lec.24may - 6? :4? “04am...” ' m“_ 74", ‘4...

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Unformatted text preview: 6? :4? “04am...” ' m“_ 74", ‘4 M VF 1L I ‘{¢¢¢C,&- (<: K(Q.P,+) :. W MW (5'11, " M w 89 = as “P. .—_- _ 25 T 3?: a _ TL; W .; A, L W! 5‘} -BMWW4 .. ‘1‘ (Psi.-H):P;Q;—l<+4_lf {WW _j hi A+ :1 FU’H —-FU:..) no Vauiutuu 62(- W :it‘ d—h MAI-M (1.919.) 1‘- FUQJ’: W .i 9—“! and (5 m M a “M iféuyéw” ?‘- W wad W éo ’“ Sun ‘Qflfiflf& .__—— ci; P" ._—_ vpc ——--J— _—-_ ¢+Lwé K’(a',?‘3 =/uv 24%.?) 91$: 1—; 3K! 8.3,! -_-_- DKI | 3/10” __._———~ __—————_— 23?; 312; DP; 81>; v ' Bf, 51; | Ejt gkow MOM . €(s3- C9 P. - " Pc — at P‘ :-Q.-P¢ “K *QEE: 1"“: ct.*?—E=P 3+ 8?; . __ 4' _ .— aFL .- ci. P‘ —Pt [Q‘ a”). —M 4' K * at}: :0 c 1n = 25:: QL -: 2.51 K = 3” “2‘11 ach- ‘D ‘ 34: AC3) F: = F3 (P. Quiz) -+ «P: <2P‘ Dal ole @ F :— Fq (I), P, + C( P! “ (Qt ‘Pc __-:7 q : -9Fq Qt : (DE-5 [(1. %;Dfl 3?: apt' Di: 0 F: 2-" ilQ, P. :: QE' ’2 Q. P' ~ ’Et 2'2‘ 3“" 93. avg: P 17; 1' — 5L" I: H A m+ subs-Julio“ CL": ‘1)" us? H: 323' ’« £4412 PJQ" 1"” 1. -n K 2-. (32“2M + J; RT" 9.1 The Equations of Canonical Transformation 373 rather will be functions of q, P, and t. We would then seek a generating func- tion that is a function of the old coordinates q and the new momenta P. Clearly Eq. (9.13) must then be replaced by an equivalent relation involving Pi rather than Qi. This can be accomplished by writing F in Eq. (9.11) as F = F2(q, P, t) — QiPi- (9-15) Substituting this F in Eq. (9.11) leads to . - d mw—H=—@fi—K+EHmPn. aw) Again, the total derivative of F2 is expanded and the coefficients of q,- and P; collected, leading to the equations 3F p1- = —2, (9.17a) 361i 3F2 ~ = —, 9.17b Q. aPi ( > M K=H+3f @flo As before, Eqs. (9.173) are to be solved for P,- as fianctions of q j, p j, and t to cor- respond to the second half of the transformation equations (9.4). The remaining half of the transformation equations is then provided by Eqs. (9. 17b). The corresponding procedures for the remaining two basic types of generating functions are obvious, and the general results are displayed in Table 9.1. It is tempting to look upon the four basic types of generating functions as being related to each other through Legendre transformations. For example, the m...» a new" Wummw .W.» TABLE 9.1 Properties of the Four Basic Canonical Transformations Generating Function F = F3(P. Q. t) +qipi F = F4(p. Pit) +4iPi — QiPi «(7)6 7‘32; zfii Q"‘3F ‘— 3‘!" t 5—}. ‘if '9: =?: V?“ 1-: ~1>' " t _ f 2?;90' (F: r; 4-31‘Pt) PC ’Dc :3 _' cl; Fir—Pup" (‘1: IP11 77: (31‘ 4' 7, P I 7— ’- _. [:2( ———— 1H,.PU‘E) 4' F3(P1-»11})"QIPI "' 11?). ——-> ?. = 25' Q = M” 31, 5—15, ixz~52£l g-PLZ’DFI P» 3697. M + apt/‘04: ._ 35:7. - ._ -_... LA - M ED; H ‘1‘“: (imt'z?) "L 7‘5 Fa: ("Uh mi)? + 63%.- (6,, H 91‘ = aF" ‘— 57:. " “it and Made- PJ : E1: 7' Pf 4’ a 3‘63 3%; P: :C¢P‘+ be} "N-J; a2 ...
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This note was uploaded on 06/13/2011 for the course PHYSICS 821 taught by Professor Eric during the Spring '07 term at Ohio State.

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lec.24may - 6? :4? “04am...” ' m“_ 74&amp;quot;, ‘4...

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