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( !F % (( !( 4+" " "+  B  "@ 5 8 %(! ) *4I 88 :" $ $$ $ $4δ  * 2(% mω65 7.+ 86 89: 4525 15: ;92:55 4657 62:52 8:5(445 9254 48229;2 65:#5 5+8 4:9 568: ;5 ,8 ; ;5) ;2 ;5 423 27 45% 8 *27 :55% 8% : 87 !842 ;57:5. 485 " 5 8 8;457 127 ( 5 7;87 ;82 # *2 ";5 48 57 69 7 234 "8 48 ,:2 952 , 5 ,249481 8% 8;:7 27% 8:2 "4%75 :72 :952 892 ; 492 92 "42 527 4,42+ 87 +( 7 ,42% ;523 6:5%" 6:852 4895 :92 48 *8 9$ 8$ 7 4$ = ±λδλωt + ϕ + δaba×baa[a, b]bαba × b ≡ [a, b] = c, c ⊥ a, c ⊥ b, c = a b sin α .ab:5 ()5589 ":9;52 *42: 89854* 545 16:2 577 ;89 4:52;; 6:56:52 89(228 82 42 ;922 $[a, b] ex ey ez [a, b] = ax ay az .

bx by bz cez × ex = −ex × ez = ey ,[a, b]x = ay bz − az by , [a , b]y = az bx − axbz ,[a, b]z = axby − ay bx .Mp = mv1:9 87427 "!95 4 #5.2%791;5287 a:9 87 ^427  95 6 8:5 *24 ^9 1;52 6: 22  8 FKxyzK = r × F,M = r × p = mr × v .dp= F.dtdM= ṙ × p + r × ṗ = r × F = K .dt:48r %ex × ey = −ey × ex = ez , ey × ez = −ez × ey = ex ,@5ez822867ex ey+ :.4%8 @56Fc7M56:52 M" 22 4(2 8 *7 27 52 85 1 4 %27 5 5 8%9 5 86 452 +  .2% 42 7' 574 5% 8 817 1 2 "&8! 8!7 57 ;857 "8( 6:22 *;2%22 4/ ;2 4482 452 *28 57 8 #6 7 67 95 $4 442 ;2 :5 8 46 : ;% 53 : 9;2 ;92 95 22 :( 9 854: 4:5 "# 848 "95 4822 488"26 655 572% "14 9 ;92 6:2:85 :5 #92 6: :5 6429(2 89(2 ;:5 995 9 8;482 482 4 95 845 "8:3 4:28 65 #5 +9 27 5%481 2( *2 8 :5 654" 45 48 ;% ; ;577# ;9;2 ( ;2 46: ;92 153 895:5 5 :5 92 :955 "92 ;5292 8 "#9 "44 65 ;5 :9514 ;5 #9 5 ":"25 65 5 43 8+ $cba b6 47 56:52 81452 27 5 252 445 15% 87 8!7" ";; 84(2; 48225248$6571225;524%82a842 ! 8: :5%45% :57877 "4522 8257 71/ 878 .276 978 445287 5242 818298(7 45 456:52 ;528  62 ;4528928)75545,4825*456:2789;;%J%:0L&C)?))592 ; 88182 9(274 4542 6:8( 8!422 8;87 17 265) ;5%5 (282 427:5+ ;27 695:95 :22427 8292 2 ax ay az ([a, b] , c) = bx by bz .

cx cy cz ([a, b], c) = ([b, c], a) = ([c, a], b) .(22.1)r(22.2)(22.3)(22.4)657 :52 44 " 8%( ;5 64542972#:#/;952:952;8   4 45%7 "4% ;45:7 ;:5 2 754 857 :52 4:#582 812 4827 ;7 ( 8% 592 "7 42::87& 87 87 6 1 65  ' 7 ( 7; 8! 5;; 727 645 *455 87 827 68:2 8 ;( ;9$2 15$ 8:5$ 4$rdtdr = v dt:57 ;92 ;57 579 :5 :5 6:94( 17 "83 4827 8! 92252 "9  ;545 2 84 :72 84;88 - 5 !6 9 65 8:7 :22 . 4557 48 ( 5%745 ;2 ;:5,248 :5 22 8572 8 ( 4 +: 2;5$7 8$7 65$7K = r×F = 0 ! 655) 57 :# ! 44822 5 42 ! 86 7 5 41( 12 5 F k ±r 2;45 4:5:9 +94 821849:P: 6 dtdS/dtM = 2mdS,dt+49:P@F568:"26 4:95 65..8 8( 72/ ;52445 :% 53: :5(87 499;$2 455:55 ;9217 :58! 8" 8(:9$2 (22.5)4("9%J%:2K&?G@M = mr × vv8 6 ;5 aa7 *2 m5:7 1 #42 5( 7 ; :5 45 89( %7 54 872 "65 17 :92 8! 482 645 5 "% 65 5 45 42527 8 &' 2 ;: 8 : 45( 2 972 57 87 48 ! ;8 482 : 87 7 895 6 8 6: 1 ;7 42 1/ " 8 53 ; % ;2  82 % 7 9$72 92 2 2 8 6 8 8 6 :! ! 5( ;; 6:5 8:5 9%7 415( ;:5 482 15 45% 45 :5 87 8(2 452 47 5 8 1 + 2 5 : ; 6 ;2 5 4 8 87! ! 42 95 87 85 .

1 8;7 42 ;52 :2 :9 8:5 42 +7 6 855 8 8: ; 482 81 8 6 7 425! 4: 76"2 "8 4 42  17 :2 87 6 1 8:5 82 87 6 86: 6 8(8 9 87 8 ;5 1 7 7 81 6 "! 7 :52 : 42 :  ( ! 67 8! 7  6 :3 5 +: 982282 44 7 7 42 87 17 "! "9 "8! 77 ;557 :52 95( 97( 7 +2 95) 9553 77 9 8! 5 2 57 695 ; 97 488 "4 55 2 " 57 7 5 2 ;7  & 97 * .2  17 955 457 !72 92 47 :5 6: 497 47 :557 +4 '"53 :922 2 : : 4 7 95 5 5% 5 6 :5 #57 6: 49 : 6 5 45 8 86 :9 17 9 4 ! 4: 5 482 45 ;557 42 4 482 7 42 8! *"4 1 452 $ % + %$ " 2% 5 % 8 rz = z0 + c ,ax = x0 + a, y = y 0 + b,5:84 492;952 .65: 8 6: :5 4 525 65 : 8:5( 92 577 482 :45588;4882 4;:541 452 5 957 "531/ : 4 12 68: 56:2 17/5:3 )5(945 8%7788 289 "68:7$2.r × (r + dr) = r × drr r + dr1dS = [r, dr]2rdS 1 [r, dr] 1== [r, v] .dt2 dt2r + drv = p/m(22.6)(22.7)9' 3' ' 89%8 8 8 ;524:526:5 O O 89(22482dM=Kdta = (1, 0, 0)a = (1, 1, 0)a = (1, 1, 1)b = (1, 1, 0)b = (2, 0, 0)b = (1, 1, 0)a×b27827 8597 2 587 ; 57 9287 4 : 72 5 42 717 877 488 22 645 542 17 55 ;9 8 5 2727 42 :8 495;4 6 45 :7 15 7 85:89 95572$2 $526:268$2Fik = −Fki*72&53'(24597:276:"72"#892zk6=i.4 87 517 2 428 97 86;4 8 75 4 8772877 442 1 :9487 7577 52:455 2" 4972 "##57 52 4 5$ 9" # % ! (+ .

# % "+ "2 !S "(" + "  ""% #   "   )% / % + . %  ( $( $ " $  S:85 95 4857 449 472 45 ;5 72 ; 4 52 : 4;4 55 4 7 42 86 57 8: 8 7 45 ; : 4 4 5 42 88 6:5 2 6: 41 42 ;89 85 4 92 5 86 89 7 9 46 ;55 7 2 5 ;5575772 :952 6:5 5 8:2 42 :5748 8; 52 15 4 84:72 8 452 ;85 5 894 57 95 4 1;5 2;5 8 45 8572 8 88 974 8 8 7 :5 5Fik .%J%:8KC?#%.:;7 7659982 :28;45+45Q&'(%(ri × Fik + rk × Fki .$;;;92#+48; :27& 9 1= 7')559":*482:944(22.8)Ma49 "4 :57 41 488 :2 892 : 27 442 27 + "4 8 814 4922 8 8# 8: % 5 "2 57 6:87 7 ;95 52 92 8:( 5%.5/: ;8 6:2 445 7 4 83 48 8:2 82 42 ,4 ; 7 ;2 +4 :( 3 45 " 8 52"  %4 5 8;"52541( 95 97 84: 8:97 8 ;; 23482 "7 ;:5 :9 .2 6:7 :55 "84:57 . 4 5 5% 2 8 56 597% 42 : 8:7&*42 8;: *4585 ,8 .482 7 Q'"42 6:2 6:245 7( 3 955% 4: "42 97,4822 17/ 8:$2 ;5$7 97 5$7 8$7 89 % (  # %+ &'! * ! ( % +% Fi =XmdM Xr i × Fi ,=dti§v = 1a = 3m = 50ω/ω0M = 3mxωb = 4ambω0may(22.9)ri × Fik + rk × Fki = (ri − rk ) × Fik .Fik k ± (ri − rk ) .88rdr2=±[E − U (r)]ṙ ≡ṙ ≷ 0dtmrmdrp,dt = ±2 E − U (r)M2U (r) = U (r) +.2mr2(25.4)m(25.6)(25.7)7 8:2 88: 7 2 = 98 8 8%482 45 423 6:27 ;7 8;6 18 452 ;42 145 86#6:2 45 87 8!:423 :55 ;57 21:2 "4 :57 "46#, ::5 6 2: / :#87 4 54 522 8$ $m(25.3)! &" '+ .% ! 511E = mṙ2 + m(rϕ̇)2 + U (r);22M = (0, 0, M ), M = mr2ϕ̇ .ϕMϕ = ±√2mZr0r2mv0:944282:2;588:2(25.1)rpdr+ ϕ0 .E − U (r)dt%/8;7+188(&>'84:268279&'9:52r Z rmdrp+ t0.t=±2 r0 E − U (r)mr2dt =dϕ,Mqρ$ $ $'48 9557 6 65 8(7 5( 1429 82 97 8(;45 *482 187 7:4#5 "8!482 !4924:)2 :88 458 65252 54$ $3E?S2':7&&<='Sx;5 86 5  12 (889757!t(r)#8;7+188(&xyMr(25.2)%J%9/KC?#%.91(&'675;57 $ 6:2 (87& 657 642 :9442 ;57 8; &8' :465 95 5% 7<'12 65 657 2 :4 445 12"2; 5 :55:855 ;92 834 :5 4#5" 88 67 79) ) 695% 5 ;5765 3 572 8$7 1E = mv2 + U (r) = const;2M = mr × v = const.:9 5 + 4142 8(3 8 ;5 ; 57 87 ! 845 ;)2 + 857 1 6:4 : *12 ;57 87 22" 8 168: ;527 87 :57 ! 87! 52 + " 4*248  6:92 86 7 8422 ; 5 !: :( ,2 89 4 8482 1 42 5 :717 465 " 4 58! ;855 6: 2 48286 ,9 2 + 44:223+7 :5 .8 ;55:12 1 4 (23 :528 27 : ;57 68:r(ϕ)85 ;865 89 53482 482 :5945 :4:2 ;82 888 495 4529685 *565 2 482248 9722 57 452 452 :2 ;57 *.

4525 62 4:22 5$ 5$ *!(+)%) 1E = mṙ2 + U (r) ,2ϕ̇2;8& 8:5 7 ' 8%2 2 8:5 987 89( 532 7 2 8 45 57 28(& 45':4# 8& ;2 8 85 8:5+' 1758* !45 "4% 888 :( $&' 2M§(25.8)(25.9)(25.10)v0ρR(25.5)U (r)M / (2mr )2U (r)ϕ(r)%J%91vAA)&=r*rm axU (r)65518E2Ur1M2(r) = − αr + 2mr2+U (r)8495+2:(3(4:2+6:2224468:2 7 5*482 8:8*94822178!"7";94955;5765$!S95 27 273 : 3 8 ;45 4492 87&1452 7 82 8 :2 48 15 84& ;5 452 7 8 88: 89 8! 48482 *42 867 6 5282'" 8: 8989 68: "92 *42<&: 822'='842";52348SSr ≥ r1rmin ≤ r ≤ rmaxrMrr2ϕ=∓Zr=dr2mα M 22mE +− 2rrE1pdue=e2 − (u − 1)2r1+ϕ = ± arccosu−1+ constep.1 + e cos(ϕ − const)"445565ϕ=±Z:853"%FPF4P7899278;4("92αM2U (r) = − +,r 2mr29228184r1 rm in5E1 ≥ 02; 45 281:2;57 "7 65 6:52 88 8945 *482&5  295 89:55 *89 :28 89' 4 !) 8;:2 445545 6592) !α = ke2(4298545152728;8249654!8244:)28`_ !tα = Gm mm44 "9(2: 28:5 74892 8949" @65 @P42 47 :) :22 8534 "4:5285(:3*4248:7αU (r) = − ,r884!82 187 ! 46" : 8#3 :2 .5:5 8( 273;459 1421;52 85787 23495 5 :*% 2/ 4 4 65$mE1 ≥ 0;E2 < 0E2 < 0.+ const.M2p=mαpu= ,r+ const ,2EM 2mα25 42 6: ,5% 7 4!2 " 2 65 12 7 9  577 *8 :5 57 ;:2 8:59 67 :2 + :5 53 2 5) ;5 8 2572 8 88: -) 65: 67 86 :53*482 9 77 4952 *4& !544:;P87 :5 "B7:2H %: ; <57'5J 6% F5 5 5 5 19: :8%"F #;577L4 :5;+7 654 5 1P)F 5 53*56452 $ 8 $ 52p=M=r ϕ=.mα2a = CA = (1/2) DAp = OP242:456;5(15Cϕ = π/2πE<0Ob = CBDACa=11(OA + DO) = (rmin + rmax) =22α1ppp==+.=2 1+e 1−e1 − e2 2|E|C O = ae OB = appb = CB = (OB)2 − (CO)2 = 1 − e2 a ,bb=pM2m|E|.2bB8655%J%9.vBEr&#@const = 0e=0E = −mα2/(2M 2)e<1E<0e=1E=0e>1E>0p " , *" .%  & B * 2:P(: ( " J F 9: $ # 8 L4  54P F'  )" (! 2 ( %  "$ " $ : 68:28926 482 5  6"  312 42 8: ":9 )442 582 1$ &(  *@4':2 ;4 82 M;5 "( "988: 92 8899 7 182 P4 6: 5@H 845p.1 + e cos ϕ52 5 ;95 6:5 87 4 62 2 ;5 452 ;5578 5 ;8 :5 6:7 ;6 65 8 :9 9 :7 57 22 45 82 748 57 9: 5 ; 8:98897 897 : 4557 522 8:95 : 62;5 :5 442 52;5 6 9 8 95 42 ; ;72 24:2 95 8:5 788 2 82 4 4245 52 : 57 :42 84 49 ;:5 45 2568: 6 ) 5 4 45 897 5% 1:2 ;72 "92 !5,285; 22 8( 8%6::628 :523 53 867 *4 57 88 57 8) :92 77 ;5 65 89(275 , 6 5 4 8 48 "4571852 4 2 8 4 :;45    65 6: 55 657 ;5 ;5577 "6 1 ;:2 2 5 5 892  6 9& 4!9;&71"2256+15 & 6& 8: 2 4 : 17 4:# 48 76: 68: 8: 6:& 1 48 2 45 572 87 57 1;27 9558 5 5 4 7 142 72 6 8 27':2 ' O O ;:2 455% 87 1 823.; :5 81722 +95 :3 ;2-2 2 6 ;5 48 .8:9:8 2 ( .

5 57 : ;42(82;7 .+ 6 : 6:2 ;57 5 84 485412 2 * 3 5 ( 127% 524854'"7O5;94;;;7555 57 % 62 4 + 84+:7 + : :#5 67 7 172 9' 69 5 85 :22 ;8: 48287 ( 3  $ $ $ $ $r=ϕ = const4:)2 ;588 745 285 252 154 8 66 572 5798(78425;55mtr = rminr = rminϕ = 0yPOx2aaSSS = πababmT = 2π= 2πa3/2Mrm.α4π 2T22m= 4π,=a3αGmz = x3RXY Zb45 48 : ;7255 +6 87 :5 :2:553 557 ;9$2 8 855 455 45% :34:52 %7;2 6 ;57 .482::85 "42 857 5 ( *45% 5 5 445;:57 845 4 8%57 :53 65 878 5 . 65:9 (4 46 ( 65:92 8:5( 2 57/4 92 445$ $2 957dSM·T =T.dt2mS=! !65 57 :5 %8 78 4:55 "2 )*292 865) 185 47 26:2 685 55:3. ).482 8677T7 5 8: ;57  15 : B:.8P4:%*45 45 2 ;5 5" 9 5) &3 57 4# 45 6:2 @84 8: *(" 853 2 8;2 ;52 8# 423 @J 6 92 ($)7:" 4 8 6246:6:2 57 53 83 1 2 8 F4 5 &'(2 86 573*8 # 97 @PF BA$2P5259 5 + 48 ; 8 B 86 I*52 42847 4 95/ 95( 64:2 89 P:4 : 54, 7 :5: 6284:4 8 * 354:"I >9 2+252B76 :2 2 8 *2 5 @ PF "34:5+8:95( 157 6:5 ,9 8692 !8 8@4 8: 2*5 2 89 4:2 45 8 ! @ 65$55/45% 65 :2 2 :2 :9/ 55( 4 :(&7;5 .5 18 889 ( 55 41I 92:5$ + 5$2 $7 45 $2 2 9G: P4@"JF9:8L454PF27+722dS/dt+*;95 48 2" ; ;3 953 6#72,5% 4265 5587 ,45:2 4822;5 ;988: : ;5( 62"92 8:5 7 53 :845 .2$ $RSTSSmarmax = (1 + e) a,877 8: ) ;) ;2 22 ;55 877 5 :55 55:8 2 4 2 84 *424 ;5 1  828 ": :5 5 68: :9/8: 8 84 9 83 5 55 $7 5 * 12 87 45 2 9:I ;2;5 ;5 @B 72 ( 9:"52 :8: 8:7& ; 7 4;52" 55 2 5(:5%:5 1' 58 .+ 8 ; + 8 458: 9 ,27 "3 85 4!8:242 84:2 92 2 :(3 4! !8+ 8 81 "4( 5 5%7572 45 84 9 7 872$% %-65 8: *2 ;5 92 7 :55  818 48 " 9% 42 6: 8( 95 676 2 :7 523 482  ;9/  9 4 2, 85 ;57 49/ :2 .::(3 :57 87 .+8995 867 5 2 %7 62,8 ;:5 4 47 8 6 ; 787 8 : 65 9552 995 ;57 2 :5%4 22 :57 522 8232 42 2 8: :, 2 5 87+ 2/ 5$ % rmin = (1 − e) a,p = (1 − e2) a .XY Zx = x1 y = x2OO;4!529"68,2654"2(255745,48227 89 ;;  7 @I 5;952 .: 9557 8:5( 84'R ;92 8F 1;:2: 2 ;6 95 :5 8@4 B 42457 45 J4 94 9 459;2 5 2 5 8 PF: *4:5 8: #9 *892 *82  8 "%42 : 6:57 482 4(2 @P @F: 8( " 9/ BI 422 5F4 8#9 :5 97@4 42% 8795 6: *482 52 452 8@@ @B:I,52542 89-6% 5 6:9 PBA8:" ;6: *482 57 *45 482 ::HI @ 192 88 6 6:2 2&4 4 548 (2 2 7 45 5F P:4 4 4 992 5 8 42 #5 9:: @4P;$ 5 2G @xa, ya, zararaRraXraxyzxyzOyXY ZxyzNaa = 1, 2, .

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