II-

a -

sheet

Exercise I :

-

tous: in - Xr"

= o

Tints : X

.

. X'

= o

2t 72=0

general periodic solution :

4444 =

xrtdpktifn-2.tn/qe-ink-rltIIeinkerYk--DXoCo,rl- o -

-

xotifEE.tn/xoneinr-a:ein7

⇒ Ko=o,

I

In = Lon-

-

X.Tort-

- R -

- a' pot if off . inxoneinina: e- in

]-

- image- inn

= a' pot FIE,

Chon eine + ageing

⇒

xiA°yTogether : /X°C4o1=R#

IFIXYorkxi-ifEE.aafxieinr-E.EEI Rz ( ear+ e

- is)

⇒ X , d^n=I^n=ofrnI#

Erleiree-irt-fweir-aie-ioy-GIe-ifa.net/IitE=feirca:-aIite-irca

: - a :D if

⇒ { irala - II ) -

- R( * )

i Rai C -

an to ? ) = R

X.

tort= x'

p 't iFE±o£ find einr.in Eine-

ing

= I pie FE ? ,

lanein '

+ IIe- into

⇒ II = - x ?,

I ? = - LI, p

In CH : iFai=ER= - ifa ;-

if a :-. - IR = - i FIE :

-

Altogether :

XY-e.at -

- FR ( eir 't e- it

; e- irteiot )

= ER ( cost - r ) t cos ( etat )

µarl⇐RaszcTT-

- 3 -

Similarly

X'

= ER ( sin lttrl - sink - o ))

⇒

IXKRca-cs.in#-/XiC-er ) so

,

i = 3,

.

- .

,D - a

-

check constraints

X '=

- R sine cost,

X''

= - R cos c- sins

A2

= - R sine sins

, X' '

= R cost cost

X °

= R,

X"

=o

*i

= O

,Xi

'

=0 i =3,

. - -

,D -

a

o• Xoxo'

=o

,X.

n

Xi'

= R2 sine case sins cost

X.'

x"

= - R' sine caesuras a/⇒i.

• . Ho )'t IY 't CX42 = - R 't R-sinkcosk-RS.int sink

= - R2 cos-

I

- IX.'

I't Hd'

I 't #'

II otrcaksinrt R2 carcass

= R'

cost

⇒ (ill'

-11×42=0 ✓

five : - 4 -

Cal First

X.r = a'

pint fE¥.

Carne- ink -

Hearne - incurs )

IT'

= FINE.

( - ah e- ink - H

+ Inn e-

in leery

⇒ flirtXr

'

=a' p

-+ Fa Io Eze

-in " " I

f⇒£zeincurs

Kr - X'

= x' pre Fa ¥ anne-

ink - o '

=ff÷,

In e- into '

-

rain :

We have :

{ Me..at

,T like

.r 's I = z~S.ca - o

')-

puce.

a '

)

⇒ { Xi , I 3 = I y~ Fr Sca- a's GI

Moreover :

{ Ncaa,

X'

Car'll- - o ⇒it '

,x

"

} ⇒ El

{ p-

c -44,

p"

CT,

or93--0⇒Hr,X.of = o Cs)

calico ,⇒

Wtxf'

,XI. ix.

'

3

=

H.

-

+ x :

:3±

Hix:S-

± Ex :c,

iv. I

⇒ Cliatt'

,iii. ex : . ,

→ -

=

t.FI#sr-IEy~darAr-o7=oe*s

I Since dot ,Scr - e

' I = dffo Sto - at = - Str - on 't

da.

Sir - o 't =dcrd-EY-s.to - on 't =

Sir- ⑤

'I ]

Ex :±x :'

,

t.xiit.cm#IiIt.kIix:il--IEzndzSCr-o4=t-IITE;nein9¥'*,

- -

I Remember : 2-afcxt-zzeinx-sddzsco-rg-E.ie#eincr-t'

]

On the 1. L.s. of htt and Ct # I plug in the

Fourier modes for XIX ':

Htt

:{X rt Xr 't

'- XV

'

f =

2dn.qe-inkthe-imk-mknpomf.to⇒ { Inn,

Lum } = o for men E I

given that e- inkedand e

-innit - o "

arelinearlyindependent .

CAN : { Art XM,

XIX"

f-2dzme-inktrk-imktmgynn.gg

.

This should be compared with( using T= ¥ , ) :

Lia'

you En ein to - r' )

- 6-

= - List y~§n e-

ink - r'

!

- zijynuqne.in#rleinCeer'I

= 2×1 Tyme- inutile

- ink 's')

( -

ing~

fun

!⇒ V-n.me,

: { Inn,

I'm } = -

ing 'Smen ,

Analog Sir { In ,NmI= - iny~dm.in ,. from

He lower sign of c** ).

(b) { Lm,

In ) =o is obvious from Cal

Now first calculate

[ Lm, Ne } -

- ZE { dm-n.tn idk }

-

- EEKm-nbkk.akli-EEE.mx :3 Kalu- -

- ing" Snee

,.

-

ilm-nlzkfm-neqo-EKm.ie/uirzurtEEilCm-lmtrDYrKmodo

-⇒ I Elm,are } = iron

.

-

tElm

,ae.n.gl- it -

-

-

- E E Elm, de

.

is knbtEECG.io C Lurid-

- E I ice - uld knlui-EEKe.lu in EmeaMtr -

re

-

Ikea -utilitarianNIM -1M

= I ⇐ file - nltiln - ml ) the + m . n

'

An

= i Ce - m ) Lstm

=film-elLm#-

Similarly for { Em,

Ee ) =- i I m - et Emer

.

(c) { Meps,

like .es/--2a-a1z~SCo - o' I

Integrate both sides over a Elo, it )

.

On the e. b. s.

the terms with oscillators

in Xr IT,

r ) give no contribution and

on the v. e. s.

the delta function disappear

⇒ { x-

+ WE,

X.

heir = atyr

'

Given that IT is a sum of terms

involving an and II,

in view ofC 2.5 ) we have far

,it } = o

Thus we have - 8-

⇐ ffxn, Nn 's FEE- ink - "

+ exr.TL/Ee-inHtol ]= x

'

qmvG- ran that e-

in-

and e- in .tt

arelinearlyindependent ,we team

thro : { x-

,a :3 =o= { Xn

,I :}

and { Xm,

pv } = z~ .

(d) HE.#

xrtrk.in#irE,.E.Efxnne-inoi-IIe-iwJ⇒ flu

,

XY-flmxntrxkn.at/ttiFIEEee-imIn.xnu3Mto

( Now : { Ln ,xrl=EE{ an . e. he,

N }

= EE Exile,

xnxx.lu -1 EEK .eh Cai,XY

Bat : {In:÷'IE?"

} ⇒ cnn.xrt-sm.VE

⇒ { Ln ,x- I

-_⇐II. THE - Elan )uy~FI

rt

=- A In2-

and { Ln, If =

irate ]

- q -

Thus :- Fait if E. Fee

- in-

im Em

=- FIE

,

e- int

-

Ahem G1

-

On the other hand :

Xr -

-

xrtfdx.EC#.rTtiEE.oEfaze-inIzne-inryZE--Ezaki FIE.io#-ixlaie-inr-= FI 2- e

- ins-

armMete-

- FI ?,

e-

ilm-nlr-ynmn.LI-

FINE,

e-

in --

drain CH

-

(a),

(2) ⇒ ( 2. ro )