Hamiltonian Approach-Fundamentals of Classical and Relativistic Mechanics-Lecture Handout, Exercises of Classical and Relativistic Mechanics

Download Hamiltonian Approach-Fundamentals of Classical and Relativistic Mechanics-Lecture Handout and more Exercises Classical and Relativistic Mechanics in PDF only on Docsity! 0 2 Mey 2005" Fron the Ley corgi b He Hansthnion Approach Ceant) ~ Give Ls TQ oR, we noe know 2 cosrdmebe-free wey of describing he mop 1: TA TO Ga > (asp? given Mm local coordinates by ei = at i 34! We aay L ry regular if dA is 2a di Fbeomerphisin Crom TQ h Some open subsek XS TQ. ER try cage YE con descri bbs. what one system is domg eqrelly well by speciFyna posi He & veloeihy . ~~ (4,4) € TQ Or position Ke momentuun Gypy= Ag ae X. We call X the plese space, of we sy Sten « hn pre-bee X= T*Q ) & He L B sack bbe shangly veguler . Exongles : A particle M 2 Reenann ian moar bold (a,9) ae pole-tial Vi QR hes Leg ro~gien (4,4) = M9: 99 -VG@) Hee . Pi* gt? gag! docsity.com Me 6] So hCq,4)= Ca, mg (4,-?) Se tl is Skool regular shce. 7a — Ta yr 3(¥,-) is I+ & anbe , LQ, He mebie rs rovdege-erdle, Exasple: A gereral relehvishe parlile wild, chee em a elechronreamel c Vee lye polenb- ial A has Lagrangion Ula.) = 5695.94 _ As & thas fis L 3 shil skmoly regular , bul haw eacl, meq0 . —_—__> TF Nee’ 1,@ PQ 4 -}-—~-> rq (4,-) + AG) ts allue robber Hen lmea, Exvenple : The free geeal welelwishe poicle with repeereteiichnn~ hvesa_b Legve-gie : (4,4) = ™ Vg 4'@ 9 Thrs iy lewible Crane. oe perspective ot regelacthy prape-Le s gl delhreteble shen a34'2 vanishes 2 cdeolrod wiben docsity.com [dus i = ‘dy - pide: | docsity.com Me Assume the Lagrmaian Li TQ@—?R is regulon, se 6S 4 Moy 2005" A: TQ -*>XET#Q Caqdbo> Carp? is 2 diffeomorphism, This lets us regard betty L aul the Homil anim H= pighl as fuvchions on fe phase space X, and use. (gi) as local coordinates on X. Ag we Saw last time, phic gives us dH = qdp, — pide: But we con olso werk ont abl divectly, this time asing local coovdinahes a, pi), +o get = ot dH oy dH o Ap; + gi da! . Pi Since ap; dai fron a basis of I-forms we conclude : ato. OH Se aH Hametonl s L OP; Pi = og EQuations docsity.com 66 Thongh 3 and Ps are just bnchons of x, wher the E-L equabions hold fr some path 9:[t,,4.]—> Q, they will be the time derivahves of 9 and Pi « the E-L eguehons bold Harvil dm's eguabons desenbe the woken of a pemt xt) = (olt),pMEX. Tn fact. Hamil tor's equations are. jest Hoe Enter -logronge equaling wn dlisquise : the equation -i _ 3H 1 ~ Op; Se wiben really yest lets us recover the velocity 9 as 4 foneon of zg & Pp; inverting fhe Farmula - ob Pi = agi for ao _. hiX- 9 TAR Cap) | ae (np Given fais, fre other Hamilton eqn —. oH Pi ~ ~ qi 1s secretly the. E-L eqn du. | lw dt d4i Ogi PE 3a These are the same. because 5 = $3 (pgi- L) -- a whieh qeve. P as a fa. of q & q- So we get a Gemuls docsity.com 6y (we virihe. ay! nslead of 3 h emphasize Hb we wean fe time CS derivalve vather than 9 coordrale in phose space.) Let's show 8S =O <> Hai Yan's equations : §S= §| (ai) ae ~ [Gpai + piSii - SH)at San Tv. arts = |lipai— pba -sHae 27 = j (Spa pio9' - ge bq - uu 5p: ) at = | (Sp (BE) © bef 28)) a This vonistes WY Se = (Sn,8p) iF ad oy Hola’ e ushers © v i, OH a ~ Opi Pi = 34! hold. We've seen two prvciples of least action: r) 6 paths mM conkqurakion Space a ; $S=0 => EL egns 2) G- paths Dat phese. spece x > S$S=0 => Hawatl hes § eqns Addibomlly shee Xe TQ, we. might consider a phire versign based on paths in posi ho~~ velocity space TR. But wher our ® docsity.com 40 Lagrangian is regular, we have @ dF bea mor phisie \: TQ 9X, %e Hoax thaed prnciple of feast actor js jest 2 rebrwulabion of (2) However fle. really jnfereshog porciple of least achon involves paths m tle exle-ded phase space where we. have an add ihnead coord hale &- time : XxR. Local fhe ache SG) = {Coa - H) at =| pdat ~ Hat -| pide! - Hat We Can interpre! te inhegra-al as 2 a B= pide! - Hat on. X*R whicda has coardivcle 5 Pig t. Ss ony pall x:l,41-> X aves 2 path ec: [t,t] 2 XR £ H— > (xlt) avd He acho becomes the trhegeol of 2 (Ben over 2 curve: Ste) = | pity’ - Hat = [p docsity.com |