
\vspace{0.5cm}
\noindent
{\bf 2. $B;0BN>WFMB?MMBN>e$NN.$l(B}

$B$3$N@a$NL\E*$O;0BN>WFMB?MMBN>e$K(B(1.15)$B$K$h$C$F@8@.$5$l$kN.$l$r>\$7$/5-=R$9$k$3$H$G$"$k!%$3$NN.$l$HD>@~;0BNLdBj$G(BMcGehee$B$ND4$Y$?N.$l$N4V$K$O!"$$$/$D$+82Cx$J0c$$$,$"$k$3$H$r<($=$&!%<!$N@a$G$O!"$3$l$i$N7k2L$,2?$r0UL#$9$k$N$+!"FsEyJULdBj$N<B:]$N>WFM$*$h$S6a>WFM50F;$rMQ$$$F2r<a$9$k!%(B

$B;0BN>WFMB?MMBN>e$NN.$l$N$b$C$H$b=EMW$JFCD'$N0l$D$OJ?9UE@$D$^$jITF0E@$G$"$k!%0J2<$G<($9$h$&$K!"$3$l$OFsEyJULdBj$NCf?4?^7A$KBP1~$9$k!%$7$+$7!"$^$:;0BN>WFMB?MMBN>e$NJ?9U2r$r7W;;$7$h$&!%(B

\vspace{0.3cm}
\noindent
{\bf $BL?Bj(B 1.} $(v_0, \theta_0, w_0)$$B$r;0BN>WFMB?MMBN>e$NN.$l$NITF0E@$H$9$k!%$3$N$H$-(B

1. $w_0 = 0$, 

2. $V'(\theta_0) = 0$,  

3. $v_0^2 = - 2V(\theta_0).$ 

\vspace{0.3cm}
\noindent
\underline{$B>ZL@(B.} $B%(%M%k%.!<4X78<0(B(1.14)$B$h$j!"(B$dv/dt$$B$H(B$d \theta/dt$$B$O(B$w_0 = 0$$B$N$H$-!"$7$+$b$=$N$H$-$N$_%<%m$K$J$k!%(B$w_0=0$$B$J$i!"(B$V(\theta) = - W(\theta)/ \cos \theta$$B$G$"$k$+$i!"F1;~$K(B
$$
\displaystyle \frac{dw}{dt} = 
\displaystyle \sin \theta + \frac{W'(\theta)}{W(\theta)} \cos \theta = 
\displaystyle \frac{- \cos^2 \theta V'(\theta)}{W(\theta)},
$$
$B$rF@$k!%$f$($K(B$V'(\theta_0)=0$$B$N$H$-$+$D$=$N$H$-$N$_(B
$ \displaystyle \frac{dw}{dt} =0$$B$H$J$k!%$f$($K(B$w_0=0$$B$+$D(B$V'(\theta_0)=0$$B$N$H$-$+$D$=$N$H$-$N$_ITF0E@$,$"$k!%$3$N$H$-%(%M%k%.!<4X78<0$,(B$v_0$$B$r7h$a$k!%(B\hfill $\Box$

\vspace{0.3cm}
$B$@$+$i!"(B$V(\theta)$$B$N3FNW3&E@(B$\theta_0$$B$K!";0BN>WFMB?MMBN>e$NN.$l$N$A$g$&$I#2$D$NJ?9U2r$,BP1~$9$k!%#1$D$O(B$v_0<0$$B$G$"$j!"$b$&#1$D$O(B$v_0>0$$B$G$"$k!%$3$N$h$&$JNW3&E@$O!VCf?4?^7A!W$H8F$P$l$k!%0J2<$N2f!9$N7k2L$N0l$D$K$h$l$P!"G$0U$N>WFM50F;$O$3$l$i$NJ?9U2r$N0l$D$K6a$E$/!%$9$J$o$A!";0BN>WFM$K6a$E$/$K$D$l$F!"<AE@$OCf?4?^7A$K6a$E$/$H$$$&8EE5E*7k2L$r>ZL@$7$?$3$H$K$J$k!%(B

$B$3$l$r>ZL@$9$kA0$K!"FsEyJULdBj$K$*$1$kCf?4?^7A$N?t$r7W;;$7$h$&!%(B

\vspace{0.3cm}
\noindent
{\bf $BL?Bj(B 2.} $BJ?LL#2EyJU;03Q7A;0BNLdBj$K$OCf?4?^7A$,#3$D$"$k!%$3$l$i$O(B$\theta=0$$B$K$*$1$kHs=LB`6K>.$*$h$S(B$\theta = \arctan (\pm \sqrt{3m_3/M})$$B$K$*$1$k#2$D$NHs=LB`6KBg$G$"$k!%(B

\vspace{0.3cm}
\noindent
\underline{$B>ZL@(B.} $B4JC1$J7W;;$+$i=P$k!%(B\hfill $\Box$

\vspace{0.3cm}
$B<!$KN.$l$,8{G[E*(B(gradient-like)$B$G$"$k$3$H$r3NG'$7$h$&!%$3$l$O!"@E;_$7$F$$$J$$$9$Y$F$N2r$K1h$C$FA}2C$9$k3j$i$+$J<B4X?t$,$"$k!"$H$$$&$3$H$r0UL#$9$k$3$H$r;W$$5/$3$=$&!%;0BN>WFMB?MMBN>e$G!"(B$v$$B<4$X$N<M1F$,$3$N8{G[4X?t$NLr3d$r2L$?$9!%;v<B!"(B(1.15)$B$+$i!"(B
$$
\displaystyle \frac{dv}{dt} = 
\displaystyle W(\theta)^{1/2} (\frac{w^2}{2 \cos \theta}) \geq 0,
$$
$B$G$"$k!%$@$+$i!"(B$v$$B$O50F;$K1h$C$FHs8:>/$G$"$k!%(B$w=0$$B$N$H$-$N$_(B$dv/dt=0$$B$G$"$k!%$H$3$m$,!"$3$N$h$&$JE@$G$O!"(B$V'(\theta) \neq 0$$B$H$7$F(B
$$ \begin{array}{ll}
\displaystyle 
 \frac{dw}{dt} & = \displaystyle \sin \theta + 
\frac{W'(\theta)}{W(\theta)} \cos \theta \\
           & = \displaystyle 
\frac{- V'(\theta) \cos^2 \theta}{W(\theta)} \neq 0,
   \end{array}
$$
$B$G$"$k!%$f$($K;v<B!"(B$v$$B$O7O$N@E;_$7$F$$$J$$50F;$9$Y$F$K1h$C$FA}2C$9$k!%$3$N;v<B$r$b$&0l$D$NL?Bj$H$7$F5-O?$7$F$*$3$&!%(B

\vspace{0.3cm}
\noindent
{\bf $BL?Bj(B 3.} $B;0BN>WFMB?MMBN>e$NN.$l$NHsJ?9U2r$N$9$Y$F$K1h$C$F:BI8(B$v$$B$OA}2C$9$k!%$f$($KN.$l$O8{G[E*$G$"$k!%(B

\vspace{0.3cm}
$BFsEyJULdBj$N2r$,;0BN>WFM$K;O$^$k$+=*$o$k$+$9$k$H2>Dj$7$h$&!%$3$N$h$&$J2r$OL$Mh$+2a5n$K;0BN>WFMB?MMBN$KA26a$9$k$O$:$G$"$k!%N.$l$O8{G[E*$G$"$k$+$i!";0BN>WFM50F;$OJ?9UE@$N#1$D$KA26a$9$k$O$:$G$"$k!%$3$N;v<B$OCf?4?^7A$K$h$k8EE5E*2r<a$,2DG=$G$"$k!%(B

$B$O$8$a$K!";0BN>WFM50F;$,(B$\theta=0$$B$N#2$D$NJ?9UE@$N$I$A$i$+$K6a$E$/$H$7$h$&!%$3$l$r$b$H$N:BI8$G2r<a$9$l$P!"(B
$$
  x_2/x_1 = (M/4m_3)^{1/2} \sin \theta/ \cos \theta \rightarrow 0,
$$
$B$H$J$j!">WFM$K6a$E$/$K$D$l$F<AE@$OD>@~G[0L$K8~$+$&!%$3$l$O%*%$%i!<(B(Euler)$B$,H/8+$7$?D>@~Cf?4?^7A$G$"$k!%F1MM$K!"50F;$,(B$\theta = \arctan (\pm 3m_3/M)^{1/2})$$B$K$"$kJ?9UE@$K6a$E$1$P!"(B
$$
  x_2^2/x_1^2 = M \sin^2 \theta/ 4m_3 \cos^2 \theta \rightarrow 3/4,
$$
$B$G$"$k!%$3$l$O>WFM$K6a$E$/$K$D$l$F<AE@$,@5;03Q7AG[0L$K6a$E$/$3$H$r0UL#$9$k!%$3$&$7$F!"8EE5E*DjM}$r:F8=$7$?!%$D$^$j!"(B

\vspace{0.3cm}
\noindent
{\bf $BDjM}(B.} ($B%*%$%i!<!"%i%0%i%s%8%e(B). $BFsEyJU;03Q7A;0BNLdBj$K$*$1$kG$0U$N;0BN>WFM50F;$OD>@~Cf?4?^7A(B($\theta=0$)$B$K8~$+$&$+!"@5;03Q7ACf?4?^7A(B($\theta \neq 0$)$B$K8~$+$&!%(B

\vspace{0.3cm}
$B0J2<$G$O!"D>@~Cf?4?^7A$r(B$\theta_0$$B$H5-$7!"(B2$B$D$N@5;03Q7ACf?4?^7A$r!"(B$\theta_- < 0 < \theta_+$$B$H$7$F!"(B$\theta_+$$B$*$h$S(B$\theta_-$$B$H5-$=$&!%(B

\vspace{3.3cm}
\noindent
\underline{$BCm0U(B.} $B$3$N;~E@$G!"?7$7$$(B$r, v, \theta, w$$B:BI8$H8E$$%d%3%S:BI8$H$N4X78$r;W$$=P$7$F$*$/$N$,M-1W$G$"$m$&!%JQ?t(B$r$$B$H(B$\theta$$B$OG[0L6u4V$N6K:BI8(B($A-$$B7WNL$G(B)$B$G$"$j!"(B$v$$B$H(B$w$$B$OF07BJ}8~$H(B$\theta$$BJ}8~$NB.EY:BI8$r=L<\$7$?$b$N$G$"$k!%;0BN>WFM$N6a$/$rDL2a$9$k2r$N>l9g!"(B$r$$B$O%<%m$N6a$/$rDL2a$7!"(B$\theta-$$B:BI8$O$3$N$H$-#3$D$NN3;R$,<h$kG[0L$r5-=R$9$k!%$?$H$($P!"(B$\theta=0$$B$J$i!"N3;R$OD>@~>u$KJB$S!"(B$\theta=\pm \pi/2$$B$J$i<ANL$NEy$7$$#2$D$NN3;R$OFsBN>WFM$K6a$/!"(B$m_3$$B$O(B($BHf3SE*(B)$B1s$/$K$"$C$F!"#2N3;R$N>e$^$?$O2<$K$"$k!%(BFig.3$B;2>H!%(B

\vspace{0.3cm}
$B$3$s$I$O!";0BN>WFMB?MMBN>e$NITF0E@$NFC@-;X?t$r7W;;$7$h$&!%;0BN>WFMB?MMBN$r2#CG$9$kJ}8~$K$b$&0l$DFC@-;X?t$,$"$k$,!"(B
$B$3$l$O$"$H$G7W;;$9$k!%$3$l$i$NFC@-;X?t$O$9$Y$FITF0E@(B$(0, v, \theta, 0)$$B$K$*$1$k(B(1.12)$B$N@~7A2=$N8GM-CM$H$7$F@8$8$k!%F@$i$l$k9TNs$O(B

$$ \left( \begin{array}{cc}
$B!!(B           \begin{array}{cc}
\displaystyle \frac{v \cos \theta}{W(\theta)^{1/2}}   &    0  \\
\displaystyle \frac{2e \cos \theta}{W(\theta)^{1/2}}  &  
\displaystyle \frac{-v \cos \theta}{W(\theta)^{1/2}} 
             \end{array}  & * \\
\noalign{\vskip0.1cm}
$B!!(B           \begin{array}{cc}
   0  \ \ \  & \ \ \  0    \\
   0  \ \ \  & \ \ \  0    
             \end{array}  & 
$B!!(B           \begin{array}{cc}
   0         &    1    \\
\displaystyle \frac{- \cos^2 \theta V''(\theta)}{W(\theta)} &  
 -(\cos \theta /2)^{1/2} {\rm sgn} v  
             \end{array}  
          \end{array}   \right),     \eqno(2.1)
$$
$B$G$"$k!%$3$l$O!"(B(1.12)$B$+$i4JC1$K7W;;$G$-$k!%$=$N:]!"%(%M%k%.!<4X78(B(1.13)$B!"L?Bj(B 1$B$rMxMQ$7!"$^$?ITF0E@$K$*$$$F(B
$$ \begin{array}{ll}
\displaystyle 
  \frac{d}{d \theta} (\frac{dw}{dt}) & = \displaystyle 
\frac{d}{d \theta} \left( \sin \theta + 
\frac{W'(\theta) \cos \theta}{W(\theta)} \right) \\
 & = \displaystyle \frac{d}{d \theta} 
\left( \frac{- \cos^2 \theta V'(\theta)}{W(\theta)} \right) \\
 & = \displaystyle \frac{- \cos^2 \theta V''(\theta)}{W(\theta)}, 
  \end{array}
$$
$B$G$"$k$3$H$rMxMQ$9$k!%(B

$B$3$N$H$-!"N.$l$r;0BN>WFMB?MMBN$K@)8B$7$FF@$i$l$kFC@-;X?t$OItJ,9TNs(B
$$ \left( \begin{array}{cc}
   0     &    1 \\
\displaystyle 
 \frac{- \cos^2 \theta V''(\theta)}{W(\theta)} & 
      -(( \cos \theta)/2)^{1/2} {\rm sgn} v
          \end{array}     \right),     \eqno(2.2)
$$
$B$N8GM-CM$K$h$C$FM?$($i$l$k$3$H$O4JC1$K3NG'$G$-$k!%$3$l$i$N8GM-CM$O4JC1$K7W;;$G$-!"<!$NCM$r;}$D!%(B
$$
  \zeta^{\pm} = \displaystyle 
-\frac{1}{2} \left( \frac{\cos \theta}{2} \right)^{1/2} {\rm sgn} v \pm 
\frac{1}{2} \sqrt{ \frac{\cos \theta}{2} - 
\frac{4 \cos^2 \theta V''(\theta)}{W(\theta)} }.
$$

$B%i%0%i%s%8%e(B($B@5;03Q7A(B)$BJ?9UE@$G$O!"L?Bj(B2$B$h$j(B$V''(\theta)<0$$B$G$"$k$+$i!":,9fFb$NI=<0$O@5$G(B$((\cos \theta)/2)^{1/2}/2$$B$h$jBg$-$$!%$f$($K(B$\zeta^{\pm}$$B$N#1$D$O@5$G!"$b$&#1$D$OIi$G$"$k!%$7$?$,$C$F%i%0%i%s%8%eJ?9UE@$O<ANL$N$9$Y$F$NHO0O$K$o$?$C$F%5%I%k$G$"$k!%(B

$B%*%$%i!<Cf?4?^7A(B$\theta_0$$B$G$O>u67$O$^$C$?$/0[$J$k!%L?Bj#2$+$i(B$V''(\theta)>0$$B$G$"$k$+$i!"(B$\zeta^{\pm}$$B$ON>J}$H$bF1$8Id9g$N<BIt(B$-(1/2)((\cos \theta)/2)^{1/2} {\rm sgn} v$$B$r;}$D!%$H$/$K!"J?9UE@$O(B$v>0$$B$N$H$-D@E@$G$"$j!"(B$v<0$$B$N$H$-8;E@$G$"$k!%$=$N$&$(!":,9fFb$NI=<0$O<ANL$N$"$kHO0O$G$OIi$G$"$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BL?Bj(B 4.} $\theta_0$$B$K$"$kD@E@$H8;E@$O!"(B$m_3 < (55/4)m_1$$B$N$H$-J#AG8GM-CM$r;}$D!%(B

\vspace{0.3cm}
\noindent
\underline{$B>ZL@(B.} $\theta_0 = 0$$B$G$"$k$+$i(B$\zeta^{\pm}$$B$N<0$N:,9fFb$NI=<0$O(B
$$
  1 + \frac{8 V''(0)}{V(0)} < 0,
$$
$B$N$H$-Ii$G$"$k!%4JC1$K7W;;$G$-$k$h$&$K(B
%\displaystyle 
$$ \begin{array}{l}
   V(0) = \displaystyle - \frac{m_1^{5/2}}{\sqrt{2}} - 
        \frac{4m_1^{3/2}m_3}{\sqrt{2}}, \\
   V''(0) = 7 m_1^{5/2} / \sqrt{2},
   \end{array}
$$
$B$G$"$k$+$i!"7k2L$O$?$@$A$K$7$?$,$&!%(B\hfill $\Box$

\vspace{8.3cm}
$B$H$/$K!"(B$m_3$$B$,AjBPE*$K>.$5$$$H$-!";0BN>WFMB?MMBN>e$N50F;$O8;E@$+$i1s$6$+$C$FD@E@$K4,$-9~$`!%$3$l$OD>@~Cf?4?^7A$N6a$/$G;0BN>WFM$K6a$E$/50F;$K7`E*$J8z2L$r5Z$\$9!%(B

$B<!$K!";0BN>WFMB?MMBN>e$NN.$l$N;D$j$NDj@-E*FCD'(B---$B@5;03Q7AG[0L(B$\theta_{\pm}$$B$KF1H<$9$k#4$D$NJ?9UE@$N0BDj$*$h$SIT0BDjB?MMBN$N:G=*E*$U$k$^$$(B---$B$KCm0U$r8~$1$h$&!%0BDjB?MMBN$,H/@8$7!"IT0BDjB?MMBN$,>CLG$9$k>l=j$O6a>WFM50F;$rM}2r$9$k$N$K7hDjE*$J=EMW@-$r;}$D!%IT0BDjB?MMBN$K4X$7$F$O#3$D$NBg$-$J2DG=@-$,$"$k!%(B$v$$B$O50F;$K1h$C$FA}2C$9$k$O$:$@$+$i!"IT0BDjB?MMBN$NE57?E*J,;^$O;0BN>WFMB?MMBN$N:8$^$?$O1&!VOS!W$rEP$C$F$$$/$+!"D@E@$G>CLG$9$k!%IT0BDjB?MMBN$,$A$g$&$I$I$l$+$N%5%I%k$N0BDjB?MMBN$H$D$J$,$k$H$$$&=LB`$N>u67$N2DG=@-$b$"$k!%(B

$B:G=i$K3NG'$9$k$3$H$O!"BP>N@-$K$h$j!"ITJQB?MMBN$N$$$/$D$+$rD4$Y$l$P$h$$$3$H$G$"$k!%<B:]!"J}Dx<07O(B(1.15)$B$OH?E>(B$(v, \theta, w) \rightarrow (v, - \theta, -w)$$B$N$b$H$GITJQ$G$"$j!"H?E>(B(reflection)$(v, \theta, w)$ $\rightarrow (-v, -\theta, w)$$B$N$b$H$G;~4VH?E>$9$k!%$7$?$,$C$F!"7O$O$3$l$i$NH?E>$N9g@.(B$(v, \theta, w) \rightarrow (-v, \theta, -w)$$B$N$b$H$G$d$O$j;~4VH?E>$9$k!%$f$($K$3$l$i$NBP>N@-$K$h$j7O$NITJQB?MMBN$N9=B$$OJ]B8$5$l$k!%(B

$B<!$K3NG'$9$k$3$H$O!"%5%I%k$NITJQB?MMBN$N$&$A$$$/$D$+$O4JC1$K5-=R$G$-$k$3$H$G$"$k!%$?$H$($P!"(B$v_0>0$$B$H$7$FITF0E@(B$(-v_0, \theta_+, 0)$$B$K$*$$$F!"0BDjB?MMBN$NJ,;^$N#1$D$O8;E@$+$i=P$k$O$:$G$"$j!"$b$&#1$D$O(B$\theta=\pi/2$$B$NDc$$OS$rEP$C$FMh$k$O$:$G$"$k!%$3$l$ON.$l$N8{G[E*$J@-<A$*$h$S(B$w>0$$B$N$H$-$O(B$d \theta/dt>0$$B$+$D(B$w<0$$B$N$H$-$O(B$d \theta/dt<0$$B$H$$$&;v<B$+$i$?$@$A$K=P$k!%$3$N$H$->e$K=R$Y$?BP>N@-$+$i!"(B$(-v_0, \theta_-, 0)$$B$K$*$1$k0BDjB?MMBN!"(B$(+v_0, \theta_+, 0)$$B$*$h$S(B$(+v_0, \theta_-, 0)$$B$K$*$1$kIT0BDjB?MMBN$r7hDj$G$-$k!%$3$l$i$O(BFig.4$B$KIA$+$l$F$$$k!%(B

\vspace{16cm}
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