelaw

force:F

=−

Ar2

.

•A

ngularm

omentum

:J=

mr2φ

Energy:

12m

r2

+J

2

2mr2−

Ar=

E.

•S

lightlyeasierto

work

with

u=

1/r:r

=dr

dφ

φ=

−φr2du

dφ

=−

Jm

du

dφ

.

•S

ubstituteinto

theenergy

equation

(du

dφ

)2

+u

2−2mJ

2(E

+A

u)

=0.

Com

pletesquare:

(du

dφ

)2

=e2

r20

−(

u−

1r0

)2,w

heree2

r20

≡2m

E

J2

+1r20

andw

ehave

definedr0

=J

2

mA

,theradius

ofthecircularorbitw

iththe

same

J.

Standard

integral:du

√

e2

r20 −

(

u−1r0

)2

=dφ⇒

u=

1r0

(1+

ecos(φ−

φ0 ))

•E

quationofconic

section:r0

=r(1

+ecos

φ)

•Fora

repulsivepotential

r0

=r(e

cosφ−

1)

ClassicalD

ynamics

P31

KE

PLE

R’SL

AWS

•150A

DP

tolemy

-Earth

atcentreofsolarsystem

,with

Sun

rotatingaround

it,andthe

planetaryorbits

describedby

acom

binationofcircles

(epicycles).

•Tycho

Brahe

(1546-1601)made

observationsofplanetary

andstellarpositions

toan

accuracyof10

arcsec(the

resolutionofthe

eyeis

1arcm

in).

•Johannes

Kepler(1571

-1630)spent5years

fittingcircles

tothe

Brahe’s

dataforM

ars’sorbit

andfound

differencesofthe

orderof8arcm

in.R

ejectedm

odelbecauseofknow

naccuracy

ofBrahe’s

measurem

ents.

•E

ventuallyK

eplerconcludedthatthe

orbitsw

ereellipses.

•K

epler’sLaw

s

•FirstLaw

:P

lanetaryorbits

areellipses

with

theS

unatone

focus.

•S

econdLaw

:The

linejoining

theplanets

tothe

Sun

sweeps

outequalareasin

equal

times.

(Implies

conservationofangularm

omentum

.)

•Third

Law:

Thesquare

oftheperiod

ofaplanetis

proportionaltothe

cubeofits

mean

distanceto

theS

un(itis

proportionaltothe

cubeofthe

lengthofthe

orbit’sm

ajoraxis).

ClassicalD

ynamics

P34

INV

ER

SE

SQ

UA

RE

OR

BITS

—A

LTER

NATIV

E

(See

Section

4.3.1)

•S

hapeoforbit.

Since

thevectors

J,v

and˙er

havem

agnitudesm

r2φ

,A/m

r2

andφ

respectivelyand

arem

utuallyperpendicular,w

em

ayw

rite

J××× ××v

=−

A˙er

(thesign

isobtained

byinspection).

•S

inceJ

isconstant,the

equationm

aybe

integratedto

giveJ

××× ××v

+A

(e

r+

e)

=0,

where

eis

avectorintegration

constant.

•Taking

thedot-productofthis

equationw

ithr

gives

J××× ××v·r

︸︷︷

︸+

A(r

+e·r

)=

0

=J·v

××× ××r

=−

J2/m

•Therefore

r(1+

e·er )

=r(1

+ecos

φ)

=J

2

mA

=r0 ,

which

isthe

polarequationofa

conicw

ithfocus

atr=

0(K

epler’s1stLaw

).

•The

majoraxis

isin

thedirection

of e;e

isthe

eccentricity:a

circle(e

=0);ellipse

(e<

1),

parabola;( e=

1)orhyperbola(e

>1).

ClassicalD

ynamics

P33

INV

ER

SE

SQ

UA

RE

LAW

—E

LLIPTIC

AL

OR

BITS

(E<

0)

O

P

fr

sem

i-latu

s re

ctu

m0 r

rm

in=

r0

1+

e

rm

ax

=r0

1−e

•E

llipseofeccentricity

e(0

<e

<1).

Centre

ofattractionatone

focus.

•Polarequation:

r0

=r(1

+ecos

φ)

r0

iscalled

thesem

i-latusrectum

.

•C

artesianequation:

r=

r0 −

ex

y2

+x

2(1−e2)

+2er

0 x=

r20 .

Setx

′=

x+

r0 e

1−e2⇒

y2

+(x

′)2(1−

e2)

=r20

1−e2

Ellipse:

(x′)

2

a2

+y2

b2

=1

;a

=r0

1−e2

;b

=r0

√1−

e2

O

P

r

rφ•

Area

ofanellipse

isπab

=πr20

(1−e2)

3/2

•Period

Tis

Area

Rate

ofsw

eepin

gou

tarea

.

•R

ateofsw

eepingoutarea:

12r2φ

=J2m

,

henceperiod

T=

2πr20 m

J(1−

e2)

3/2

=2π

√

ma3

A(K

epler’s3rd

Law).

ClassicalD

ynamics

P36

ELLIP

TICA

LO

RB

ITS—

IMP

OR

TAN

TT

HIN

GS

TOR

EM

EM

BE

R

OC

P

sem

i-majo

r axis

sem

i-latu

s re

ctu

m

fsem

i-min

or a

xis

b

a

0 rr

•The

equationofan

ellipsein

polarcoordinates

r0

=r(1

+ecos

φ)

•The

distancesofclosestand

furthestapproachfollow

fromthis:

rm

in=

r0

1+

eand

rm

ax

=r0

1−e

•The

semi-m

ajoraxisa

satisfies2a

=rm

in+

rm

ax

⇒a

=r0

1−e2

sothatr

max,m

in=

a(1±

e).A

lsob

=r0

√1−

e2

•The

semi-m

ajoraxisa

determines

theenergy

andthe

periodofthe

orbit

E=

−A2a

;T

=2πω

;ω

2=

A

ma3

•The

semi-latus

rectumr0

determines

theangular

mom

entumofthe

orbit:J

2=

Am

r0

•Ifyou

needto

deriveany

oftheseform

ulaein

ahurry

considera

sillycase.

Use

thesim

plebalance

offorcesargum

entforcircularmotion:

Ar2

=m

ω2r

.

ClassicalD

ynamics

P35

ALTE

RN

ATIVE

—E

NE

RG

YO

FTH

EO

RB

IT

•To

gettheenergy

takethe

scalarproductofAe

=−

(J××× ××v

+Ae

r )w

ithitself

(notethatJ

andv

areperpendicular).

A2e

2=

J2v

2+

2J

××× ××v·e

r︸

︷︷

︸A

+A

2

=J·v

××× ××e

r=

−J

2/mr

Therefore

A2(e

2−1)

=J

2

(

v2−

2A

mr

)

=2E

J2

m=

2AE

r0

where

Eis

thetotalenergy.

Them

ajoraxisofthe

orbitisgiven

by

2a=

r0

(1

1+

e+

1

1−e

)

=2r

0

1−e2

=−

AE

i.e.E

=−

A/2a

,independentofeccentricity.

•The

distanceofclosestapproach

occursatφ

=0:

rm

in=

r0

1+

eand

thedistance

of

furthestapproachoccurs

atφ=

π,atr

max

=r0

1−e

.

•The

semi-m

ajoraxisa

satisfies2a

=rm

in+

rm

ax ,so

thata=

r0

1−e2

.

•S

ubstituting,we

findthe

usefulrelationsrm

ax

=a(1

+e)

andrm

in=

a(1−

e).

ClassicalD

ynamics

P38

ELLIP

TICA

LO

RB

ITS—

AN

OTH

ER

(NO

N-EX

AM

INA

BLE)

WAY

•R

eturnto

theenergy

equation:12m

r2

+J

2

2mr2−

Ar=

E.

•C

hangethe

independentvariable,defininga

radially-scaled“tim

e” rds

=dt

sothat

drdt

=1r

dr

ds

We

thenget

(dr

ds

)2−

2Emr2−

2Amr

+J

2=

0

•D

efiningΩ

2=

−2E

m(rem

emberE

isnegative)w

egetthe

same

formofequation

as

before,butfor rinstead

ofu.

Thedistances

offurthestandclosestapproach

area(1±

e),

sothe

solutionis

justr=

a(1−

ecos

Ωs).

•W

ecan

nowintegrate

r=

dt

ds

tofind

anice

parametric

formforthe

time

t

t=

as−

aeΩ

sinΩ

s.The

periodis

T=

2πa

Ω=

2π

√

ma3

A.

•There

isalso

asim

pleclosed

formforφ

(s):tan

(√

1−e2φ2

)

=

√

1+

e

1−e

tanΩ

s2.

•This

isrelated

tothe

“squareroot”ofthe

Keplerproblem

,which

transforms

theorbitto

a

centralellipse(2-D

SH

M).M

anyessentialfeatures

ofthissolution

were

known

toN

ewton,

buttheprocedure

iscalled

the“K

ustaanheimo-S

tiefel”transformation.

ClassicalD

ynamics

P37

TIM

EIN

OR

BIT

—(D

ETA

ILSN

ON-E

XA

MIN

AB

LE)

•W

ehaven’tso

fargotaform

ulaforthe

coordinates(r,φ

)as

afunction

oftime.

Thereis

an

easyw

ayofdoing

this,which

iscom

inga

bitlater,butwe

canin

factgetaform

ulafort(φ

),

ratherthanφ(t).

It’sjustnotvery

pretty...

•From

theequation

oftheellipse

andh≡

Jm=

r2φ

we

get

dφ

(1+

ecos

φ)2

=hdt

r20

•The

integralisfound

asform

ula2.551

ofGradshteyn

&R

yzhik:

t=r20

h(1−

e2)

(−esin

φ

1+

ecosφ

+2

√1−

e2tan

−1

(tan

(φ+

14π)+

e√

1−e2

))

OC

A

P

•You

canalso

getaform

ulaforitby

subtracting

thearea

ofthetriangle

CP

Ofrom

thesectorC

PA

.

ClassicalD

ynamics

P40

GR

AVITATIO

NA

LS

LING

SH

OT

•The

escapevelocity

ofaspacecraft

fromthe

Solarsystem

attheradius

oftheE

arth’sorbitis

42km

s−

1,

which

shouldbe

compared

toits

orbitalvelocityof30

kms−

1.

•W

ecan

usea

gravitational‘slingshot’

aroundplanets

toincrease

kinetic

energyand/orchange

direction

inorderto

visitotherbodies

inthe

Solarsystem

.

•Voyager2

made

a‘grand

tour’.

ClassicalD

ynamics

P39

EX

AM

PLE:

TH

EH

OH

MA

NN

TR

AN

SFE

RO

RB

IT

Dv

2

2Dv

1

1

aa

•The

Hohm

anntransferorbitis

onehalfofan

ellipticorbit

thattouchesboth

theinitialorbitand

thedesired

orbit.

•Forgravitationalcase

putA=

GM

m.

•In

circularorbitsT

=−

E=

−12U

,forelliptical

orbits 〈T〉=

−〈E

〉=

−12 〈U

〉(V

irialTheorem).

•The

initialenergy(fora

spacecraftofunitmass)is

E1

=−

GM

2a1

,andthe

velocityhas

tobe

increased

untilthespacecrafthas

theenergy

ofthetransferorbitE

t=

−G

M

a1

+a2

.

•The

importantthing

toknow

is∆

v1 ,since

thatdetermines

theam

ountoffuelused,butwe

canw

orkallthatoutfrom

theenergies:

Et=

−G

M

a1

+a2

=−

GMa1

+12v2t1 ⇒

12v2t1

=G

Ma2

a1 (a

2+

a1 )

•The

restofrelationsare

easyenough,butnotparticularly

informative...

•The

Hohm

anntransferis

them

ostfuelefficientorbit,unlessthere

areotherm

assivebodies

inthe

vicinity,inw

hichcase

youcan

usethe

gravitationalslingshot.

ClassicalD

ynamics

P42

TH

ET

WO

-BO

DY

PR

OB

LEM

AN

DR

ED

UC

ED

MA

SS

•The

two

masses

M1

andM

2orbitthe

centreofm

ass.

•E

achorbitis

anellipse

ina

comm

onplane

with

thecentre

ofmass

atonefocus.

Theellipses

havethe

same

eccentricityand

phase.

•The

importantcase

iscircularm

otion.

Mass

M1

isdistance

M2 r

M1

+M

2

fromthe

CoM

.

•B

alanceofforces

forM1 :

GM

1 M2

r2

=M

1 ω2

M2 r

M1

+M

2

⇒ω

2=

G(M

1+

M2 )

r3

•This

isthe

reallyim

portantresultandis

usuallythe

beststartingpoint(e.g.

inproblem

Q12).

•You

getthesam

eresultby

consideringthe

balanceofforces

forM2 .

•W

ecan

rearrangethe

resultasµrω

2=

GM

1 M2

r2

,inw

hichw

ehave

thetrue

separationr

andthe

actualforceG

M1 M

2

r2

,butam

odifiedreduced

mass

termµ≡

M1 M

2

M1

+M

2

.

•Ido

notadvocatethe

useofreduced

mass,despite

itsw

idespreaduse

inthe

textbooks...

ClassicalD

ynamics

P41

GR

AVITATIO

NA

LS

LING

SH

OT

—V

OYA

GE

R2

ClassicalD

ynamics

P44

HY

PE

RB

OLIC

OR

BITS

OC

P

impact p

ara

mete

r

sem

i-latu

s re

ctu

m

sem

i-majo

r axis

ff

Path

for re

puls

ive fo

rce

b

a

0 r

r

8

c

•A

ttractivepotential:

allpreviousform

ulaestillvalid,

bute>

1so

a<

0and

energy

E=

−A2a

=(e

2−1)A

2r0

>0.

•Im

pactparameter b

andvelocity

atinfinityv∞

determine

angularmom

entumJ

=m

bv∞

andenergy

E=

12m

v2∞

.

•M

ostproblems

(e.g.R

utherfordscattering

Q14)

requirethe

totalangleofdeflection

χ=

2φ∞

−π

(χpositive).

•A

symptotes

areat ±

φ∞

;fromthe

equationofthe

conicsection

(validfore

<1

ore>

1)

we

havecos

φ∞

=−

1/e⇒

secφ∞

=−

e⇒

tan2φ∞

=e2−

1

.•

Note

thatπ/2

<φ∞

<π

.

ClassicalD

ynamics

P43

TH

ET

WO

-BO

DY

PR

OB

LEM

•Tw

oparticles

ofmasses

M1

andM

2orbiting

eachother—

positionsr

1and

r2 .

M1

M2

2r

1r

Co

M

•The

energy,angularmom

entumand

equationsofm

otioncan

beexpressed

interm

softhe

reducedm

assµ≡

M1 M

2

M1

+M

2

andr

1 −r

2 .

•The

centreofm

assis

at R0

=M

1 r1

+M

2 r2

M1

+M

2

.

•D

efineρ

1≡

r1−

R0

=M

2

M1

+M

2

(r1−

r2 )

ρ2≡

r2−

R0

=M

1

M1

+M

2

(r2−

r1 )

•K

ineticenergy

inthe

centreofm

assfram

e:

T=

12M

1 ρ21+

12M

2 ρ22

=12

(M

1 M22

(M1

+M

2 )2

+M

21M

2

(M1

+M

2 )2

)

(r1−

r2 )

2=

µ2(r

1−

r2 )

2

•A

ngularmom

entum:J

=M

1 ρ1××× ××ρ

1+

M2 ρ

2××× ××ρ

2=

µ(r

1 −r

2 )××× ××(r

1 −r

2 ).

•E

quationsofm

otion:M

1 r1

=F

12

;M

2 r2

=F

21

=−

F12

r1 −

r2

=

(1

M1

+1M2

)

F12

=1µ

F12

;R

0=

0

•R

educedto

theone-body

problemin

thecentre

ofmass

frame.

ClassicalD

ynamics

P46

RU

THE

RFO

RD

SC

ATTER

ING

ClassicalD

ynamics

P45

HY

PE

RB

OLIC

OR

BITS

II

(See

Section

4.4ofH

andout)

OC

P

impact p

ara

mete

r

sem

i-latu

s re

ctu

m

sem

i-majo

r axis

ff

Path

for re

puls

ive fo

rce

b

a

0 r

r

8

c

Attractive

potential

•W

eneed

tofind

theeccentricity

fromthe

physicalparameters

Eand

J.

•From

thedefinition

ofr0

=J

2

mA

we

canw

rite

(e2−

1)=

2r0 E

A=

2J2E

mA

2.

•In

terms

of band

v∞

thism

eans

tan2φ∞

=e2−

1=

m2v

4∞b2

A2

⇒tan

φ∞

=m

v2∞

b

A.

Repulsive

potential:

•C

hangeAr2→

−Br2

,defineJ

2=

Bm

r0

anduse

otherbranchr0

=r(e

cosφ−

1).

•a

ispositive

againand

thetotalangle

ofdeflectionis

nowχ

=π−

2φ∞

(χnegative).

•The

asymptotes

arestillrelated

tothe

physicalparameters

bytan

φ∞

=m

v2∞

b

B.

•The

distanceofclosestapproach

isa(1

+e).

ClassicalD

ynamics

P48

HY

PE

RB

OLIC

OR

BITS

INR

EP

ULS

IVE

PO

TEN

TIAL

—A

NO

THE

RW

AY

OC

P

impa

ct p

ara

me

ter

se

mi-la

tus re

ctu

m

se

mi-m

ajo

r axis

ff

Pa

th fo

r rep

uls

ive

forc

e

b

a

0 r

r

8

c

•W

ecan

simply

usethe

resultsforthe

attractive

potentialcase(r

0=

r(1+

ecos

φ))

andletφ

exceedφ∞

.

•The

radiusr

isthen

negativeand

theparticle

tracesoutthe

repulsivebranch,getting

closest

toO

at φ=

π,so

thatr(π)

=a(1

+e),

where

ais

nownegative.

•This

works

becausethe

potentialenergy−Ar

ispositive

when

r<

0and

sorepresents

arepulsive

potential.

•This

approachhas

theconsiderable

advantagethatno

signchanges

areneeded,butithas

something

ofa“A

licethrough

thelooking

glass”qualityto

it.

ClassicalD

ynamics

P47

RU

THE

RFO

RD

SC

ATTER

ING

II

ClassicalD

ynamics

P50

GR

AVITATIO

NA

LP

OTE

NTIA

L,FIE

LDA

ND

TIDA

LFO

RC

ES

Therethree

importantaspects

togravitation

(New

tonianorG

R).

•G

ravitationalpotentialφ(r

)This

determines

energiesand

redshifts;velocitiesof

objectsand

temperature

ofgases.A

lways

relative—

can’thavean

absolutevalue.

Forpointmass

φ=

−G

MR.

[New

tonianpotentialis

onepartofthe

metric

ofGR

.]

•G

ravitationalfieldg(r

)=

−∇

φThis

determines

accelerationsand

orbits.The

fieldis

alsorelative

(perhapssurprisingly).

e.g.w

e(and

theLocalG

roupofgalaxies)could

allbeaccelerating

toa

“GreatA

ttractor”andnothing

would

changehere.

Forpointmass

|g|=

GMR2

.[G

ravitationalfieldis

onepartofthe

“connection”inG

R].

•G

ravitationaltidalfieldR

(a)

=a·∇

gThis

isallone

canfeeland

measure

locally—

itdescribeshow

thegravitationalfield

variesin

space.In

components

[R(a

)]i=

Rij a

jw

hereR

ij ≡∂g

i

∂x

j.

Thegravitationaltidalfield

variesas

1/R3.

Forapointm

assitis

R(u

r )=

2GM

R3

ur ;

R(u

θ )=−

GMR3

uθ ;

R(u

φ)=−

GMR3

uφ;

Thereis

aradialstretching,and

asquashing

byhalfas

much

inthe

transversedirections.

Thetidalacceleration

tensorRij

givesthe

coefficientsofthe

quadraticterm

inthe

Taylor

expansionofthe

gravitationalpotential.[Tidalfield

ispartofthe

Riem

anntensorofG

R].

ClassicalD

ynamics

P49

TH

ET

HR

EE-B

OD

YP

RO

BLE

M

•S

ome

hierarchicalsystems

canbe

stableindefinitely

e.g.S

un,Earth

andthe

Moon.

•A

general3-bodyencountercan

bevery

complicated,buta

generalfeatureem

erges.

•If3

bodiesare

allowed

toattracteach

otherfroma

distance(a),they

willspeed

upand

interactstrongly(b).

Eventually

theinteraction

islikely

toform

aclose

binary(negative

gravitationalbindingenergy)releasing

kineticenergy,w

hichm

aybe

enoughforthe

bodiesto

escapeto

infinity(c).

•This

mechanism

isresponsible

for“evaporation”ofstarsfrom

starclusters.(m

aybealso

invalidatesthe

virialtheorem?)

•The

planetPluto

hasa

closecom

panionC

haron,andhas

aneccentric

orbitwhich

takesit

insideN

eptune’sorbit.

A3-body

collisionam

ongstNeptune’s

moons

isthe

mostlikely

cause.

ClassicalD

ynamics

P52

OR

IGIN

OF

THE

TIDE

S

•Looking

down

onthe

orbitalplane,we

see

theE

arthrotating

underatidalbulge

ofwater,

making

two

tidesa

day(≈

1hrlaterperday).

•The

tidalaccelerationin

theE

arth-Moon

direction

is3G

M2 z

r3

,where

zis

thedistance

fromthe

centreofthe

Earth.

•Integrating

togetthe

tidalpotentialwe

findφ

tide

=−

3GM

2 a2

2r3

atthesurface

atpointA.

•There

isno

tidalpotentialatpointB,so

theheighth

ofthetide

isgiven

byφ

tide

=−

gh

,

where

thegravity

g=

GM

1

a2

.

•E

liminating

g,the

heightoftidesis

h=

3M2 a

4

2M1 r

3=

0.5m

.

•C

anbe

lessdue

tolim

itedflow

(Mediterranean),orm

uchm

orein

estuaries(e.g.

St.

Malo

andB

ristolChannel).

Can

becom

plicateddue

toam

plificationby

resonances(e.g.

Solent).

•Tide

fromthe

Sun

abouthalfthatfromthe

Moon

—explains

“spring”and“neap”tides.

•The

Moon

nowkeeps

thesam

eface

towards

us—

itsinitialadditionalrotation

was

dissipatedagainstE

arthtides

of 16m

.Itw

asonce

much

closertous,and

itisstillreceding.

(Therotation

oftheE

arthis

alsoslow

ingdow

n.)

ClassicalD

ynamics

P51

MO

RE

AB

OU

TTID

AL

FOR

CE

S

•The

gravitationalfieldneara

pointmass

isdirected

radiallyand

isproportionalto

1/r2.

The

tidalforcesconsistofa

radialstretching2G

M/r

3and

asidew

ayscom

pression−G

M/r

3.

•Forthe

two-body

potentialwe

mustalso

addthe

contributionfrom

centrifugalforce—

thisis

astretching

intw

odirections

inthe

planeofrotation,and

nocontribution

inthe

directionof

therotation

axis.This

assumes

ourstickm

anis

corotatingw

iththe

orbit(i.e.keeping

the

same

relativeorientation

with

respecttothe

mass

M).

•The

sumis

astretch

of 3GM

/R3

alongthe

radialdirection,nocontribution

inthe

orbital

planeand−

GM

/R3

perpendiculartothe

plane.

•Tidalforces

arew

eakon

Earth

andnotvery

strongin

thesolarsystem

(exception:Jupiter

andIo),buttidalforces

canbe

colossalnearcompactobjects

suchas

neutronstars.