Mechancs Physcs 5 Lecure 9 Hamlonan Equaons of Moon (Chaper 8) Wha We Dd Las Tme Consruced Hamlonan formalsm Hqp (,,) = qp Lqq (,,) H p = q H q = p H L = Equvalen o Lagrangan formalsm Smpler, bu wce as many, equaons Hamlonan s conserved (unless explcly -dependen) Equals o oal energy (unless sn )
Hamlon s Prncple Lagrange s equaons were derved from δi δ Ld = 0 Hamlon s Prncple How does work n Hamlonan formalsm? Frs, le s remember how hs worked before Consder he pah q = q() and defne varaon by q (, α) q () + αη() ( α ) ( (, α ), (, α ), ) I f q q d di( α) δ I d α α = 0 dα Hamlon s Prncple q (, α) q () + αη() ( α) ( (, α), (, α), ) I f q q d A b of work shows di( α) δ I d α α = 0 di( α) f d f f = η() d η() dα + α = 0 dq d q q dα So δi = 0 s equvalen o = 0 a, f d f = 0 Lagrange s equaons dq d q
Hamlon s Prncple Rewre acon negral usng Hamlonan ( (,, )) δ I δ Ld = δ p q H q p d Now δ denoes varaon n he phase space.e. boh q () and p () are vared ndependenly q (, α) q () + αη () p (, α) p () + αζ () Calculaon of δi goes exacly he same way Jus consder n varables q and p nsead of q only f(, q q, p, p,) = pq Hqp (,,) Ths can be omed Hamlon s Prncple δi = 0 s equvalen o f d f f d f = 0 and = 0 dq d q dp d p H H + p = 0 q = 0 dq dp Well, ha was easy One subley remans he end pons f ( qq,, p, ) = pq Hq (, p, ) Hamlon s equaons 3
End Pon Consrans Varaon n Lagrangan formalsm requred δ q ( ) = δ q ( ) = 0 The end pons are fxed n he confg space Dervaon of Hamlon s equaons requres δ q ( ) = δ q ( ) = δ p ( ) = δ p ( ) = 0 More resrcve Or does? Wha we really need s f f η() () 0 q + ζ p = Bu f does no depend on p! Ths s 0 Ths doesn f ( qq,, p, ) = pq Hq (, p, ) have o be 0 So we need only δ q ( ) = δ q ( ) = 0 as before End Pon Consrans We may sll wan o requre δ q = δq = δ p = δ p = ( ) ( ) ( ) ( ) 0 Keeps q and p symmerc Adds flexbly o he defnon of he acon negral You can add me-dervave of any funcon F(q, p, ) df( q, p, ) I (,, ) pq H q p + d d Dfference would be df( q, p, ) d = [ F( q( ), p( ), ) ] δ = 0 d 4
End Pon Consrans df( q, p, ) I (,, ) pq H q p + d d df Choose for example F = qp qp qp d = ( (,, )) I pq H q p d Ths s no longer he Lagrangan Acon negral can be defned whou referrng o L Ths allows a larger se of coordnae ransformaon canoncal ransformaon for Hamlonan formalsm Wll do hs nex me, bu le s ake a sneak peek Canoncal Transformaon Lagrangan Hamlonan formalsm mean movng from he confguraon space o he phase space (,, ) q q n n ( q,, q, p,, p ) Lagrangan formalsm s nvaran under coordnae ransformaons such as Q = Q( q,, qn, ) Is Hamlonan formalsm nvaran under smlar bu more general ransformaons? Q = Q( q,, q, p,, p, ) n n P = P( q,, q, p,, p, ) n n n 5
Canoncal Transformaon Q = Q( q,, qn, p,, pn, ) P = P( q,, qn, p,, pn, ) Well, no. I s much oo general, bu There s a subse of such ransformaons canoncal ransformaons ha work We wll fnd he rules Hamlonan formalsm s more forgvng Goal: Fnd he ransformaon ha makes he problem eases o solve We may make coordnaes cyclc, as we dd by usng polar coordnaes n he cenral-force problem Prncple of Leas Acon Prncple of Leas Acon s a confusng erm Acon changed s meanng hsorcally I would raher call Hamlon s prncple as he prncple of leas acon Leas s no srcly rue Exreme would be he rgh word Le s follow Goldsen s usage for oday Prncple of leas acon s expressed as -varaon pqd = 0 acon Wha are hey? 6
-Varaon In δ-varaon, end pons are fxed n space and me Wha f we allowed me o vary? We make a physcal sysem o move from one sae o anoher, bu allow o ake as much me as wans Ths requres a dfferen knd of varaon q(, α) does no have o be q(, 0) a =, I may ake exra me, e.g. q( +, α) = q(, 0) How can we defne such varaon? -Varaon Vary he pah n he confg space Ths me, we do no fx he end pons.e. η ( ) 0, η ( ) 0 Wha s more, we vary he range of negraon +, + We defne he -varaon by + α + Ld = L( ) d L(0) d q (, α) q () + αη () Less resrcve form han wha we need 7
-Varaon In he confguraon space End pons of negraon move by q ( ) = q ( +, α) q ( ) s -order approxmaon q ( ) = q ( ) +δ q ( ) q () q () +δ q () + Same for We wll laer mpose q ( ) = q ( ) = 0 + These pons wll be he same -Varaon + Ld = L( α) d L(0) d + Consderng only s -order erms of and η() Ld = L( ) L( ) + L( α) d L(0) d Las erms are δ-varaon wh free end pons L d L L Ld = q d + q dq d q q δ δ δ Assume Lagrange s equaon [ δ ] Ld L p q = + δ [ δ ] Ld = p q 8
-Varaon Usng q ( ) = q ( ) +δ q ( ) q ( ) = q ( ) +δ q ( ) [ ] Ld L pq p q = + [ p q H ] = Ths s a very general form q () + [ δ ] Ld L p q = + q () +δ q () + Now we examne a more resrcve case Resrced Varaon We mpose hree condons q = 0 a =, Inal and fnal saes are fxed H s conserved H does no explcly depend on Vared pahs are resrced so ha H s consan Las wo are bascally energy conservaon -varaon s now [ ] Ld = p q H = H ( ) 9
Prncple of Leas Acon Ld = H ( ) From he defnon of Hamlonan Ld= ( pq H) d= pqd H( ) Comparng he wo equaons pqd = 0 Prncple of Leas Acon A conservave sysem akes a pah ha mnmzes he negral p qd Prncple of Leas Acon Now, wha s p qd? Le s consder a smple example A parcle under conservave poenal pq = mv = T Prncple of leas acon s equvalen o = v V( x) The parcle res o move from pon o pon wh mnmum knec energy me As slowly as possble, ye spendng as lle me as possble L m Td = 0 Prncple of Ideal Commung? 0
Prncple of Leas Acon For a free parcle, knec energy T s consan Td = T ( ) = 0 Prncple of Leas Tme Ths also means ha a free parcle akes he shores pah = sragh lne beween pon and pon Smlar o Ferma s prncple n opcs Lgh ravels he fases pah beween wo pons Summary Hamlon s Prncple n he Hamlonan formalsm Dervaon was smple Addonal end-pon consrans δ q ( ) = δq ( ) = δ p ( ) = δ p ( ) = 0 No srcly needed, bu adds flexbly o he defnon of he acon negral Prncple of Leas Acon -dervave allows change of me For smple sysems, equvalen o ( ) δi δ pq H( q, p, ) d = 0 p qd = 0 Td = 0