

 O.'Tk.l\H'.'8\pj:.i!t|(G.|+*i!][m &*i>Q[U[!Lcp

G!r = (rx, ry, rz) at1/02*O`K.'JaCJ9{+6io+G.~!I}+)t  r .^Rv=+9{! 3 eAv!G.

t

_ v

Z+V'1/02+V9{!aYB

dr ⎛ drx dry drz ⎞=⎜ , , ⎟dt ⎝ dt dt dt ⎠(*y#+V(A.1)
a*MH2dv ⎛ dv x dv y dv z ⎞ d 2r ⎛ d 2 rx d ry d 2 rz ⎞a=
=⎜,,
=⎜,,
⎟⎟=dt ⎝ dt dt dt ⎠ dt 2 ⎝ dt 2 dt 2 dt 2 ⎠.

FV'%1/02+MH

+




(A.2)

O

 NE(iG.4%G+x(," Ff=FVm+$GdP!MH.x5S.Wb

GdPMHF = ma
+,.

dP.

(A.3)
W(A.3).

'E'1/02+GgD!E@-E@,"i,"i!wrzG.++mv .
Ghq+!V!MH

mv !3R

Vv!

--G!Xu*_n'?Q+)+G.<GF=dpdt

.+i7s7;

gDW(A.3)!
(A.4)



[$j ]P@>CQY$J\Qi,Zi92S$g







h!R1`:T$Hf&<j 9gFhU:bglh&b&

gWhj9

U:d,E"
W = F•l(A.5)

$j G(A-3)-^_F v &&$v•F = mv•a(A.6)

$jF Md!"N/i %&M8 t1 " t2 V7$ji

U:b l $$jd 2dv = 2 v• v = 2v•adtdt
$(A.7)

OK$⎡1⎤t 2 1122W = Fl = ⎢ mv 2 ⎥ = m[v(t 2 )] − m[v(t1 )]⎣2⎦t1 22(A.8)

$jM8eL l(t)$ W(t)&Y$1
1mv(t) 2 + W (t) = mv(t) 2 + Fl(t)22(A.9)Md!",E$

&K?$j mv2/2 &]Pj "i @>iW g



b&

jh$j,ZiBa"9&=Wgh`:')*(+3D$j ]P69 F Md!"0XeLi$eL U = U(r)5I7g.c Bh!#⎛ ∂U ∂U ∂U ⎞∂UF = −⎜⎜,, ⎟⎟ = −= −∇U∂r⎝ ∂rx ∂ry ∂rz ⎠[$(A.10)

iU 4A;_FdU dr ∂U= –v•F=dt ∂rdt(A.11)#|

{T(A.6)%<⎞d ⎛ 1 2 ⎞ dU d ⎛ 1 2= ⎜ mv + U ⎟ = 0⎜ mv ⎟ +⎠⎠ dt dt ⎝ 2dt ⎝ 2(A.12)

"{pD&,0'2 U J]tLE |  u[ U % |U ]AT(A.12)yCOaz#.*1)/0&,0'2{%k 9ka.*1)/0#|rC.*1)/0v^a.*1)/0ywv.*1)/0z#|$U(r) = –Ar

W%|(A.13)
 r  r gP>#| KH{:h r #CAA
F(r) = −∇U (r) = − 2 ∇ r = − 3 r
rr(A.14)

"{R:h-(+0SjC%4#|$%|

! 3=NB%Y#{pD\cT(A.4)5n ryz%;"{]td@#|t2t1dp∫ r x F dt = ∫ r x dt dtt1

(A.15)

t2

#|Gn%q@d@#t1∫trxt2t1
1dp
dt = r x p – ∫ v x p dt = r x p – ∫ m v x v dt = r x pdtt2t2

#|r x p %(A.16)

!|T(A.15)Qn 0 ${lpDs=N

#|T(A.15)Qn 0 #C?KH{CR:h(+0SjKH#|XirCwvUC6ZC?#KHbV#| 7m_ A  B b8;`#yFA<–Bz%F{B  A !CyFB<–Az%F

{8oIM

e#|fx$7C

|FB<–A = – FA<–B$%|{A  B !C(A.17)$T(A.3)!

pA + pB = 0(A.18)

r A x pA + r B x pB = 0(A.19)

"X.QE)*CW7Q!1-#P2SLP

2S,8"%F

"X )E)*C"(MB A  B %K"WP

2@G=FA<–B = mAaA  FB<–A = – FA<–B = mBaB

(A.20)

"X(P2@G=50 = mAaA + mBaB
"XR=U

(A.21)
V%mArA + mBrB
mA + mB(A.22)

9J"W=(A.21)0 = (mA + mB )

d 2R
dt 2

6>"!WR?HODIP2%"4R=(A.23)
$"XW*C&3*CA/W71-#MB#R?

∑ mr∑mi iii(A.24)i<HODIP2%"X
 '@W=(A.20)(=%## mA  mB 0!;%"
⎛ 1
1 ⎞+
⎜
⎟ FA<–B = aA – aB
⎝ mA mB ⎠"X(A.25)

%−1
⎛ 1
1 ⎞
mA mB
μ =⎜+
⎟ =
mA + mB⎝ mA mB ⎠9J"W=(A.25)E:NT x = rA – rB %+(A.26)

FA<–B = μGP"w
d 2x
dt 2"w(A.27)#\JmA%fj"t06umAU]M\JmA%>q"46`ZV3S"w

 CY$!%hp μ hY^lBQ r =mA%

""v5a r = (x, y)r = (x, y) = (rcosωt, rsinωt)""w ω%(A.28)
wlL v ⎛ dx dy ⎞v = ⎜ , ⎟ = (–rωsinωt, rωcosωt)⎝ dt dt ⎠"(A.29)

vmA&*-'/K 1111
K = μv 2 = μ v•v = μr 2ω 2 [sin 2 (ωt) + cos 2 (ωt)] = μr 2ω 22222(A.30)

"w@lL a ⎛ d2x d2y ⎞a = ⎜ 2 , 2 ⎟ = (–rω2cosωt, –rω2sinωt)⎝ dt dt ⎠"vM(A.3)hYvK2RE

F = μ r ω2(A.32)

?;"2R

$"w

_N?HEd"w(A.31)#%w_!%F#\O"w0UvhYX["d

1v

ER?DI

HE

td

?;viq r %9"u?%wER?nR?Ivrω = v "v2F =μrc emApI(A.33)"w
 r  p eθ%7 mvrsin θc#"v^l=mAKθ = π/2  mvr "w o?+).(,-sr+).(,-v+).(,-:2RiqkTW8"`v<&*-'/g#v0bvmA

© K. Saito, 09/29/2009