Download Lagrangian-Classical and Relativistic Mechanics-Lecture Handout and more Exercises Classical and Relativistic Mechanics in PDF only on Docsity! 18 Apel 2005"
(rive the Lagramgion he e porlicle wil mess m f cherye em
an elecbomegnebre vec toe pole bied A:
Lly,4)> Dall + Aig!
let’, work oul tee E-l equabions ;
2 hm bi
Pen ag aay tei
= mv +t eA;
where ve R™ is the velocihy normalized Se Hot lui =
~~ Nole t Vow Momentum TS ws longer Mees times velocity |
The Cece iS
~ at dA; |:
Foo ee Be leaghy = 0 9!
3s ii’) ea’
Se E-t eons Sam
_ _ partis Aclgl&Y
P ~ x , .
4 l(mv teh) = 2A; .
ae Yj ‘ esy7
q' 7
Ca OA. |: |
Mat = & 3G? - eTPA;
_ a4 q dt
maa aA; si _ 4 Webs
dt et 343 t
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Fi. —— fe @ lec bonegachic
freld Fad,
So we get a “4
he oe 6.
fm a é ij 9° | He lerente te Lb
(scaly called he "Lorena Goce lew)
We've looked al 2 pout pebrle & bred
S= meng fa
No pape NA Ifo com be
lected over a
I-densiah pelt .
T. skin Heon we boost He drensim by | 8
to-sthe 2 Sky bree ma ot » 2 sur face 2 bywne
\ becomes \\
peesee
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43
20 Apel 2005
L Startng From the Lagrangian for 2 relebwishe changed porkicle
in Ow elechomegnetic Geld
L= Amaqig’ + Aig!
let's work oub the Hamil boniaw . Recall Hoek be or
reparereleizelte, ~nveriant Lagrangians
L= Uj rede’
ne 7 te 1 Hine bronslaha wes 2 ge9e- syrmelry .
Now ‘ ‘5 nor.
H = pid -L
Bd now
= ab ‘
oT SR = mg, + eA;
Pre 43 2
So
H = (mg; +eA)g' - (40959! + eA,4*)
= $mgiq’
(vegeely like how e@ ronrela hwisk ¢ pe liele “2a pote tial V
has H=pginl = 2k-(k-wy a Kav, bE aw He
“polebiel™ V=eA.g' is Incanto velocity 34 how
NY
H=piai ml = (y-y)-(e-V) = K)
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Now H js mat tere, 2 the Cab Heb ih’s aseved
Seu 4c ts cosh a4, 2 Funcben ef t, 0
He perbele's pet By perereleited by pare bree,
up bh res calne of t. Tue. we've getting “ensevelln
\
ol ‘speed rather Her sare move Ponilier “con servelin
of oreryy.” The VC 25 a5 1s, thes ayes btow ion ene s
Frame He symmebry
9.) = g lt+s)
eT3 Ne
wee
hsheed of spece hee brenslebo
symone hry
glt=9(t)+ sw weR™
f"
One Legrangisn
Llg,4) = plait + AG
hea tre \ranslalen syrmely eft A is dranslobn,
Mueriank qT ge-eel , Yer , Here's us tnse-ve gf
ee
erergy” Ce ger pelle com. b fronsblon.s im bre.
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4s
The Ge-evel Reletivishe Parbile
~ qT GR , specebre Q Ts an (nt D-domensinn ol lwetate..
menilold romely 2 smeol (oD -dimensacd me bolf wi
e laverbziag weber ic 9 > narnely
') a exh xeQ , we heve 2 bilmeer mop
g) : T,Q * TQ — R
(v, w) '&———9 gy, w)
lon glu) Co shack)
d) Dans 2 Wik respect by some basis of TG we
— hove g(r) = 55 vin
whene ! ~ ©
q32( 7
O 7
(ot course we con wv ihe gfv,-) = 41) v' ir) mn oy
bests , but (- mosh bases 93 will hee 2 d- fleet
Cre)
3) gh) veries seeelhly
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