Mechanics Physics 151

Transcription

1 Mechancs Physcs 5 Lecture 7 Specal Relatvty (Chapter 7) What We Dd Last Tme Worked on relatvstc knematcs Essental tool for epermental physcs Basc technques are easy: Defne all 4 vectors Calculate c-o-m energy and boost Go about wth busness Eamples: Partcle creaton Elastc scatterng Partcle decays Today s Goals Relatvstc Lagrangan formulaton Two dfferent approaches: practcal and truly relatvstc Nether s perfect Wll cover both Wll do a few easy eamples n the process

2 Lagrangan Formulaton Proper Approach Set up a covarant form of Hamlton s prncple Keep everythng n clean tensor forms Practcal Approach Buld a Lagrangan that reproduces 3-force n a frame May or may not be correct n other frames Works OK pretty often, but no guarantee Practcal Formalsm For a sngle partcle of mass m L β V = ( ) β = reduced velocty Let s check f ths works L β β Space component p = = = good. But no tme v β β component 3-d equaton d L L V p p F = + = = of moton dt v Looks OK for the 3-d part Try to push ths path Generalzed Potental Epand the defnton to allow v-dependent potental Consder the EM force = β ( v, ) = β φ+ A v L U q q We know that U gves us U d U + qe q( ) = + v B dt v Dd ths before Stll works fne Only dfference s the defnton of the momentum L Same thng happened P = = p + qa wthout relatvty v Canoncal momentum Classcal 3-momentum No bg deal

3 Energy Functon = β ( ) L V Energy functon h s defned by conservatve L β v β β h= L= + β + V = + V Ths s total energy It s conserved f V s tme-ndependent Proved ths before No changes by gong relatvstc Smple Eample Partcle acceleratng under constant force Electron n an electrc feld E L= β + φ = e Lagrange s equaton d L L d β = = dt v dt β d β = dt β Integrate twce, assumng =, v = at t = t β = = + ( ) t ( t) + ( ) β φ = V Smple Eample ( ( ) t ) = + nonrelatvstc = ct t Relatvstc soluton s a hyperbola Approaches v = c Non-relatvstc soluton (parabola) accelerates faster 3

4 Smple Eample β = t = ( + ( t) ) + t ( ) Low-velocty lmt t lmt Look at t n terms of energy = ( ) = ( γ ) γ LHS V( ) v= t m β ct = = t m RHS = pc = T All as epected Energy conservaton Relatvstc Oscllator Consder a -dm. harmonc oscllator m L= β V V = k Let s use energy conservaton ths tme 4 E = + V = const β β = ( E V) > Soluton ests only when E V > Oscllaton between two ponts epected What s the frequency? b V( ) E E b Sem-Relatvstc Oscllator Integrate β for ¼ of the cycle 4 d m c τ b β = = = ( ) cdt E V 4 c b s gven by E = + kb Oscllaton perod 4 ( E V) E V k = + ( b ) + κ ( b ) Appromate for V << 3 E V + 4 ε = + ε ( + ε) ε 3ε ε d Nasty ntegral 4

5 Sem-Relatvstc Oscllator 3 4 b + 4 κ ( b ) π 3 m 3kb τ = d b c = + κ = π + κ ( b ) c κ 8 k 6 Perod s longer than non-relatvstc oscllator τ 3kb 3V = ma = Wrong sgn n tetbook τ 6 8 Relatvstc soluton slower than the non-relatvstc one Dfference depends on the ampltude of oscllaton Lmtatons of Practcal Approach L= β V( ) gves correct relatvstc answers for many practcal problems It s an ad-hoc technque Not Lorentz covarant by constructon Tme s treated separately from space Lorentz transformaton of Lagrangan s not gven Must redefne L n each nertal frame Truly relatvstc theory should respect relatvty from the prncple all the way up Let s see how well t works Lagrangan Formulaton Practcal Approach Buld a Lagrangan that reproduces 3-force n a frame May or may not be correct n other frames Works OK pretty often, but no guarantee Proper Approach Set up a covarant form of Hamlton s prncple Keep everythng n clean tensor forms but t quckly runs nto dffcultes even for a sngle partcle. For a system of more than one partcle, t breaks down almost from the start. No satsfactory formulaton for an nteractng multpartcle system ests n classcal relatvstc mechancs ecept for some few specal cases Goldsten, p. 33 5

6 Truly Relatvstc Formalsm Hamlton s prncple δi = δ Ldt = We want the acton ntegral to be Lorentz scalar Integraton should not be by t, but by a Lorentz-nvarant varable Proper tme τ could be a good choce? Lagrangan L must then be a Lorentz scalar Lagrange s equaton should look lke L d L µ µ = u Soluton s not unque. None of them perfect Let s look at one Goldsten Secton 7. for more Symmetrc for tme and space components Free Lagrangan We try a force-free Lagrangan Λ= muν uν Looks lke the non-relatvstc knetc energy Lorentz scalar d Λ dmu ( µ ) Lagrange s equaton would be µ = = u Conservaton of 4-momentum Tme component s conservaton of energy Energy functon doesn t gve total energy, though µ Λ µ h= u Λ= mu uµ = µ u Conserved, but not energy EM Force We know only one force n 4-vector form EM Potental was gven by qu µ Aµ µ µ µ µ Lagrangan can be Λ (, u ) = muµ u + qu Aµ Lagrange s equatons d Λ Λ d A ( mu qa ) qu µ µ ν ν ν ν = + ν = u ν dmu ( ) A µ µ da ν = q u = K ν Ths looks promsng 4-force found last week 6

7 Lmtatons of Purst Approach We don t know 4-force for anythng but EM Most real-world problems cannot be solved ths way What to do wth mult-partcle system L d L δ I = δ L = µ µ u Proper tme of what? Lagrangan formalsm allows coordnate transformaton Each coordnate does not correspond to a sngle partcle Problem wll be solved only when we gve up the partcle pcture Summary Constructed Lagrangan formulaton Practcal approach provdes useful tools Relatvstc solutons can be L= β V( ) found for many systems Not really relatvstc at heart Purst approach can be bult only for lmted cases µ µ E.g. sngle partcle n EM feld Λ= muµ u + qu Aµ Done wth specal relatvty Net: Hamltonan formalsm 7