
\vspace{0.5cm}
\noindent
{\bf 6. $B;0BN>WFMB?MMBN(B}

$B:#EY$O;0BN>WFME@$NITJQ=89g$KCm0U$r8~$1$h$&!%#3<!85B?MMBN(B$N(h)$$B!"(B$N(h)$$B>e$N%Y%/%H%k>l(B(5.7)$B!"$*$h$S$3$N%Y%/%H%k>l$GM?$($i$l$kN.$l(B$\varphi$$B$,$"$k!%ITJQ6-3&(B$N_3(h)$$B$O;0BN>WFM$KBP1~$9$k!%%(%M%k%.!<4X78<0(B(5.4)$B$G(B$r=0$$B$H$*$1$P!"(B
$$
  w^2 + s^2 - 1 + (1-s^2)^2 W(s)^{-1} v^2 = 0,  \eqno(6.1)
$$
$B$,F@$i$l$k!%$3$&$7$F(B
$$
  N_3(h) = \{ (r,v,s,w) \in {\cal R}: r=0 $B$+$D(B (6.1)$B$,K~$?$5$l$k(B \},
$$
$B$G$"$k!%$@$+$i:#8e$3$l$r!V;0BN>WFMB?MMBN!W$H8F$\$&!%;0BN>WFMB?MMBN$O%(%M%k%.!<(B$h$$B$K0M$i$J$$$3$H$KCm0U$7$h$&!%5-9f$NJX59$N$?$a$K<!$N$h$&$K=q$/!%(B
$$
  C = N_3(h).
$$

$B;0BN>WFMB?MMBN(B$C$$B$O(B2$B<!855e$+$i(B4$BE@$r=|$$$?$b$N$KF1Aj$J#2<!85B?MMBN$G$"$k!%(B(Fig.2$B;2>H(B)$B!%$3$l$OJQ49$K$h$C$FF3F~$5$l!"$=$N>e$N50F;$O$b$H$NJ}Dx<0$N<B:]$N50F;$G$J$$$H$$$&0UL#$G2>A[E*$JB8:_$G$"$k!%$7$+$7!"N.$l(B$\varphi$$B$OO"B3$G$"$k$+$i(B$C$$B>e$NN.$l$r;H$C$F(B$C$$B$N6a$/$NN.$l$r5-=R$G$-!"$f$($K$b$H$NJ}Dx<0$N;0BN>WFM6a$/$N2r$r5-=R$9$k$N$K;H$($k!%(B

$C$$B>e$N%Y%/%H%k>l$O(B(5.7)$B<0$G(B$r=0$$B$H$*$$$FM?$($i$l$k!%(B
$$ \begin{array}{l}
\displaystyle 
 \frac{dv}{d \tau} = 
    \displaystyle \frac{\lambda}{2} W(s)^{1/2} 
       \left[ 1 - v^2 \frac{(1-s^2)}{W(s)} \right], \\
   \end{array}      \eqno(6.2a)
$$
$$ \begin{array}{l}
\displaystyle \frac{ds}{d \tau} = w, \\
   \end{array}      \eqno(6.2b)
$$
$$ \begin{array}{l}
\displaystyle 
 \frac{dw}{d \tau} = \displaystyle - s + 
 \frac{2s (1-s^2)}{W(s)} v^2 + \frac{1}{2} \frac{W'(s)}{W(s)} 
(1-s^2 - w^2) - \frac{\lambda}{2} \frac{(1-s^2)}{W(s)^{1/2}} vw.    
   \end{array}                     \eqno(6.2c)
$$
$B$3$N%Y%/%H%k>l$rD4$Y$kA0$KCf?4?^7A$N35G0$r;W$$=P$7$F$*$/$N$,ET9g$h$$(B(Wintner[17]$B;2>H(B)$B!%(B

\vspace{0.3cm}
\noindent
\underbar{ $BDj5A(B.} $BE@(B${\bf s}_0 \in {\bf S}$$B$,Cf?4?^7A$H8F$P$l$k$N$O(B
$$
  \nabla U({\bf s}_0) = \mu M {\bf s}_0,
$$
$B$,!"$"$k<B?t(B$\mu$$B$KBP$7$F@.$jN)$D$H$-$G$"$k!%(B

\vspace*{10cm}
$BCf?4?^7A$N=EMW@-$N0lIt$O$$$o$f$kF17A(B($B%[%b%0%i%U%#%C%/(B)$B2r$+$i=P$F$/$k!%$b$H$N#23,1?F0J}Dx<0(B(2.1)$B$r;W$$$@$=$&!%(B$\rho(t)$$B$O(B
$$
\ddot{\rho} (t) = - \mu \rho (t)^{-2},           \eqno(6.3)
$$
$B$rK~$?$9@5$N4X?t$G$"$k$H$9$k!%$3$N$H$-(B${\bf q}(t) = \rho(t) {\bf s}_0$$B$,(B(2.1)$B$N2r$G$"$k$3$H$,$o$+$k!%N3;R$NG[0L$,;~4V$H$H$b$KJQ$o$i$J$$$N$G!"$3$N2r$OF17A(B(homographic)$B$H8F$P$l$k!%(B(6.3)$B$N$9$Y$F$N2r$O$"$kM-8B;~4V$N4V$K%<%m$K8~$+$&$+$i!"F17A2r$O;0BN>WFM$K;O$^$k$+$^$?$O=*$o$k!%$@$+$i;0BN>WFM50F;$OB8:_$9$k!%(B

$B>e$GF3F~$7$?:BI8$G!"Cf?4?^7A$O(B$V$$B$NNW3&E@$KBP1~$9$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BL?Bj(B6.1.} ${\bf s}_0 = S(s_0)$$B$,Cf?4?^7A$G$"$k$N$O(B$V'(s_0) = 0$$B$N$H$-$^$?$=$N$H$-$N$_$G$"$k!%(B

\vspace{0.3cm}
\noindent
\underbar{ $B>ZL@(B.} ${\bf s}_0$$B$,Cf?4?^7A$G$"$k$H$9$k!%$3$N$H$-(B(4.15)$B<0$O(B
$$
  V'(s_0) = \lambda DU({\bf s}_0) A {\bf s}_0 = 
\lambda \mu {\bf s}_0^T M A {\bf s}_0,
$$
$B$r0UL#$9$k!%$@$+$i@-<A(B(4.3)$B$h$j!"(B$V'(s_0) = 0$$B$G$"$k!%(B

$B5U$K(B$V'(s_0) = 0$$B$H2>Dj$9$k!%$3$N$H$-!"$U$?$?$S(B(4.15)$B<0$h$j(B
$$
 {\bf s}_0^T A^T \nabla U({\bf s}_0) = DU({\bf s}_0) A {\bf s}_0 = 0,
$$
$B$rF@$k!%(B$\nabla U({\bf s}_0) \in P$$B$G$"$k$+$i!"$"$k<B?t(B$\alpha$$B$*$h$S(B$\beta$$B$KBP$7$F(B
$$
  \nabla U({\bf s}_0) = \alpha M {\bf s}_0 + \beta M A {\bf s}_0,
$$
$B$H=q$1$k!%$7$?$,$C$F@-<A(B(4.1)$B$*$h$S(B(4.3)$B$h$j(B
$$
 0 = \alpha {\bf s}_0^T A^T M {\bf s}_0 + 
\beta {\bf s}_0^T A^T M A {\bf s}_0 = \beta,
$$
$B$rF@!"$h$C$F(B$\nabla U({\bf s}_0) = \alpha M {\bf s}_0$$B$H$J$k!%$@$+$i(B${\bf s}_0$$B$OCf?4?^7A$G$"$k!%(B\hfill $\Box$

\vspace{0.3cm}
$B;0BNLdBj$NCf?4?^7A$O$h$/CN$i$l$F$$$k(B(Wintner[17]$B;2>H(B)$B!%$$$o$f$kD>@~Cf?4?^7A$G(B$s_1<s_2<s_3$$B$H$J$k$b$N$O$R$H$D$7$+$J$$!%(B

\vspace{0.3cm}
\noindent
{\bf $BL?Bj(B6.2.} $BM#0l$NCf?4?^7A(B${\bf s}_c \in {\bf S}_1$$B$,$"$k!%(B

\vspace{0.3cm}
\noindent
\underbar{ $B>ZL@(B.} $BA0$NL?Bj$h$j(B$V$$B$,(B$(-1,1)$$B>e$KM#0l$NNW3&E@$r;}$D$3$H$r<($5$M$P$J$i$J$$!%(B$s \rightarrow \pm 1$$B$N$H$-(B$V(s) \rightarrow \infty$$B$G$"$k$+$i!"(B$V$$B$ONW3&E@$r;}$D!%(B(4.15)$B!"(B(4.10)$B!"(B(4.4)$B$*$h$S(BEuler$B$N8x<0$r;H$C$F(B
$$
  V''(s) = \lambda^2 [ D^2 U(S(s)) (A S(s), A S(s)) + V(s) ],
\eqno(6.4)
$$
$B$H7W;;$G$-$k!%(B$U$$B$NDj5A$h$j(B
$$
  D^2 U({\bf q}) (\xi, \eta) = \displaystyle \sum_{i<j} 
\frac{2m_im_j (\xi_i-\xi_j)(\eta_i-\eta_j)}{|q_i-q_j|^3},
$$
$B$H7W;;$G$-$k!%$f$($K(B$D^2 U({\bf q})$$B$O@5DjCM$G$"$k!%(B$V(s)>0$$B$G$"$k$+$i!"(B(6.4)$B<0$h$j$9$Y$F$N(B$s \in (-1,1)$$B$KBP$7$F(B$V''(s)>0$$B$,=P$k!%$@$+$i(B$V$$B$OM#0l$NNW3&E@(B($B:G>.(B)$B$r;}$D!%(B\hfill $\Box$

\vspace{0.3cm}
$B$$$^$d(B$C$$B>e$NN.$l$NITF0E@$r7W;;$G$-$k!%(B${\bf s}_c = S(s_c)$$B$r(B${\bf S}_1$$B>e$NCf?4?^7A$H$7!"(B
$$
  v_c = (2V(s_c))^{1/2} = \displaystyle \left( 
\frac{W(s_c)}{1-s_c^2} \right)^{1/2},  \eqno(6.5)
$$
$B$H$9$k!%(B$C$$B>e$K<!$N(B2$BE@$rDj5A$9$k(B(Fig.2$B;2>H(B)$B!%(B
$$ \begin{array}{l}
 c = (0, -v_c, s_c, 0), \\
 d = (0, v_c, s_c, 0).
   \end{array}
$$

\vspace{0.3cm}
\noindent
{\bf $BL?Bj(B6.3.} $C$$B$K@)8B$7$?N.$l(B$\varphi$$B$O$A$g$&$I#2$D$NITF0E@(B$c$$B$H(B$d$$B$r;}$D!%(B

\vspace{0.3cm}
\noindent
\underbar{ $B>ZL@(B.} $BE@(B$x = (0,v,s,w) \in C$$B$,(B$\varphi$$B$NITF0E@$G$"$k$N$O!"(B$x$$B$,%Y%/%H%k>l(B(6.2)$B$N%<%m$G$"$k$H$-$^$?$=$N$H$-$N$_$G$"$k!%(B(6.2b)$B$*$h$S(B(6.2a)$B$h$j!"(B$x$$B$,%<%m$G$"$k$N$O(B$w=0$$B$+$D(B
$$
  v^2 = \displaystyle \frac{W(s)}{(1-s^2)},   \eqno(6.6)
$$
$B$N$H$-$N$_$G$"$k!%$@$+$i(B(6.2c)$B$h$j(B
$$
\displaystyle s + \frac{1}{2} \frac{W'(s)}{W(s)} (1-s^2) = 0, 
$$
$B$,F@$i$l$k!%$H$3$m$,(B$W$$B$NDj5A$h$j(B
$$
\displaystyle \frac{W'(s)}{W(s)} = 
   \displaystyle - \frac{2s}{1-s^2} + \frac{V'(s)}{V(s)}. 
$$
$B$f$($K(B
$$
 \displaystyle \frac{1}{2} \frac{V'(s)}{V(s)} = 0,
$$
$B$,F@$i$l$k!%$7$?$,$C$F(B$s=s_c$$B$G$"$k!%$3$N$H$-(B(6.6)$B$h$j(B$v = \pm v_c$$B!"$9$J$o$A(B$x=c$$B$^$?$O(B$x=d$$B$rF@$k!%(B\hfill $\Box$

\vspace{0.3cm}
$B0J2<$G(B(6.2)$B<0$N2r$K1h$C$F:BI8(B$v$$B$,A}2C$9$k$3$H$r<($=$&!%$@$+$i(B$C$$B>e$NN.$l(B$\varphi$$B$O!V8{G[E*!W$H2f!9$,8F$V$H$3$m$N@-<A$r<($9!%(B

\vspace{0.3cm}
\noindent
\underbar{ $BDj5A(B.} $\psi$$B$O40Hw5wN%6u4V(B$X$$B$NN.$l$G$"$k$H$9$k!%O"B34X?t(B$g:X \rightarrow R^1$$B$,$"$C$F!"(B$x$$B$,ITF0E@$G$J$$8B$j(B
$$ \begin{array}{lll}
  g(\psi(x,t)) < g(x) & {\rm if} & t>0,
   \end{array}
$$
$B$,@.$jN)$D$H$9$k!%$5$i$K!"(B$\psi$$B$NITF0E@$O8IN)(B(isolated)$B$7$F$$$k$H$9$k!%$3$N$H$-(B$\psi$$B$O(B($g$$B$K4X$7$F(B)$B!V8{G[E*!W(B(gradient-like)$B$H$$$o$l$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BL?Bj(B6.4.} $C$$B$K@)8B$7$?N.$l(B$\varphi$$B$O(B
$$
  g: C \rightarrow R^1: (0, v, s, w) \mapsto -v,
$$
$B$K4X$7$F8{G[E*$G$"$k!%(B

\vspace{0.3cm}
\noindent
\underbar{ $B>ZL@(B.} (6.2a)$B<0$O(B$s = \pm 1$$B$N$H$-(B$\frac{\displaystyle dv}{\displaystyle d \tau}>0$$B$r0UL#$9$k!%(B(6.2a)$B$r(B(6.1)$B$H7k9g$7$F!"(B$s \neq \pm 1$$B$N$H$-(B
$$
\displaystyle \frac{dv}{d \tau} = 
   \displaystyle \frac{\lambda}{2} W(s)^{1/2} \frac{w^2}{1-s^2}, 
$$
$B$rF@$k!%$7$?$,$C$F(B
$$ \begin{array}{ccccc}
\displaystyle \frac{dv}{d \tau} > 0 & 
{\rm if} & w \neq 0 & {\rm or} & s = \pm 1.
   \end{array}
$$
$w=0$$B$+$D(B$s \neq \pm 1$$B$N$H$-(B
$$ \begin{array}{l}
\displaystyle 
 \frac{dv}{d \tau} = \displaystyle \frac{d^2v}{d \tau^2} = 0, \\
\displaystyle 
 \frac{d^3v}{d \tau^3} = \displaystyle 
 \frac{\lambda (1-s^2)^3}{W(s)^{3/2}} V'(s)^2.
   \end{array}
$$
$B$H7W;;$G$-$k!%L?Bj(B6.1$B$h$j!":G8e$NI=<0$O(B$s = s_c$$B$N$H$-$r=|$$$F@5$G$"$k!%$@$+$i(B$v$$B$O#2$D$NITF0E@$r=|$$$F$$$?$k$H$3$m$GA}Bg$9$k!%(B\hfill $\Box$

\vspace{0.3cm}
$BL?Bj(B6.3$B$*$h$S(B6.4$B$+$i;0BN>WFMB?MMBN(B$C$$B>e$NN.$l$,5-=R$G$-$k!%(B10$B@a$G$O!"$"$k<o$N<ANL$KBP$7$F$b$C$H40A4$J5-=R$rE83+$9$k$,!"$^$:;0BN>WFM$K=*$o$k50F;$N=89g$K4X$7$FCN$i$l$F$$$k#2$D$NDjM}$K$D$$$F5DO@$7$h$&!%(B

\vspace{0.5cm}
\noindent
{\bf 7. $B;0BN>WFM50F;$NA26aE*$U$k$^$$(B}

Sundman[15]$B$O#3<!85;0BNLdBj$K$*$$$F;0BN>WFM$K=*$o$k50F;$,A26aE*$KCf?4?^7A$K6a$E$/$3$H$r>ZL@$7$?!%$3$N@a$G$O!"D>@~LdBj$GJL$N>ZL@$rM?$($h$&!%(B

(2.3)$B<0$N2r(B$({\bf q}(t), {\bf p}(t)) \in {\bf Q} \times {\bf P}$$B$O!"$"$k<B?t(B$t_1$$B$KBP$7$F!"(B$t \rightarrow t_1$$B$N$H$-(B${\bf q}(t) \rightarrow 0$$B$H$J$k$J$i$P;0BN>WFM50F;$H8F$P$l$k!%(B
$t \rightarrow t_1-$$B$N$H$-(B${\bf q}(t) \rightarrow 0$$B$H$J$k$J$i;0BN>WFM$K=*$o$k$H$$$o$l!"(B$t \rightarrow t_1+$$B$N$H$-(B${\bf q}(t) \rightarrow 0$$B$H$J$k$J$i;0BN>WFM$K;O$^$k$H$$$o$l$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BDjM}(B7.1.}(Sundman). $({\bf q}(t), {\bf p}(t))$$B$r;0BN>WFM50F;$H$9$k!%$9$k$H(B$t \rightarrow t_1$$B$N$H$-(B
$$ \begin{array}{ll}
\displaystyle \frac{{\bf q}(t)}{r(t)} \rightarrow {\bf s}_c, & $B$*$h$S(B
   \end{array}                \leqno(a)
$$
$$ \begin{array}{l}
 r(t) \sim \kappa (t_1 - t)^{2/3}.
   \end{array}                \leqno(b)
$$

\vspace{0.3cm}
$r(t)^2$$B$,47@-%b!<%a%s%H$G$"$C$?$3$H$r;W$$$@$=$&!%$@$+$i(BSundman$B$O;0BN>WFM50F;$,Cf?4?^7A$K6a$E$/$3$H$P$+$j$G$J$/!"47@-%b!<%a%s%H$,(B$(t_1-t)^{4/3}$$B$N7A$G%<%m$K6a$E$/$3$H$r>ZL@$7$?!%DD=R(B(a)$B$H(B(b)$B$O$H$-$K7k9g$7$F(B
$$
 {\bf q}(t) \sim \kappa (t_1 - t)^{2/3} {\bf s}_c,
$$
$B$H$b=q$+$l$k!%Dj?t(B$\kappa$$B$NCM$O0J2<$N>ZL@$GM?$($k!%(B

$BD>@~$N>l9g$N(BSundman$B$NDjM}$r>ZL@$9$k$N$K!"$3$l0JA0$N@a$GF3F~$7$?JQ49$r;H$$!"5wN%6u4V$NN.$l$N0lHLE*9M;!$r4pAC$K$9$k!%40Hw5wN%6u4V(B$X$$B>e$NN.$l(B$\psi$$B$KBP$7$F!"E@(B$x_0 \in X$$B$N(B$\omega$$B6K8B=89g$r(B
$$
  \omega (x_0) = \cap _{t>0} \overline{\psi (x, [t, \infty))}.
$$
$B$GI=$o$9!%$3$3$G>e$D$-$N@~$O0LAjJDJq$rI=$o$9!%DjM}(B7.1$B$O<!$NJdBj$N5"7k$G$"$k!%$3$NJdBj$N>ZL@$OJd0d$GM?$($k!%(B

\vspace{0.3cm}
\noindent
{\bf $BJdBj(B7.2.} $\psi$$B$O6I=j%3%s%Q%/%H$J5wN%6u4V(B$X$$B>e$NN.$l$H$9$k!%(B$x_0 \in X$$B$O(B$\omega(x_0)$$B$,6u$G$J$/%3%s%Q%/%H$J=89g$G$"$k$h$&$JE@$H$9$k!%(B$\psi$$B$r(B$\omega(x_0)$$B$K@)8B$9$k$H8{G[E*$G$"$k$H$9$k!%$3$N$H$-(B$\omega(x_0)$$B$O(B1$BE@$G$"$k!%(B

\vspace{0.3cm}
$B;0BN>WFM50F;$N(B$\omega$$B6K8B=89g$,6u$G$J$/%3%s%Q%/%H$G$"$k$3$H$r<($;$l$P!"DjM}(B7.1$B$O$[$\D>$A$K>e$NJdBj$+$i=P$k!%0J2<$N5-K!$,I,MW$G$"$k!%$3$N5-K!$O(B9$B@a$G$bMQ$$$k!%(B

$\nu > \alpha > 0$$B$KBP$7$F!"JQ49$5$l$?Dj%(%M%k%.!<B?MMBN(B$N(h)$$B$NItJ,=89g$r0J2<$N$h$&$KDj5A$9$k(B(Fig.3$B;2>H(B)$B!%(B
$$ \begin{array}{l}
 {\bf B} (h, \alpha) = \{ (r,v,s,w) \in N(h): r \leq \alpha \}, \\
 B_0 (h, \alpha, \nu) = \{ (r,v,s,w) \in {\bf B} (h, \alpha): 
|v| \leq \nu - r \}, \\
 B^- (h, \alpha, \nu) = \{ (r,v,s,w) \in {\bf B} (h, \alpha): 
v \geq \nu - r \}, \\
 B^+ (h, \alpha, \nu) = \{ (r,v,s,w) \in {\bf B} (h, \alpha): 
- v \geq \nu - r \}, \\
 b^{\pm} (h, \alpha, \nu) = \{ (r,v,s,w) \in B^{\pm} (h, \alpha, \nu): 
r = \alpha \}, \\
 \sigma^{\pm} (h, \alpha, \nu) = B^{\pm} (h, \alpha, \nu) \cap 
B_0 (h, \alpha, \nu).
  \end{array}
$$
$B50F;8I(B$\varphi(x_0, [\tau, \tau'])$$B$,JD=89g(B$K$$B$K$*$$$F6KBg$G$"$k$H$O!"$=$l$,(B$K$$B$NItJ,=89g$G$"$C$F!"G$0U$NBg$-$J6h4V(B$I \supset [\tau, \tau']$$B$KBP$7$F(B$\varphi(x_0, I) \not\subset K$$B$N$H$-$G$"$k!%(B

\vspace{5.3cm}
\noindent
{\bf $BL?Bj(B7.3.} $\nu > \alpha - h$$B$H$9$k!%(B$\varphi(x_0, [\tau, \tau'])$$B$,(B$B^- (h, \alpha, \nu)$$B$K$*$1$k6KBg50F;8I$J$i!"(B$\varphi(x_0, \tau) \in \sigma^-(h, \alpha, \nu)$$B$+$D(B$\varphi(x_0, \tau') \in b^-(h, \alpha, \nu)$$B$G$"$k!%(B

\vspace{0.3cm}
\noindent
\underbar{ $B>ZL@(B.} (5.7)$B<0$N:G=i$N#2$D$r2C$((B(5.4)$B$r$=$l$KBeF~$9$l$P!"(B$s \neq \pm 1$$B$KBP$7$F(B
$$ 
\displaystyle \frac{d}{d \tau} (v + r) = \displaystyle \lambda 
\left[ \frac{W(s)^{1/2}}{2(1-s^2)} w^2 + 
r \frac{(1-s^2)}{W(s)^{1/2}} (h+v) \right],
$$
$B$rF@!"(B$s = \pm 1$$B$KBP$7$F(B
$$ 
\displaystyle  \frac{d}{d \tau} (v + r) = \displaystyle 
\frac{1}{2} \lambda W(s)^{1/2},
$$
$B$rF@$k!%(B$(r,v,s,w) \in \sigma^-$$B$J$i(B$v = \nu - r$$B$+$D(B$r \leq \alpha$$B$G$"$k$+$i!"(B$v \geq \nu - \alpha$$B$G$"$k!%2>Dj$K$h$j(B$\nu > \alpha - h$$B$G$"$k$+$i(B$v + h>0$$B$G$"$k!%$7$?$,$C$F(B$(r,v,s,w) \in \sigma^-$$B$KBP$7$F(B$\frac{\displaystyle d}{\displaystyle d \tau}(v+r)>0$$B$G$"$k!%$@$+$i(B$\sigma^-$$B>e$NE@$O(B$B^-$$B$KF~$k$+$i(B$\varphi(x_0, \tau') \in b^-$$B$G$"$k!%(B$(r,v,s,w) \in b^-$$B$O(B$v>0$$B$r0UL#$9$k$+$i!"(B(5.7)$B$NBh0l<0$+$i(B$b^-$$B>e$NE@$O(B$B^-$$B$r=P$F$$$/!%$D$^$j(B$\varphi(x_0, \tau) \in \sigma^-$$B$G$"$k!%(B\hfill $\Box$

\vspace{0.3cm}
$BF1MM$J5DO@$K$h$j<!$,>ZL@$G$-$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BL?Bj(B7.4.} $\nu > \alpha - h$$B$H$9$k!%(B$\varphi(x_0, [\tau, \tau'])$$B$,(B$B^+ (h, \alpha, \nu)$$B$K$*$1$k6KBg50F;8I$J$i!"(B$\varphi(x_0, \tau) \in b^+(h, \alpha, \nu)$$B$+$D(B$\varphi(x_0, \tau') \in \sigma^+(h, \alpha, \nu)$$B$G$"$k!%(B

\vspace{0.3cm}
\noindent
\underbar{ $BDjM}(B7.1$B$N>ZL@(B.} $B;0BN>WFM$K=*$o$k50F;$N>l9g$N$_$r9M$($h$&!%;0BN>WFM$K;O$^$k50F;$N>ZL@$bK\<AE*$K$*$J$8$G$"$k!%(B

$x_0=(r_0,s_0,v_0,w_0) \in N(h)$$B$rJQ49(B(3.1)$B!"(B(3.4)$B!"(B(4.13)$B!"$*$h$S(B(5.3)$B$N$b$H$G$N(B$(q(0),p(0))$$B$NA|$H$9$k!%$@$+$i50F;(B
$$
  \{ (q(t), p(t)): t \in [0, t_1) \},
$$
$B$O50F;(B$\varphi(x_0, [0,\tau_1))$$B$K<L$5$l$k!%;0BN>WFM50F;$NDj5A$h$j(B$\tau \rightarrow \tau_1$$B$N$H$-(B$r(\tau) \rightarrow 0$$B$G$"$j!"$^$?(B$\{ r=0 \}$$B$ON.$l$KBP$7$FITJQ$J=89g$G$"$k$+$i!"(B$\tau_1 = \infty$$B$N$O$:$G$"$k!%(B

$\alpha > 0$$B$r8GDj$9$k!%(B$r(\tau) \rightarrow 0$$B$G$"$k$+$i!"(B$\tau_2>0$$B$,$"$C$F$9$Y$F$N(B$\tau \geq \tau_2$$B$KBP$7$F(B$r(\tau) < \alpha$$B$G$"$j!"$f$($K(B
$\varphi(x_0, \tau) \in {\bf B}(h, \alpha)$$B$G$"$k!%<!$N$h$&$K(B$\nu$$B$rA*$V!%(B
$$
  \nu > \alpha + \max ( -h, |v(\tau_2)|).
$$
$B$3$N$H$-(B$\varphi(x_0, \tau_2) \in B_0(h, \alpha, \nu)$$B$G$"$k!%GXM}K!$K$h$j(B
$$ \begin{array}{lcc}
 \varphi(x_0, \tau) \in B_0(h, \alpha, \nu) & {\rm for \ all} & 
\tau \geq \tau_2,  
   \end{array}     \eqno(7.1)
$$
$B$r<($=$&!%(B

$B$"$k(B$\tau_3 \geq \tau_2$$B$KBP$7$F(B$\varphi(x_0, \tau_3) \in B^-(h, \alpha, \nu)$$B$H2>Dj$9$k!%$3$N$H$-L?Bj(B7.3$B$*$h$S(B$r(\tau) < \alpha$$B$G$"$k$3$H$O$9$Y$F$N(B$\tau \geq \tau_3$$B$KBP$7$F(B$\varphi(x_0, \tau) \in B^-$$B$r0UL#$9$k!%$H$3$m$,(B$B^-$$B$NE@$KBP$7$F$O(B$v>0$$B$G$"$k$+$i!"(B(5.7)$B$N:G=i$N<0$h$j(B$\tau \geq \tau_3$$B$KBP$7$F(B$\frac{\displaystyle dr}{\displaystyle d \tau} \geq 0$$B$,=P$k!%$H$3$m$,$3$l$O(B$r(\tau) \rightarrow 0$$B$KL7=b$9$k!%$7$?$,$C$F(B$\tau \geq \tau_2$$B$KBP$7$F(B$\varphi(x_0, \tau) \not\in B^-$$B$G$"$k!%L?Bj(B7.4$B$+$i!"(B$\tau \geq \tau_2$$B$KBP$7$F(B$\varphi(x_0, \tau) \not\in B^+$$B$,=P$k!%$@$+$i(B(7.1)$B$,<($5$l$?!%(B

$B_0$$B$O%3%s%Q%/%H$G$"$k$+$i!"(B$\omega(x_0)$$B$O6u$G$J$$%3%s%Q%/%H$J=89g$G$"$k!%(B$r(\tau) \rightarrow 0$$B$G$"$k$+$i(B$\omega(x_0) \in C$$B$G$"$k!%L?Bj(B6.4$B$h$j(B$\omega(x_0)$$B$K@)8B$7$?N.$l$O8{G[E*$G$"$k!%$7$?$,$C$FJdBj(B7.2$B$h$j(B$\omega(x_0)$$B$O@53N$K#1E@$+$i$J$j!"I,A3E*$KITF0E@$G$"$k!%L?Bj(B6.3$B$h$j$?$@#2$D$NITF0E@$O(B$c$$B$H(B$d$$B$G$"$k$+$i(B
$$ \begin{array}{lcc}
 (r,v,s,w) \rightarrow (0, \pm v_c, s_c, 0) & {\rm as} & 
\tau \rightarrow \infty,
   \end{array}
$$
$B$G$"$k!%$@$+$i(B$\tau \rightarrow \infty$$B$N$H$-(B$r^{-1} {\bf q} = {\bf s} \rightarrow {\bf s}_c$$B$G$"$j!"DjM}(B7.1$B$N(B(a)$B$,3NN)$5$l$?!%(B

(5.7)$B$NBh0l<0$+$i(B$v(\tau) \not\rightarrow v_c$$B$G$"$k!%(B$\frac{\displaystyle dr}{\displaystyle d \tau}>0$$B$@$+$i$G$"$k!%$7$?$,$C$F(B
$$ \begin{array}{lcc}
 \varphi(x_0, \tau) \rightarrow c & {\rm as} & 
\tau \rightarrow \infty,
   \end{array}
$$
$B$G$"$j!"(B
$$ \begin{array}{lcc}
 \frac{\displaystyle dr} {\displaystyle d \tau} \sim  - 
\frac{\displaystyle \lambda (1-s_c^2)}{\displaystyle W(s_c)^{1/2}} 
v_c r & {\rm as} & \tau \rightarrow \infty ,
   \end{array}
$$
$B$rF@$k!%;~4VJQ49(B(3.7)$B$H(B(5.5)$B$O(B
$$ 
 \frac{\displaystyle dr} {\displaystyle dt} \sim - v_c r^{-1/2},
$$
$B$r0UL#$9$k$+$i!"(B
$$ \begin{array}{lcc}
 r(t) \sim (\frac{3}{2} v_c)^{2/3} (t_1-t)^{2/3} 
& {\rm as} & t \rightarrow t_1-,
   \end{array}
$$
$B$G$"$k!%$@$+$iDjM}(B7.1$B$N(B(b)$B$b3NN)$7$?!%(B\hfill $\Box$

\vspace{0.3cm}
$B;0BN>WFM$K;O$^$k50F;$O@-<A(B
$$ \begin{array}{lcc}
 \varphi(x_0, \tau) \rightarrow d & {\rm as} & \tau \rightarrow \infty,
   \end{array}
$$
$B$r;}$D$3$H$KCm0U$9$Y$-$G$"$k!%(B
