Least Action and Least Time-Classical and Relativistic Mechanics-Lecture Handout, Exercises of Classical and Relativistic Mechanics

Download Least Action and Least Time-Classical and Relativistic Mechanics-Lecture Handout and more Exercises Classical and Relativistic Mechanics in PDF only on Docsity! 25° Ape 200s Leas} Acho. vs. Least Tre We reckned Fermat's pers ple of leash Pe ophes as draley on to we povciple ol least achion noe petele mechgate 5. Thee oualegy fix He is shrage shee Re priciple ol leash acho we tang wberval a: [o, ij— Q. Alse: light Pale, light slower ~ Nowetleless, Sacchi wea able be verlepat the mechanics of 2 perbele m2 pelebil 26 on optics prablen Be Maniby® Ye hug minimize, ciples. Fret |el's corsthe- fe, ht m6 redinn wih « weyrg mie of eelaclan nm Crecell OC speedt of Iyhh), Srppse tbls mm IR™ uth its usol Eaclideas webic, TE He light cs Hy my Ip mmmize Ke tre) ths bying b wainize te erclenglh of sks pol wm He rebre Si = wf. — he dex of rebracten ni: Rk 6, oo) tmes He ususk Encldenn nelric _[° °) R a ' | ° : docsity.com 50 Tl is gest lite the Gee perkicle m quest velebvily (mmimramg ibs eroper hee) except Pel nae 955 ts 2 Riern~sem ion rebrc: 9 (vw) = 95 vw We glv.v) Zo, Co eh use He seme. Lagrangian: (4,4) = J 936 4'63 8 get the Sarre. Exle~~Legremge eguelims : 2; ; ie Puget = 0 * | fa ne Cai Te derivehves of He melve 5) ®@ As, whel Joecbi ded chow hoo the molio~ af a portrele Me prolantiol could be. Viewed as 2 special cese of fea. Conse 2 perhele ab mace mm Eneldeo RO oth pcb VR RR. ~~ Tt sah Ges Fama, Le. ay 2-9, w at? a, V tk & Ito aid Jacobi see AA as > special case of a He onsiderd a pe-hele of nergy FEF g te chose pdex of velox by be. _ & nlg) = [Eu E-VG)) docsity.com 53 We've seen: ‘) Du the geowebvic ophics oppriximabin | light m™ Q=R* ack, like perlicle s hacing out geodesres Be HK. mebweic Si = n (9) $i; hee. n. Q—? (0,0) is dex ol reba bon, 2) Jacobi sau ot e parbele tn Q=R* WK Same poletiod YeQoRrR traces ouk geodesis mr He mekic Le 2 a= ZED §, if pre po hele hor enersy E (ubere V<e) “ i , 3) A free pelle s gerered relelivihy hreces anh o Geode si« on 2 bove-beien manitale (@,4). dn feck all 3 com be ge-evelized by cover every patle~ we've Atyscussed ! 1) Light cn ony Riemann ion yeni ld (0,4) with dex ok refreche~n ni Q=— (0, oe) traces out geodesic S mw He refric h=n'g. 2’) A pekele on 2 Rinewnion memibyld (Q,4) wikk polrbiol ViQ OR trace ant gecdesse 5 wed fe mebrre — h= R(Eg iB ik hes energy E. Lobe ab physical sychms con’ docsity.com 54 be described thes Weng ew Abd medre oP | rigid rebabsy body - {Q= $0(3)) ; hops eke, —_— all of hose hewe Lagra~gin whi kh is queda c ke of velocihy + seme Panchion of postin, so Hey all Ar ab Hos Cramebbog k . 3') Keluze ~Klein Heany a pehicle wilh, charge e@ mea Loventaien men itold (2,9) wR a elecbonegrele vecly~ pulesbiat Frlloos 2 pel wilh r 4; = “Ti aigk + EF 3) ~~ "y m iy here Fi = 0,43 OA; . faa 13 2c hl geodess« mohtn on QxUC') Where Uld<= fe® gekS is 2 cle, & Q*UMD 8B given a clove designed nebrre buslt bar 5 £A QxtO) TUS) a gesdesi< Q — te appt pall ® docsity.com 55 hee Ue rot of ‘Spirallag” — He velecihy s UO drecter is £ The webeic ho wm Ax UL) fs belt ba g dA I ~ a very si—ple Wes - Lek’; pick coord m cheg zim Q where. LE fo,...,93 Ghee ve ‘ve oe (nel)-day sperehme , 2B % ove lock coardmbe on S' The compares, ot h ere hi; = Ou + AA, Y a hia > hg; =A; bog 7 JF hy = I i Werkns oat the eguodions f,- a geade sve wm pars nwebnre we, : get : ches syle of g eo hog é : a in ggk + & at if a =& she F fs pat of the ClurglaEPel sybel Co h . S> ~ every preblen we ‘ve discussed Can be seer as geades< rohien | ® docsity.com 58 We wart fp debre x jane cood mete Fee. wey 5 a 4 fhe “Bibbentiol of Lom the verbreal direclin” — ie, Me gi diveclnns, We bee ~ q:TQ—7a (4,9) 4 dx : TQ) TO hoy kernel consthry of verbical veclws: VTQ = ker dr < TTQ. The Aillewtil of | ot some pont Ge TQ & Oa a5 © TQe TQ CL fe, dLgar? TqyT& OR We cn vestrrt His bh VIQE TTA, gts f. Venan 1& oR But wake Vaar1@ = TC,Q\ & Siege TQ (5 a veclge spere , lag @ = GQ L 8 he 2 cenentead med = docsity.com 57 So f gives a (mee mop pi QR Le, k pe Q f Ths sy He momectumn ae ® docsity.com