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2 5+: 5472553 8 8 6:5 7 1;241( @854 442% !845 49 8488 4" 5%  8  845 " 4# @ ;!5 82% :27 55 ( ?8 8 $ 5$ 5 !U (x) = βx488842 5 48 88 652 972 7 :4 3 485 48 48 ; ;4 82;5 6 2 ;5 45 5 9 997223 :2 9 6 68: 2 548 7 ;5 52 5 2 592 7 93 87 84 44 15 9 8 : 42 ;5;; 4 22 8 7 ;2 2354 852 44 848 : 523 48587 53 2 89 48 152 27 ;5 65 9 9 4245 5 65 42 8 45 96:5 4 :57 8 482 8; 47 ;52 5 4 12 8527 3 :2 45 87 8652 948 2 :2 9 18 % ! ""#  (" ( (" " +  # * %  "#%" " ( " "  %   ( *!* *"" +  % % . ) ! ," +$ $ $mT (E)E4174"8589781;5 655 1;85 57 48 7 ``;55 4822 1895 49 6 ^442 8: : 95 :5 ;52 52 8 495 48 49 :5 48 5 927 ; 952 55 8 8 7 87 15 4 41 65 8 ;2 4 482 827 7 ;; 8:2 ;55 4 :2 ;:8 4 1 68:4 5 827 15 8 4:2 57 9 ; 2 8 12 4:2 ;5 9 5 :9 ;:55 1 452 27 ;4 65 84 87 48 159 9 5 ;5 8  " %* ( ! ( %*+ )"  !  ! .+  )2" # +    "%! #$ % $x(0) = aU 0(0) = 0U 00(0) = 0U 000(0) = 0U (x)x4Fx ∝ x3U 00(0) 6= 0xU 00(0) > 0U 00(0) = k > 0 .Fx = −ẍ + ω 2x = 0 , ω 2 =k>0mx(0) = x0 , ẋ(0) = v0 .(21.2)x1U (x) = kx2 ,2(21.3)11E = mẋ2 + kx2 .22(21.4)dU= −kxdxkmẍ = −kx(21.5)(21.6);455124"59%28289ωx(t) = x(0) cos ωt +.x(t) = a cos (ωt + ϕ) .x(0) = x0 = a cos ϕ , ẋ(0) = v0 = −aω sin ϕ ,v0sin ωt .ωb)txZ±1 x1dxx√t(x) ==∓ arccos − ϕ ,ω x0 a2 − x2 ωax0ϕ = ∓ arccos .a(21.9)(21.10)(21.11)248 4:2 *27 @5 7 288 452 ;57 I 5 1 !842 8 + @5 ; 4245% 4 8 @: 15 55%755 7 8 2 52378 5% P4H 5 44!4 5@ P7@ @ 5: 59:4 9::I+8;5 8;8 8@F 5F4 442%23 42 8" 8@ 55 8+5187 ;27 6:2 J4 !8577 2 @FB 1422 27 @ :582 : 95:9$2 ;5 GF $2(21.8)q414822/(15798 :456:72 985 175 5 8! 2&8; 1/812 57;57% 4(17 "955 +6%:755 55745 17,482 55% (2 85897/(4%kν1 2π2π= 2πν , T = == 2πω=Tνωrm.kmU = kx2/ 21U = mω 2a2 [1 + cos (2ωt + 2ϕ)] ,41K = mω 2a2 [1 − cos (2ωt + 2ϕ)] ,4E = K + U = mω 2a2/ 2$'ωT0(21.7)5:7427+ v 2"248!:"538;75248%3178!(ϕ=−#x20 )a=r9: 5871:2 84992( %2 4452241(48224:)2889:944282&<="( 1U (±a) = mω 2a2 = E ,26544:)28889757/(7+4%−a ≤ x ≤ a;55 65 ` 484: !82 ;52 n  5 " 4 53 3 42 4) .8 5 :2 65(3 8 2 9( 14245 887 "4282 :25(+ .2% ;; :( 5$% "24: 853 4757 553 13 %. 484 452 ;5" % 21/6:224142"58979844 29 6:9 4822187 2 :8 44924:2 2 88 6: 2: ;524886755 565 641 554:7;5:94428498726518484 "8 "" %, @5" @ H F"P : 8+)..

& @74 '9+@ +  : % @55  P:):+( (" # +5" @ $(Gx1 7/ 9 41 8  4" 5 887 4-2 8 : 5 48 7 6 42% 5897 5%& 6 ;( 8 555 '  5 !7 882 41 23 ;5 8 " 48%O 423 :5  ;54 23 8%O 6:48;84( %:55 4 9;* 1 7256:2 4 157 ;( .+22 "# 52 32  :52 57 ;4 9 2 87 8% 43 %O 84$ $ 8%- 455$ x11E = mv02 + mω 2x20 ,22aωωt + ϕϕv0.ωx0(21.12)(21.13)K = mẋ2/ 2(21.14)88ẍ + ω 2x = 0 , ω =sk→kkm(21.18)tm ẍ = −k xm= ml2 , k(21.19)T = 2π2(21.17b)r x2 ,%J%;2B#@B(21.17a);824"9278*27;57:94"1ẋ2 + k2= f (q0) .)ẋ2 + .

. . ,5d2U (q0)=.dq 29 482 87:3 4 7&42:985 7 527 187 5 ! '57 ; 72 718 12 *: ""92 972 42 ;5 5 28" 53 4 35%.423"9 2765 45% 65 995 ;5 2 :2*482 8:5 *2 8;:9 4 877 45%495 " 7 6827 4553 ;:5$2 ;55$7"xx 65 495 ;5 7 89 x?72 827  642 "4 % 27848 5;845 1;2881279 :( 9 ;95;57 *5 22 :5) 8 54%:821455 65 ;5 6553 4!82 452 4!2(9. 488878#  82  6:2 4 5 45 6+5))65 42 5 428̇5 8: *42 8:*88 482 +9 8: 82# : 29 $ x = q − q0x;52348%1E= m2818"915q̇ = ẋ(21.15)(5*8654!824+4:)28+:965f (q)2;8249*27;57)1U (q) = k x2 + . .

. ,2q;8:5(356512 2:96572642  (1K(q, q̇) = f (q) q̇ 2 .2"#m→mq6:285m1K(q, q̇) = m2 8 252 ;2849"#;5 72348%6544:)288289)k58 95 ;55:2 "::9 482 5%442 65 *8 8182 5) 27 29 ;57 578*482 8 8 4% 2 :24 4! 445288145 :) 45 "9552& 5 ,2) 8 42' :942 8:2 ,(482 4 4265 92( 5%4$ 887 "7# q0Q&'&'5:2,48228289U = U (q)mlgϕ(21.16a)(21.16b)x1K = m(lϕ̇)2 , U = mgl(1 − cos ϕ) .2ϕ1U = mgl(1 − cos ϕ) = mglϕ2 + . .

. .2mk= mgl ,= 2πslg(21.20)dnxdx= Re {γ eγt} , . . . , n = Re {γ n eγt} .dtdt(21.24)x(t) = x(t) + x@an $j=1nX44282 ;95 84252 45%55!87 5;2 :5:485 #&':;:82765) 81275 ;5 2 :9$8: :5 `8:92 ;52 2 ^553 ; 3 2 27 : 4 657 523 557 75 :23 42 4827 :918 4427 85 ;5: 5"%8 @8 8  4 " @ 8:94: :I8 * :.@ P : @4 I 8 8:4 %F I @@2 4 P48  ,@F*56" 8:4 PFB 84 8@ &" '$ $G: x(t) =;52348%877285(:3*/4$(21.23)@x(t) = Re {eγt} ,C1,2(21.22)""#C1x1(t) + C2x2(t)4;2 48 1555 67 7 ;5:5% 57 86: 4125 9 44 4 4;! " 1 955788 8; 7 65) 8 22 48!8 452% 7Q&'42 5 &' 8 85 814 4!:22 85 4 458: :549 492$ $x2(t)(21.21)48 ,2 6:2 5452 .53 7 ;5 4:5 :52 92 :254 22 9, 7 455 48 55%3 482:9 45 7 42(442 :55 "982 45 9 275 #7 84:9 :I442 "2 8P::8 # 8I81-5 74 687 555 445 :2:2 95 ,42,$ 8$2 "2x1(t)% $ $4 8 ;5 843 8 ;8#!8 4:992 48+7 6:%7:2 92,482 89 8 @8P:9 @442 8 458 P:J# @ 842 @@22% 7dnxdn−1xdx+ a0 x = 0 .an n + an−1 n−1 + .

. . + a1dtdtdtm68: 48 482 672 & 8:5(2 <8 9' 44 95.2 55 4:2 2 : "8,4822 8;2 492 ;5)) 8 48755 65 8!8:9 (2 4 42442 7 #5 88 72 *299 48 ;82 .+ : *28%,2 97 82  44 274": 4:92 ,:22:2 7,4822 :94428<&'82 ;4:5 889 55):94428553.2184 ;52987 2 48385 897 77 2 45867 5 975648 82:9 ; 442442 52 882 43<= 8 6:2665 85:53 ;5 224% 483 %8% 4$)( ( (   " %" & %& '' b"#5 ;581 :5"+7 5%5 4238#7 2455 1555 65 :;5 89(657 45"445 52" ) 545 95*"82 :2 *2 48 γx$ $S6: 42 92 5 792: :5 897 2 9& F PF2 9 <' 84<& P2@' :2 4@ ! 9 78 4 : 992:4 4 45 42:3 : 82<&812 4492 4';572 +8& %/:9 2<' 1442 87 5<&82 "9 ' 65 5 x 12 42495 9(:( 8875 5*482 665 84!82 "; 5 45% 48234:)2 %88 8:65 1/272 2+.8#11U = mgl(1 − cos ϕ) = mglϕ2 − mglϕ4 + .

. .22444 8:7 %Q 8175G L&E )) E$ r + $ )B &) ),* B?&+  B+% <-@))$ϕan γ n + an−1 γ n−1 + . . . + a1 γ + a0 = 0 .Re {Cj eγj t} .γ 2 + ω2 = 0x(t) = Re {C1 eiωt + C2 e−iωt} = a cos (ωt + ϕ) ,dnxdn−1xdx+a+ . . . + a1+ a0 x = F (t) ,n−1nn−1dtdtdt(t) .(21.25)γ 1 , γ2 , . . . , γn(21.26)γ1,2 = ±iω(21.27)F (t)(21.28)(21.29)46 :9 6542 5765 ;5:892 :55; 58 :5;7 85(:452 482 487 " 53 *27 ,8: 65495 57:5 )  867 87 4;;578(65;5%6:595;5282+.2%bx2 y 2+= 1,a2 b2a % (%J%:1KC?#%.19bk46 184  "4 :9 2 " #92 248 957 *27 957& 28 ,:2 ;57 8:7 5 % 7 %8 : 6: M '" 8168 485 *"84 "7 6 57 11 2 57, 7  ;8;9;5 ;2 8: 88 "7 "#2 5 8 6: "5 ;53 23 648% 4 49 * 2388 89 ;1 "84 41 48%# 9 6 34:)2 8:2 ,;2 4:22 ,; 8478+ ; 52 *:23 6: ;845 4$ 85 *84$%J%;8KC?#%.11g = (0, −g)"927 97 4 82 " 2/ 87 1 41 " 8!742 7";55 " 84: 8% 89 :*" 2; 2798+:5 65717 ;57 8!87 "2 86;:5 7 557 88( 2%$ $a5:57 45 4:2 6572 45 868: 8:( *2 4 55 7 48 87 9 12 ;12 ;6:5 : 8:5 %7 2 *82 7 97 ;:57895 "65 485 9 1 .+2 488 68: 4452 681565 127 84 % 6 9/ ;57 8( ;275 8(2 ;5 82 : :5  42 8#* 94 7 557 :2 : 5 *42 8 877+ 4 ! 47;5 15 8 8 # 2 887 ;5 2:57 5  47 9 6:5 23 #98 87 :5 9553 ":7 *8; 8;7 48 729 : 685 4 57 + 278*482 482 "97 ";52 8:5 88 .8#7 ;:2 :46 52 23 5 8 9 :7 4828 : 5 48  (2 :/ 8 268: 92 ;55 5 ;5 "# : 452) 44: 2 845$ 9 8$ 5$$  5$ 4%J%:/KC?#%.1;;:(2 "7 :55 .+ 8 4 62 4: 52 5 4 4 7 57 % ;2 ;5 17;8%2 4:#92 45 45%3 81 " 19 58277298255:( ; 5 57 "8; 4!5 5 "% ;#57 7 " " 232847 :9 :2 ;# ;# 4" :52 52 8%*.722:87& 45 48 44 483 483 52 8 % %; 8 ( 48"5=' 5 ;* 924 :242;6587 2 ; 8;  :55:9 4:( 4!52 6:2 "2% 5.482 $7 ;55$ 9$7 8:2 :9$Mxx§mhω022OmmxyF = −krv(0) = (0, v0)r(0) = (r0, 0)xẋ8:5 482 @7:I 6;5 812 5 94 ( 8:;572 8145 @J 52 87:2 5 F 8: 45,4822 4 8@F4 !2 4:2 4884 " 1 3  7;"49 1 5232( ;57 ; *89 53;5 :57 5 4822 #2 4 4( :3 :48% 49 1 "9 ;2 7: 4 452 27 88:72 7+87 62 ,22 @PF4 1;827 $ $ G #x(t)γ1,2 = −λ ±qx(t) = Re C1 eγ1t + C2 eγ2t .(λ > ω0)ẍ + 2λẋ + ω02x = 0 .γλ2 − ω02 .λ(22.4)x(0) > 0 x(0) = 0(λ < ω0)-̇%J%:.&O?#?)D&(22.2)x(0) = 0 x(0) > 0-̇α.2m$λ=;9:452:;#2k,m442 9 282 82 79573 18454 17" 45# ;5 "9( 9723 27 553 48% 9 4"( :68 @ ;#5,27 @84 423 899 : 9%82 FJ 57 7F 98 8@ 2: 48:29$ (285428992/55341(482m%J%:;&O?#?)D&x = Re {eγt}8:7 812 5;572 :944282 6512mẍ = −αẋ − kx .

 *  +&" % , )( , % '  * #+ &. + :2,482;5:5557ω02 =8497 2 695( 557- 5 732 +5 6548:2 *85 68(7 45: 42 7 41(4492 ; 8 488282 457 4:2 ;579 4 8 :578*482 # 899 86:82 ;: 2 6:552!: 884598 (+.4284%!4$25x5 7 5 1 ;458: 2 81 4 4523 68:5 45 42 5272 8:77 5 4 86 5 42 <53 85 84 8:3 17 8:2 :42 4;55 1 82 62 495 8:788 23 452 482 8:5 4482 72 ;4 57 6:27482:1482 5 987#:; ' 84 53 445 4 4522 ::827 1 .27 645 ;:57 ,422812 47 1 ;552 57 8;2;872 ;:55 ;2 23 85 565$ 7 5 482 14 4$3.2:2,4822278424%6:7265(8!89#:2,482%2Fx= −αẋ ,α(22.1)tω0x(22.3)γ 2 + 2λγ + ω02 = 0 ,txx;9 "9 45 45 42 2& 1: 87 55 *5 8(2 9 845545 42 *45 " ;56:5 17 23 4( 7 6 46:5 8 48 9 ( :2 "(! 9 845 4 ;9 6 :2 42 +95 PB 5 * :854" 82 " 5 45 17 44;55 ;5% ;5 2 848) 65 84: 23 523 @8J4:2 57 " 48 48 9 @58 44 87 : 65 812 !87;5 " ;:5 42 5) 84 23 ! 57 82( :548% 8:7 8 6 2/ '24 5 8:2 7 6153 *"$ 82 45 5 95$ 8λ ω0e−λte−2λt1λa e−λtẍ + 2λẋ + ω02x ="!8:2,482282894 5 927 48 ;5 42 8 23 :92 % 757 4 2 8 !8! 88 9% 48 ;55 : :5 5 2 87:5 7 557 4 7187 892 .83 !57 48:2  8 2+ :7 482 8;57 5 6;589( :57 : 8527 8 74 ;) 2 69" 5 1 8 8 7 87 55! 842 5 419 *48825 9573 4 :9( 5" 4$ % 5$$ "# # (22.6)ω02 − λ2 t + ϕ ,F (t) = f cos (ωt + ϕ)fcos (ωt + ϕ) .m$ $τ=ϕacos442 2 82 492 79 ,42 58*482 :% 8:9(58468 812 *42,27 ;57 852 % !799 87 882 " 7: & :57>& : 42'82'65 2: 9%798x = aeqm(22.5)2 225573 482% 7 55) 42 595 4# :7 45 52 5 " 8# 4<& 452% 5'- :6857 4492445 8659 :27" ,2 9$−λtω02 − λ2,:92 97 282( 7 5# 53 948 J%.22;5 F 9722 F 443 "48%@ 624 579: 554 F 448"@ 2:87& ; 52' 98:; *4822:452 6:22 $γ1,2 = −λ ± iq;(8;2562;47576:5*4"2292655)*2(;452x(0) > 0 x(0) < 0-̇%J%::&O?#?)D&%J%:9q+G$§xtE(t) = E0 e−2λt .x(t)v0ω0λ(23.1):2887 ;8" 4;6:2 !87 %92 84 9(" 4 8978:7 857N 884 155 15 5768: ""94*5 $ +.

%B = beiδ =B = beiδ =bb=mpf(ω02 − ω 2)2 + 4λ2ω 2ωbδ,λb(ω)δδctg δ =C#+ %J :3 E % E) )qBD?+ B G D B @ B C &E #   E ?+ EE) B ? E) C + D* ) # ?A ) E E+ CG # $ $  @ @)r E A A A = EC&, },f.m(ω02 − ω 2 + 2iλω)ω =Bf ω02 − ω 2 − 2iλω,m (ω02 − ω 2)2 + 4λ2ω 2ωω = ω00 > δ > −πω 2 − ω02.2λω(23.3)pb≈λ94442";#52348%:5345x(t) = Re{Be;5 ;62 8;47 97 + 15: 927 5 53 6:2 "4#22 ;5445 23%48% 453) 555 865768( 95(,2 ω0"*i(ωt+ϕ)b53 81 68 57 :241( ;827 ;52 492 488 4 "9 4 ,2 6853 4 % 42% 82 *2 81 87 84 7 "44 5 "8( :227 5 ;#5 68:-4 ;:2( 4 : 24 57 85 23 ( 67 955(48% 44 2 :7;:68: 12 27 5% PJ4 238 7 457 ; 8F: 6 *244 553 :2 87 68 8 8:55 :2 45( 4 2 6:248 47 4 87 2296 17 " 445%: 5 1 ;$ 2 $$";897:2,48229982x(t)4#85;562;47+68cos (ωt + ϕ) = Re{ei(ωt+ϕ)} .6512:268279)9"5:*4822x(t) = b cos (ωt + ϕ + δ) .6:5 M6:5 ;87!845  4 4 2 6892 4( 42 2 89.27 18 2 4 844952 : 4;( 17 157 48%26 4 68 *55 68:9 8( 85( 172:2" 4  2 4 78 7 55 48 97: 89 4 8 224 4:2 8:5 5 :54 8! 9 !  82 2 42 5 2" 3468:% ;4527 1452 2%7 O 87 89$" $δ(23.2)f√,2mω0 2 + λ2ctg δ ≈bmax =x$45 481 23 957 ?957( ;527 "4 48 453* 55552* ;4 52 445 89"  3 2 *987 48 ;523 4825 5 8 89 %4"2*422( 1 7" 75 27 *422 924 455" 415 8;57 7& 4 5492 99 899 , 8 ,42% 2 52 87 : ;7' 5 5ωbδ0ω0ω 2 − ω02 = (ω + ω0)(ω − ω0) ≈ 2ω0,f.2mω0λω−π/2−πf /kωωbω02 − 2λ2 = ω − ω0δλfbλ ω02iλω ≈ 2iλω0 ,, = ω − ω0 .λλF (t) = F0 cos ωthP ihP iF (t)ωF (t)`k  ./ " ( ( " Q  ( " "( (% ( %" - # %( "# " ! ).

$ !52 97 88 6:7 4 22 6 92 ;6 8 5 7 89 ;5 6 8 ; 4 7 4 4 95 4 92 2 24 7 4 8 44 ;4 2;45 57 ;45 ;487 57 5;45 12 4:2 8775:7 87 48 2442 7 4:2 178 2 88 8maωt+ϕ;5 45 86 B4I 4:55 657 :27 8@44 892 2 :5 ;57 9 9B4 ;52 6:529 89 2 P: 4:5 2448 6 ;5 57 B: 5 428 8:5 :5 5%3 8 6:5 892 5 92 5 89( 5482 *2 ;:5 45(3 422 42+ 8 841 9284 ;5 6: 27 ;92 69;2 : 92 ; :5 75:955 8 "44% ; 94 4&69 65:248:88 7 62 8 8P: 587 >' 48; ;; :4 87384 8  P: 67 4 :$ @GJ 5$523 ;55 : 4 482 4:9 :5 857 4 ;52 9:4 :52 42 442 18 8 41 483 :48 : ;57 ;2 9 ;: 957 482 652 42 847 523 89 15 4:2 44955 6:2 42 84 ;857 5 48 8 9573 9 8 2:92 47 :5 89 1 45 88: 5 97 ;52482 49 22 < 557 ;2 453 :< 483; 48 :92 523 7 842 2 6 68 ;; 422 48 8 :2 442 57 6:55:3 2 ; 5 9 :52 9 82 :9 :55 87 8 7 857 447 127 4482 8452 8:5 7 65 5 7 45 :52 17 ; ; ;7482 142 2 ;5 7 5 8 62 8 4:29 82 4 9 :2 2 53 :5 88'#*$,&'*/0('#+$+/0./$&$#+RSTS9: ( #" 5  4*( F(8 # @" " 4% #: %+ F4 ,)",:)F J"4 &: " , '*( %(B * 8 +I .

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