Download The lagrangian Approach-Classical and Relativistic Mechanics-Lecture Handout and more Exercises Classical and Relativistic Mechanics in PDF only on Docsity! 28 Mel. 20057
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The Prchislary of Ke Lagrangian Approach
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t
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Olen , bak mol ahooys, the achon 5 wirimized se Px
is called Ae Prarciple of Least Ack. Why dees
newre lke by minimize aca, ? = And why Hs ach:
jk-vat ? Noble K+V és conserved, So energy
sloskes back ad Cort bebreer “setual® Chnnete)
& pole-tial barns. So the Lagrangian L=K-V
Me asureS “happeningness > potential b heppen , £
we're. seeing natree likes. & minimize te btel of
thes over hime :
Fo exomple : Proveelile neha
i KV sel on. good!
Spend as mack bre os passble 9
doing Hairs!
oN ev bi we bad !
get Uris ove HIM quickly!
Abbe reloted gresha, 5 bow ded peeple discover Kes
prsccple ? Lel's book of bo. predecesses
I, He " Pemeiple of Least Energy
Bele dynamics , ply scx iste shdiecd stobres : ~
the shedy ot obecks at rest a "Mh eqilideiun,”
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Archimedes shdted @ See-Sav », ow lever
He Sans Hop hr wold be mh eqes liberties if
mL, = wo,
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Lobe DiAlenbo t undestad His using bis Prirciple
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werk dove by tas mahion ts zero,
ait
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The Persigle of Leash 7 ime
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Hab nabare likes by minimize thas, Ligtt DM 2 Vacuum maves
Aw sheaighl les , tubick minimize distarce | Bat moe’
inberestrgly , consider 2 mirroe
A c
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~B
Wht polh does the light chose? — Enpiical orsner
if: chooses B gnk Hel 6-6: "angle ot Metderce
(6) equals angle of retlecbion C6’). Ths is precisely wo
te po. phat princnizes the dalare subjecl by Ke
comrstrant Het if torches He miro Cot of heocl me posh)
Zn Fock Uhl dofes blk He shentt pot AC & He
pet ABc., Why dees pe pol ABc Winhinni 9 @ He
distance Conbject Lb He emnzhrert ) vber o =Q’ é
B riimizes ABTBC E> Bo mrinrees AB+BC’ ~~
—& ABC’ Vie ma Ine
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The real clue , however , come Fam He shedy af velrechon
wrediumy
mediun 2
Snell (4 predecessors ) nohked Ht ea median hes come
h i, He “ade ot rebreckio”® sh
nuevber n asseciahed
n an &, = n, &~ 6,
CWe norne/ ine. n sf, nv al a Vecune IS 4.5
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ok lig M 2 thediuen fs Bo pom Lin ol ty 4
pp n>?
~ de light will sobs Snell's lew if it's byry bh
BM imize tre. éb hehe s b gel fea. A b&b Cc.
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How D’Ale—be + 2 eters gueesed the buh.
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D'Ale~be f ge-erelized His by dynamics by Movewbug
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My dyno ies , egeilibrinen oceunre whe. Uy ip bol be, —
Fo+ hebig fece > verishes | om
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(Flg(t)- malt)) + $q() =0
New, leks imeyme patts
qs) = alt)+ s Soft)
where
Salt.) = S$(E) = 6
QR Coe ay Panel fF on He spare of pots |
watt delve He veriahenol derivative.
~s
sf = ge lg.)
3S=O
Tro. DAb be t's prv<vple implees
I (Fg)-mi ): $4 dt = 0
Gh all Sq, 50 if Fa -VV we get
{" (- VVlq)- mg) Sq At = 0
t ah by ph
= y (-V Wa) * bq + ing 84, ) at ~
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