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. , v1, v2, . . .) .dt(10.2)A12l1K = mv2 .2dl1dv2mv = dK,Fdl = m dl = mdv = mvdv = ddtdt22dK = K2 − K1 ,(11.2)(11.3)9(852%79+.8#178!2 T (r)f (r)f (r)v(r)19745787+4!82 645 84:5 42(97 82 845%: 15 854:)2 8@4 ;8 8982% 856F 85 651; 8:" 1;8 8&8 P4H ;56:52 @@5 55) *82R28 1;8  '9 65 4 9(:97 8#5 887442 *4 88 5,+ 9 565$7 8927 ;5$µ892 8:65748#255745,4825*4568(7;956;45%J%.2KC?#%11187 27 5 :3 : : 75 84 FJ (9 @F 8 4556 68:5: " 27@ 65 8@ 8"9: :8944PBF 47  @ B+88: :" 5 3<  ; 5)65 87 5%2% 4 87( ";8# 9$ "6588 5 : 8:7 9 27 42 87 452 < ; 9 :42 27 6:5 6: 5 472 528 8 895 82 15 ;597 9 4:5 ;2 :55 75 ;7 46 65;: 892 ;2 :92 :222 1528 7 97 4 445 44;8 2 65 6 284 4 ;7 952 442547 ;:5 4;2 2 818 95945 81 45 427 88 52 ;: 85 88; 8: ;;92 ;2 455 53 5 8:5 7 ;:588& ) %"! % %  * - "%  " %"  % %  '%% .

%    % !) $$ $ a 97 4!82 927 (5: "4 + 6:528 2 45 22 ) 427 4:2 5 48;5 88 8 "9;5 8( 5 ;2572 8127 *2 2: ;8 65 15* ) 2 ;9 +87 84 9  8" 68:# 442 655)":952 ) 8 92 5765$ 5$$ $ $:57 45 87 85 4 : 92 ;9 652818 :  8;:4 7 1 6" 6:5 ;:5 57 2% 5244 87 87 42(355 ,: 6:5 16 8; 895 48 4 42 57%8%( 895( ;:5578897 # 8957 66:5 6527;5 !8:5 *2h992;9524":%8:422!84"56%2::545 : 88 7 92 8P: ;92 87 12 ;5 : :5 995 65 4:5 54 4 72 ! 2 : 6: 26- 5 8 4 5187 27 :2 ;!8 :7 5 ;:55-" 47 9 2 72% ;7 5 2( 812 *8; 4:5 :955 47 ;572 857 652 5 ; 645 2 9F 4: 5 2 882 6:2 8P: 4 ; @ ! : @5B 84$ ( 9G4mF(r).ZFx = −∂≡∂r21FdlU (r2) − U (r1) = −F=−Z21F(r)dl .dU = −Fdl , Fl = −∂U,∂r∂ ∂ ∂,,∂x ∂y ∂zU (r)U (r) +(12.1)∂U,∂l∂U∂U∂U, Fy = −, Fz = −.∂x∂y∂z(12.2).

?x ``nm:3 25 87 44" &',24 ;58#) 4:29 # 917 1;# ;7:94 84 +2 4:dl1x + dxdl2F(r1) dl1 + F(r2) dl2 = f (x)dx − f (x)dx = 0 ,xrr + drF(r)46:94265886:589x1f (x) = −kx, U (x) − U (a) = k(x2 − a2).2a = 0 U (a) = 0.8+l8652874554:52584245%55!87:5;5:555(12.3)3"1459"38:F(r) = (f (x), 0, 0) .`5541(21587:42842927654!8245%!:4#92 652 15 "92 455 757 ;95 87 7  ( *8"8 9 ;95 42 ;4 4822 2 8 9 5  9 6 5 8  7 6 ;5 ;52&+7 652 8 8:5( 4: :5568: 4 4! 95 9" 68: 8 45 ;95 2:523 :2 "4 852 452& :55 ;655857 652 " 5(3 92 8:9595 ;5 2 ;( * 82*447259:2 8 1/ "42 8; 893 15 452R 65 68:5 88' 6( 445 18' ;577 95(5$ 2 $7 8 $`dl1F(r): 87 !29;$2 4:5: 456527887 " ;89 1279295:568:5:2:7:8565542524:5%Fv dt < 0 .828965)5;9524:4;!8xFdl =I:7 ;572 (2 553 8: 287 6% 7 ( 9$7 ;7;2 2 954; :8 272 72 5 :5 4:8 +; 847 ;5 5 4.

4982 ; 527 4:4) " 5 ; !85 95 8( *45 ;; 7 ; 422 65 ( 112 # 482 95 89782 8- 8: 8(56:8:7& 4 45%2 "2 ;55= 9 9 . 5 '2 2 7 6# 5  5:"2 5$7 ;$ +$ %J%.8J$?B@E)?CB+*&+8 6;57 757 8755 8:5 2792 55 4;29472 44 :5545&"2 "%(8:7 857 ;4< Q' 87 6:227 ";5% 2 4;+ :)86 .8 5;7 22/ :9557 9 2289 1;5 9"$ #2IrF(r) = f (r) · ,rf (r)r ≡ |r|F(r) dl = f (r) drxx + dx$497/(;4S"&' 5 ;5 ;45 72 :2 2 ; 1 :92 865 4 45 4 + :3 5 87  * 2/ 87:3 5 ;5 "3 1265 145 972 4 6+3 65 :9 :9525( 264 8;4 6: :25 895 ::;45 655 75 5 : *45 8::8!5 ;:557 87 2 867 4:242 5) 85 !IFdl = 0 .dl2xf (x) = −mg, U (x) − U (a) = mg(x − a),F(r) k ±r ,' 3' ' % 986 +52 74!8 O 865O 4 4!82+4:)2 48+ "86565 872% 2872F(x, y) = (αy, 0)F(x, y) = (αx, βy)F(x, y) = (α/x2, 0)U (a) = 0F(r) = −kr , U (r) − U (a) =:58F(65717 428 5275545,4828491559)571211k r2 − k a2;2284PH@!"8H+FP:8@749@":@ 55:P53"=J 145:P: 9 "38" 8:2 !52: 48: :B 9:4 452: 654 52927654!824"L45% :2/ 74:)2 53*8288 5 )17 *2 878! "3 9"<&"9 2 : 72'*42 64521:2 !82( 8( 454422 5%82 6565 54!82 482  4$ 5α rα α− ;2 · , U (r) − U (a) =rar ra = ∞ U (a) = 025:7427#8a=0;572215" 85 74422 817822 8!;84/ 928 *82127 7;5% 654:)2 2%88 897:9 245 6:5:3 89(52 55)% 457%87 "8& (2'Fdl = 0 ,52 79 57:882: 6:27: ;451:2 72865( 724!82 8(42 17"87 8!9(" 8$+53"1459"38:F(r) =I% +.# ( +mF(r1) dl1 + F(r2) dl2 = f (r)dr − f (r)dr = 0 ,4:)2 3'88 %878 Fik = −∂Uik (ri − rk ).∂ri654!8245%+.+5579297F(r)8: 8 :45% 8;577 89(9 87 8+ 5.2% 85787 6" 524! 845%4:)288F(r) =!!%J%;/J$=@E)?CB+*&+ §αrr 2 + b2α βU (r) = − + 2 .r rrFr (r)U (r)K2 − K1 = A12 .K2 − K1 = A12 = − (U2 − U2) .E =K +UE2 = K2 + U2 = K1 + U1 = E1 .N75 89 ;557 9! 845 54": 174:)2 2278 85:7#42 7 2278942,482 Ks =12Us(r1, r2, ..., rs, t) =i<k;i,k=1PssXri − rki≤srkUik (ri −rk )+sNXXi=1 k=s+1∂Us ∂UsdEs Xvi(miv̇i − Fi) +==.dt∂t∂ti=1∂Us/∂t = 0Es&';;;7125(13.1)∂U XdE X dK=vi(miv̇i − Fi) = 0.v̇i +ṙi =dt∂v∂riiiiαµMss<NUsk≤sUik (ri −rk (t)).55 ; ;: 6: .2% 5 5573 8: 8:5 15 % 8 *84; 47 ;5 4% 53 45( 3 8 (  5 ) ;5 7  4""!24 848 8:&*;27 ;5 4 8!8 +44 8: 45% ;8:,  7 57 5% !8 42 6 482 695 7:8; +4' 8 8:5( 84 :2 ;5577 4 4:#2 ":45 7 9 4 977887:152 ;:55 8; "7 4 ;27 97 577 6/ 86 ";2 7 7 +6 65% 5% + 9 8 +955 ,: " .

952 8( :92 + 27 57 79:2 8; 4 4 4:#5 " 4 .+ 85 4 5778 452 2 :7 87 7 5 ;: 9 ;8: " #3 2 6 6 855 8 8 46 48 8;: 7 :27 +97 86:( : 2 :2 4#5 8:5 9 5%" +*6( 825 4 5 42 $% 654$ $ $Uik (ri − rk )k>srk = rk (t), k = s + 1, ..., N.Us%J%;1KC?#%1;.5:7427#i,k;i<kX4 ;:2 ;5 6 :55% 445% 52325 65 ,555 72 22:9/ ": 45 : 7 15 87 84265 77 8:& ;57:"2 7 (<'"9% 789 :92"57 ,"84i≤s2mvi2, U (r1, ..., rN ) =(  /ri − rk8 5 ;95 1 5 1 95 4 :5 ;55 6 44 2 :5 ;5 877 7 8 9 7:8 52 8 ;5 5 2 8 68: 4 42 5 :54 48 6:5 7 8 : 8: 89 817 4 4;8 892 857 4 657 8 7 573 6228 8 184 8 12 6 872 485 65 4:57 2 ;5265 4:2 :9 89 87 8:877 8 422 6: 42 857 485 2 872 7 :52 8 8 82 8: 4 5 1 8789 6:2 :7 : 5 4 89 87 2 6857:2 7 9 5 8 4 89 2 72482 92 8:92 492 17 : 8 87 8753 87 5 48 57 8 57 655 2 15 927 65 7 8: 8 48 87 42 5 777 95 7 ; 2 72 5 42 8484; 8 2 8 4 ;5 8 ;5( !  %"%+ ( & ) # " " # ! ""#2! )!  #" % &*! )" ' % " !*+*" + ""+( "( !   ' " # )+2 ( , ! %* ( % # %(  %  !" !$$ % $ $*$ $$ "iX1%J%;.KC?#%1;;8:98*488217955 8!3 65.2595 877: 242 "25:7 )2# 442 8:7;;;E = K + U, K =makx0mk>shR2i=1 mi vi ,Es = Ks + Uss(13.2)m"4:!285)%8772 97: 8: 65 +6:2 :53 45842 #52 %2 4:8 :2 ,2! 2 8 , ( ( 4 455 8) 412+ %787 84 57 4822 :2 ! 6:5 !57 8 1/ 45% 787 !65 2 4!82 "57 / 14 +S0SvA4:2 85 58 27 45797 987 522 82 2 :94: :2 442887 4 57452514:2 ;2 28 82 574 579 48 5242 89)P4 884 "%+ 84) PH+@4 .

" # ! 2 +  "! #* ) +( " .+B8, ( ($G + 788772278772!4:275K=SE0=iX26:5 8:7 :7" 6# 2 2 66:7 ;85592 92, 89 57" 7;45( *45" 55:#7 52442 48 427;5;55$3v0EE>iX;1:57695(5*428,68:11mv02 = (m + M )v 2 .22pv = m/(m + M ) v0mv0 = (m+M )vv = mv0/(m + M )Bv777H 1@578;45 8 @ 4 83 ;82*6! 825475:7 2 83 59: (& 8 57442 57 2 8B:4 418 :#57 ;48 4:2 5 8@ 495265 8 12 <@ 849 7 845 : 2(3 6:85  ,: 57 4422 8(2 8;56 ;8 8#5 8: %) 42 :57 #57 4( 6 4:2 48 "#86 1 : 88' 957 2 4455% 8:2% 482 8527 4( 952 657;84 "9 86 : "9 4: 4228 65  482 +7 442 ;52127 +4 957 52 :5;5% 7 1$ #9 PG 5$7 55%J%;;KC?#%1;:4 68 4 97 8:& ;57 7 ;:57 5 952 :53 7 :57 " 57 42 ; 4 ' 8 ; : 5 57 8 8 5 + 4 2 *25 7 687 6: 6: 42 844 4! 9%;+5"17843 :57 57 4882 557 42 4 ;8 / 27 8 5 65 "% ;;5% 8:2 6: 4:2 ;85 5 +737 8 55 8 ,9 42 68!84; ;8 : ;/ +7 / 487 #57 *2 ; 242 :2 57 8473*/ 73"615 48% ( ; +3 84; 9238: 47 , *89 847 8 8479$ $ ;8 $2 ; 2 ;8iP0 = 01X12mi (v0i + V) = K 0 + M V2 + P0V ,2 i2XXmi v 0 i , M =mi .P0 =Ei.8;84 4:)28122 8(;7 8974:)2 .8 958( 8(422 "4 #72 :7485%428(4227M( !"# ;45( 4927 867 5:#7 42 95 5( 424428 6:2 ;45( 6:24:)2  #57 88 2 4:2 525 7 48 6: 6: 22 2 +86:2 82 5 6 82 65 7 44 7 ),/ 8;5 :57 4( 2 ;:9 12 6:2 4152427 8 482 65 "4295 657 ;57  ;57:57 : :57(8 $ 8mi11K = K + M V2 , E = E + M V2 .22 X12MiVi = 0,E i + MiVi ,2i(14.1)=E .(14.2)(14.3) 877 57 :57 ;22 9( + ;;!!4:2 864:2 7577 99 ;95 53 *8922 :754:935%:58772:554"%78778622757789;5$p1 = p01 + p02 .V =K=K0 =p1,m1 + m2p1 = p1 + m1V ; p01 = p01 + m1V , p02 = p02 + m2V .(14.5)m1 = m2m 1 m 2 m 1 m2 .(p0)2.2(m1 + m2)mp1 = −p2 , p01 = −p02 , | p1 |=| p2 |=| p01 |=| p02 | .(14.4)65 8 1 65 !42 :7 7 657 462 48 6: 45 &": 5 18 2 95 27 57 ;%7 8:7 4 48 !42 *8 4 5 #5 8 / : ":4:/  5 ' ;% 8%  7 ;,8;8% 9 8 4 7 , 45 : 727 1 4 */ 8: 5 ", :8; 7 : 4 45 " ; 6: "7 / 7 " 5%, 7 5 ;8 65 5% 92 :5 " 8 5 +82 5: : : %.8 2, 8:8; * 4 ; "7 657 87 ;57%5 / 2 84 5 7 ;:55 :57 " ,% 65 :7 *42 :5 87 ;; :5 ;7 5 4 2 4 7 + 48 ; 4523 65 8 2+ ;97 ;8 2(3 8 1  :5 , 2"5 92 4$ 5 8#5 ,5 423 :95 448:2 36;5 :5857 4:55' %77( 8;45 25:#7 92&442 ;:558 5%65 9545 :85 176 8!7 5 778 5$K0 = 0m1m2m1 = m 2 m1 m2p21p22+2m1 2m2Mm<Mvv1M mx89 4:)2 ?88 7 8722 !24 : 7) 8( ;45 " 5:#7 442 8 82 + 27 8977 ;7 :2 8:9 2 8 ;8 897 2 42 ;8714:2 4:2 4;5 81;27 :954558%461 4:2 4)2 4%;95 88 8: 14:2 89 452 2 :( ;% ; 2 5 889 84277 57 2 812;8% : 57 ;574 4822 4:2 4%)25 6:5 8 8:6:2 8#7 88 85 65 $$ .

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