\begin{flushright}
$BC+@n@6N4(B $BLu(B
\end{flushright}

\noindent
{\it Inventiones mathematicae} (1974), {\bf 27}, 191-227

\vspace{0.5cm}
\begin{center}
{\Large
$BD>@~;0BNLdBj$N;0BN>WFM(B \\
{\bf Triple collision in the collinear three-body problem} 
}

\vspace{0.3cm}
Richard McGehee
\end{center}

\vspace{0.5cm}
\noindent
{\bf 1. $B=x(B}

$B8EE5NO3X$NK!B'$K$7$?$,$C$F(B$k$$B<!856u4V$NCf$GF0$$$F$$$k(B$n$$B8D$N<AE@$r9M$($h$&!%N3;R(B$i$$B$,<ANL(B$m_i > 0$$B$*$h$S0LCV(B$q_i \in R^k$$B$r;}$F$P!"Ii$N%]%F%s%7%c%k%(%M%k%.!<$O<!<0$GM?$($i$l$k!%(B
$$
  U = \sum_{i<j} \frac{\displaystyle m_im_j} 
{\displaystyle ||q_i - q_j||} .                     \eqno(1.1)
$$
$B$3$3$G(B$|| \ ||$$B$O(B$R^k$$B$N%f!<%/%j%C%I%N%k%`$rI=$o$9!%N3;R$N1?F0$OHyJ,J}Dx<07O(B
$$ \begin{array}{ll}
  m_i \ddot{q}_i = \nabla_{q_i} U, & i=1,2,\ldots, n,
   \end{array}                                    \eqno(1.2)
$$
$B$K$h$C$F5-=R$5$l$k!%$3$3$G(B$\nabla_{q_i} U$$B$O(B$q_i$$B$K4X$9$k(B$U$$B$N8{G[$G$"$k!%(B

$BN3;R$N0LCV(B$(q_1, \ldots, q_n)$$B$O!"$"$k(B$i \neq j$$B$KBP$7$F(B$q_i = q_j$$B$N$H$->WFM$H8F$P$l$k!%>e$NJ}Dx<07O$O>WFM$r=|$$$F$$$?$k$H$3$mDj5A$5$l$F$$$k!%;~9o(B$t=0$$B$KN3;R$N0LCV$H1?F0NL$,M?$($i$l$F$$$k$b$N$H$9$k!%>WFM$+$i;O$a$J$$8B$j!"HyJ,J}Dx<0$NI8=`E*DjM}$K$h$j!"$"$k6KBg6h4V(B$[0, t^*)$$B$N>e$GJ}Dx<0(B(1.2)$B$N2r$NB8:_$H0l0U@-$,J]>Z$5$l$F$$$k!%(B$t^* < \infty$$B$J$i2r$O(B$t^*$$B$GFC0[@-(B(singularity)$B$r7P83$9$k$H$$$o$l$k!%(B

$BFC0[@-$K6a$E$/$H$-$N2r$N$U$k$^$$$O40A4$K$OM}2r$5$l$F$*$i$:!"$$$/$D$+$N2DG=@-$,CN$i$l$F$$$k!%(B$t \rightarrow t^*$$B$N$H$-$9$Y$F$NN3;R$,6K8BE*0LCV$K6a$E$/$J$i!"$3$N6K8B0LCV$,>WFM$G$"$k$3$H$r$7$a$9$N$OFq$7$/$J$$(B[12,17]$B!%FC0[@-$O$3$N$H$->WFMFC0[@-$H8F$P$l!"2r$O>WFM$G=*$o$k$H$$$o$l$k!%(B$m$$B8D$NN3;R$,0lCW$7!";D$j$O0[$J$k0LCV$K$"$k$H$-!">WFM$O(B$m$$B=E>WFM$H8F$P$l$k!%>WFM0J30$NFC0[@-$,$"$k$+$I$&$+CN$i$l$F$$$J$$!%(B

$BFsBNLdBj$N>l9g!"JQ?tJQ49$K$h$C$FFsBN>WFM$rJ}Dx<0$N@5B'E@$KJQ49$G$-$k(B[6]$B!%$3$N$h$&$JJQ49$OFsBN>WFM$N@5B'2=(B(regularization)$B$H8F$P$l$k!%$3$N$H$-2r$OFC0[@-$r1[$($F1dD9$G$-$k!%$3$N1dD9$OJ*M}E*$K$OCF@-H?H/$KBP1~$9$k!%(B

Sundman[14]$B$O!"FsBN>WFM$,;0BNLdBj$G$b@5B'2=$G$-$k$3$H$r<($7$?!%$D$^$j!"2r$,?7$7$$;~4VJQ?t$N2r@O4X?t$H$7$FFsBN>WFM$rDL$C$F@\B3$G$-$k$h$&$KJQ?t$rJQ49$G$-$k$N$G$"$k!%$3$N>l9g$b1dD9$OCF@-H?H/$KBP1~$9$k!%(B

$BFsBN$h$jB?$$N3;R$r4^$`>WFM$O$b$A$m$s$b$C$HJ#;($G$"$k$,!"$=$N$U$k$^$$$N$$$/$D$+$N6ILL$OCN$i$l$F$$$k!%N3;R$NG[0L(B(configuration)$B$r!"J*M}E*$K$O47@-%b!<%a%s%H$KBP1~$9$k%N%k%`$G3d$C$?0LCV$HDj5A$9$k!%;0BNLdBj$N;0BN>WFM$K$*$$$F!"G[0L$,$$$o$f$kCf?4?^7A$N$R$H$D$K6a$E$/$3$H$r(BSundman[15]$B$O<($7$F$$$k(B([12,17]$B!"$^$?0J2<$N(B6$B!"(B7$B@a;2>H(B)$B!%(BWintner[17]$B$,4Q;!$7$?$H$3$m$K$h$k$H!"(BSundman$B$N5;=Q$r;H$C$F(B$n$$BBNLdBj$G(B$n$$BBN>WFM$K=*$o$k2r$bCf?4?^7A$K6a$E$/$3$H$,<($;$k!%$7$+$7!"(B$n \geq 4$$B$NCf?4?^7A$K4X$7$F$O>/$7$7$+CN$i$l$F$$$J$$!%(B

$BFsBN>WFM$O@5B'2=$G$-$k$+$i!"$[$+$NFC0[@-$K$bF1$8$3$H$,=PMh$J$$$+$H5?Ld$,@8$8$k!%(BSiegel[11]$B$O;0BNLdBj$K$*$1$k;0BN>WFM$G$3$NLdBj$K<h$jAH$s$@!%H`$NH/8+$K$h$l$P!"$?$$$F$$$N2r$,;0BN>WFM$rDL$C$FJQ498e$N;~4VJQ?t$N2r@O4X?t$H$7$F1dD9$G$-$J$$!%H`$O$^$?;0BN>WFM$K=*$o$k50F;$N=89g$,Aj6u4V$N3j$i$+$JItJ,B?MMBN$r$J$9$3$H$r<($7$?!%(B

$B$3$NO@J8$G$O!";0BN>WFM$K$h$kFC0[@-$K$D$$$F9M$($k!%$3$NFC0[@-$r;}$D2r$@$1$G$J$/!"FC0[@-$r;}$D2r$K6a$$2r$b5-=R$7$?$$!%;0BN>WFM$G$$$A$P$s4JC1$J>l9g$OD>@~;0BNLdBj$G@8$8$k!%$3$N>l9g!"(B$n=3$$B$+$D(B$k=1$$B$G$"$k$+$iN3;R$N0LCV(B$(q_1,q_2,q_3)$$B$O(B$R^3$$B$NE@$G$"$k!%=E?4$r86E@$K8GDj$9$l$P!"N3;R$N0LCV$O(B$m_1q_1 + m_2q_2 + m_3q_3 = 0$$B$G7h$^$kJ?LL(B${\bf Q}$$B$K@)8B$5$l$k!%1?F0NL$bJ?LL$K@)8B$5$l!"J}Dx<0(B(1.2)$B$O#4<!85$N#13,J}Dx<07O$K5"Ce$G$-$k!%%(%M%k%.!<J]B8$K$h$j!"7O$O$5$i$K#3<!85%(%M%k%.!<6JLL>e$N%Y%/%H%k>l$X$H5"Ce$9$k!%(B

$B;0BN>WFM$O(B${\bf Q}$$B$N86E@$KBP1~$9$k!%86E@$r1_$KKD$i$^$;$k$h$&$JJQ49$r9T$J$*$&!%1?F0NL6u4V$G$NE,Ev$J:BI8JQ49$K$h$j!"$3$NJQ49$ODj%(%M%k%.!<6JLL$N>e$K!":#8e!V;0BN>WFMB?MMBN!W$H8F$V!"#2<!856-3&$rE=$j$D$1$k8z2L$r;}$D!%;~4V$NJQ49$K$h$j%Y%/%H%k>l$r<\EYJQ49$7!"6-3&$^$G1dD9$G$-$k$h$&$K$9$k!%6-3&$O3HD%$5$l$?%Y%/%H%k>l$GITJQ$G$"$k!%$3$N6-3&$NE@$O$9$Y$F!"$b$H$N:BI8$G$O;0BN>WFM$KBP1~$9$k!%;0BN>WFMB?MMBN>e$NN.$l$OA4$/2>A[E*$G$"$k!%$=$N>e$N50F;$O$b$H$N:BI8$N$I$s$J50F;$K$bBP1~$7$J$$$+$i$G$"$k!%$7$+$7$J$,$i!"Dj%(%M%k%.!<6JLL>e$NN.$l$O2>A[E*$J6-3&>e$b4^$a$FO"B3$G$"$k!%$f$($K6-3&6a$/$NN.$l$O!"G$0U$KD9$$;~4V$N4V!"6-3&>e$NN.$l$K$7$?$,$&!%$7$?$,$C$F;0BN>WFMB?MMBN>e$N50F;$N$U$k$^$$$rMxMQ$7$F!"$3$NB?MMBN$N6a$/$N!"$9$J$o$A;0BN>WFM6a$/$N50F;$N$U$k$^$$$r7hDj$G$-$k!%(B

$B;0BN>WFMB?MMBN$r(BFig.2$B$K<($7$F$*$$$?!%$3$N>e$NN.$l$O8{G[E*$G$"$j!"$^$?#2$D$N%5%I%kITF0E@$r;}$D!%;0BN>WFM$K=*$o$k50F;$O$9$Y$F;0BN>WFMB?MMBN$KA26a$9$k!%(B

$B;0BN>WFMB?MMBN$N@-<A$r$I$N$h$&$KMxMQ$9$k$+$r<($9Nc$H$7$F!"$9$G$KCN$i$l$F$$$k#2$D$N7k2L$N?7$7$$>ZL@$rM?$($k!%:G=i$N$b$N$O!"$9$G$K;XE&$7$?(BSundman$B$N7k2L$G$"$C$F!";0BN>WFM50F;$OCf?4?^7A$K6a$E$/$H$$$&$b$N$G$"$k!%;0BN>WFM50F;$O#2$D$NITF0E@$N$I$l$+$KA26a$9$k$O$:$G$"$k$3$H$r<($9!%$3$N@-<A$,D>@~LdBj$N>l9g$K(BSundman$B$N7k2L$r4^$s$G$$$k$3$H$r<($9!%BhFs$N7k2L$O(BSiegel$B$N$b$N$G$"$C$F!";0BN>WFM$K=*$o$k50F;$N=89g$ODj%(%M%k%.!<6JLL$N$J$a$i$+$JItJ,B?MMBN$r$J$9$H$$$&$b$N$G$"$k!%D>@~LdBj$N>l9g$K!"$3$l$,#2$D$NITF0E@$KE,MQ$7$?0BDjB?MMBNDjM}$+$i=P$k$3$H$r<($9!%(B

$B$^$?;0BN>WFMB?MMBN$N@-<A$r;H$C$F!"50F;$,;0BN>WFM$r1[$($F1dD9$G$-$k$+$H$$$&LdBj$rD4$Y$k!%(BSundman$B$d(BSiegel$B$h$j$b$`$7$m(BEaston[4,5]$B$N8+J}$r:NMQ$9$k!%(BSundman$B$d(BSiegel$B$OC10l$N2r$,;~4V$N2r@O4X?t$H$7$F1dD9$G$-$k$+$I$&$+$rLdBj$K$7$?$,!"(BEaston$B$O2r$,6aK5$N2r$K4X$7$FO"B3E*$K1dD9$G$-$k$+$I$&$+$rLdBj$K$9$k!%;0BN>WFMB?MMBN>e$NN.$l$r6cL#$9$k$3$H$K$h$j!";0BN>WFM$O!">/$J$/$H$b$$$/$D$+$N<ANL$NAH9g$;$G$O!"(BEaston$B$N0UL#$G!V@5B'2=!W$G$-$J$$$3$H$r<($9!%(B

$B:G8e$K!";0BN>WFMB?MMBN>e$NN.$l$r;H$C$F!"$$$/$D$+$N<ANL$NAH9g$;$N>l9g!"0J2<$N@-<A$,@.$jN)$D$3$H$r<($9!%$9$J$o$A!";0BN>WFM$N6a$/$rDL2a$7$?8e$G$O!"7O$OG$0U$KBg$-$J1?F0%(%M%k%.!<$r;}$D!%$D$^$j!"N3;R$N#1$D$,0l$D$NJ}8~$KG$0U$KBg$-$JB.EY$r;}$A!";D$j$N#2$D$,8_$$$K6a$$$^$^Bg$-$JB.EY$G5UJ}8~$KF0$/!%$3$N$U$k$^$$$O$+$J$j6C$/$Y$-$b$N$G$"$C$F!"(B$n$$BBNLdBj$K$*$$$F>WFM$K$h$i$J$$FC0[@-$NB8:_$NLdBj$K4X78$9$k2DG=@-$,$"$k!%(BPainlev\'{e}[8]$B$,<($9$H$3$m$K$h$k$H!"$3$N$h$&$JFC0[@-$O;0BNLdBj$G$O5/$3$i$J$$!%$7$+$7(B$n \geq 4$$B$G$OLdBj$OL$2r7h$G$"$k!%$3$NLdBj$K4X$7!"(B$n=5$$B$N>l9g$KM=A[$rN)$F$k!%(B

\vspace{0.5cm}
\noindent
{\bf 2. $B=`Hw(B}

$B1?F0J}Dx<0$r%O%_%k%H%s7A<0$G=q$/$3$H$+$i;O$a$h$&!%(B${\bf q} = (q_1, q_2, q_3) \in R^3$$B$H$*$-!"$^$?<!$N$h$&$K$*$/!%(B
$$ M = \left[
\begin{array} {ccc}
  m_1 & 0   & 0 \\
  0   & m_2 & 0 \\
  0   & 0   & m_3
\end{array}   \right] .
$$
$BD>@~;0BNLdBj$N>l9g!"J}Dx<0(B(1.1)$B$H(B(1.2) $B$O<!$N$h$&$K$J$k!%(B
$$ \begin{array}{l}
 U(q) = \frac{\displaystyle m_1m_2} {\displaystyle |q_1-q_2|} +
        \frac{\displaystyle m_1m_3} {\displaystyle |q_1-q_3|} +
        \frac{\displaystyle m_2m_3} {\displaystyle |q_2-q_3|},  \\
M \ddot{\bf q} = \nabla U({\bf q}) .
   \end{array}                                          \eqno(2.1)
$$
$p_i = m_i \dot{q}_i$$B$r(B$i$$BHVL\$NN3;R$N1?F0NL$H$7!"(B${\bf p} = (p_1, p_2, p_3) \in R^3$$B$H$*$/!%7O$N1?F0%(%M%k%.!<$r<!$N$h$&$KDj5A$9$k!%(B
$$
T({\bf p}) = \frac{1}{2} 
\left( \frac{\displaystyle p_1^2}{\displaystyle m_1}
+ \frac{\displaystyle p_2^2}{\displaystyle m_2} 
+ \frac{\displaystyle p_3^2}{\displaystyle m_3} \right) 
= \frac{1}{2} {\bf p}^{T} M^{-1} {\bf p}.
$$
$B%O%_%k%H%s4X?t$r(B
$$
  H({\bf q}, {\bf p}) = T({\bf p}) - U({\bf q}),        \eqno(2.2)
$$
$B$GDj5A$9$l$P!"J}Dx<0(B(2.1)$B$O<!$N$h$&$K=q$1$k!%(B
$$ \begin{array}{l}
 \dot{\bf q} = H_p ({\bf q}, {\bf p}) = \nabla T({\bf p}) = 
M^{-1} {\bf p}, \\
 \dot{\bf p} = - H_q ({\bf q}, {\bf p}) = \nabla U({\bf q}).
   \end{array}                                         \eqno(2.3)
$$

$B4X?t(B$T$$B$O(B$R^3$$B>e$$$?$k$H$3$mDj5A$5$l$F$$$k!%4X?t(B$U$$B$O>WFME@$r=|$$$F$$$?$k$H$3$mDj5A$5$l$F$$$k!%(B
$$
  \Delta = \{ {\bf q} \in R^3: q_1 = q_2, q_2 = q_3, $B$^$?$O(B q_3 = q_1 
\}
$$
$B$r>WFME@$N=89g$H$9$k!%J}Dx<0(B(2.3)$B$O(B$(R^3 - \Delta) \times R^3$$B>e$N%Y%/%H%k>l$rDj5A$9$k!%$$$/$D$+$NE@$GDj5A$5$l$F$$$J$$%Y%/%H%k>l$N$3$H$r!VFC0[@-$r;}$D%Y%/%H%k>l!W(B(vectorfield with singularities)$B$H$$$&$3$H$K$9$k!%$@$+$iJ}Dx<0(B(2.3)$B$O(B$R^3 \times R^3$$B>e$KFC0[@-$r;}$D%Y%/%H%k>l$rDj5A$9$k$H$$$&!%(B

$B=E?4$HD>?J1?F0NL$r<h$j=|$$$F7O(B(2.3)$B$N<!85$r2<$2$k$3$H$,$G$-$k!%86E@$K=E?4$r8GDj$9$l$P!"0LCV:BI8$r@~7AItJ,6u4V(B
$$
{\bf Q} = \{ {\bf q} \in R^3: m_1q_1 + m_2q_2 + m_3q_3 = 0 \},
$$
$B$K@)8B$G$-(B(Fig.1$B;2>H(B)$B!"1?F0NL:BI8$rItJ,6u4V(B
$$
{\bf P} = \{ {\bf p} \in R^3: p_1 + p_2 + p_3 = 0 \},
$$
$B$K@)8B$G$-$k!%J}Dx<0(B(2.3)$B$O#4<!85@~7A6u4V(B${\bf Q} \times {\bf P}$$B>e$KFC0[@-$r;}$D%Y%/%H%k>l$r7h$a$k!%J}Dx<0(B(2.3)$B$O(B$({\bf Q} - \Delta) \times {\bf P}$$B>e$K%Y%/%H%k>l$r7h$a$k!%(B

$B%O%_%k%H%s4X?t(B(2.2)$B$O(B(2.3)$B$N2r$K1h$C$F0lDj$G$"$k$+$i!"3F<B?t(B$h$$B$KBP$7$FITJQ=89g(B
$$
  {\bf M} (h) = \{ ({\bf q}, {\bf p}) \in ({\bf Q} - \Delta) \times 
{\bf P}: H({\bf q},{\bf p}) = h \},               \eqno(2.4)
$$
$B$rDj5A$9$k!%$3$N=89g$O#3<!85B?MMBN$G$"$k!%$3$l$rDj%(%M%k%.!<LL$H8F$V$3$H$K$9$k!%$@$+$iJ}Dx<0(B(2.3)$B$O(B${\bf M}(h)$$B>e$K%Y%/%H%k>l$rDj5A$9$k!%(B

$B$7$+$7!"$3$N%Y%/%H%k>l$O40Hw$G$J$$!%$9$J$o$A!"2r$O$9$Y$F$N;~4V$K$o$?$C$FB8:_$9$k$o$1$G$O$J$$!%M-8B$N;~4VFb$K!"$"$k2r$O>WFM$K8~$+$$!"$f$($K(B${\bf M}(h)$$B$r5n$k!%FsBN>WFM$G=*$o$k2r$O(BSundman[14]$B$N$b$N$K;w$?5;=Q$G1dD9$G$-$k!%;0BN>WFM$K=*$o$k2r$O>WFM$KL58B;~4V$+$+$C$FE~C#$9$k$h$&$K8:B.$5$;$h$&!%(B

\vspace{7.5cm}
\noindent
{\bf 3. $B;0BN>WFM$K$h$kFC0[@-(B}

$B$O$8$a$K(B${\bf q} = 0$$B$K$*$1$kFC0[@-$rD4$Y$h$&!%(B
$$
  r = (m_1q_1^2 + m_2q_2^2 + m_3q_3^2)^{\frac{1}{2}} = 
({\bf q}^T M {\bf q})^{\frac{1}{2}},
$$
$B$HDj5A$9$k!%(B$r^2$$B$ON3;R7O$N47@-%b!<%a%s%H$G$"$k$3$H$*$h$S;0BN>WFM$O(B$r=0$$B$KBP1~$9$k$3$H$KCm0U$7$h$&!%(B
$$
{\bf S} = \{ {\bf q} \in {\bf Q}: r^2 = {\bf q}^T M {\bf q} = 1 \},
$$
$B$O47@-%b!<%a%s%H$GM?$($i$l$k%N%k%`$G$N(B${\bf Q}$$B$K$*$1$kC10L1_$G$"$k$H$9$k!%(B${\bf S}$$B>e$NE@$ON3;R7O$NG[0L$H8F$P$l$k!%(B$(r, {\bf s}) \in (0,\infty) \times {\bf S}$$B$r<LA|(B$(r, {\bf s}) \mapsto r {\bf s}$$B$K$h$j!"(B${\bf Q} - \{ 0 \}$$B>e$N6K:BI8$H$_$J$9!%(B

$B$3$3$GJQ?t$rDj5A$9$k!%(B
$$ \begin{array}{l}
  r = ({\bf q}^T M {\bf q})^{\frac{1}{2}}, \\
 {\bf s} = r^{-1} {\bf q}, \\
 y = {\bf p}^T {\bf s}, \\
 {\bf x} = {\bf p} - yM{\bf s}.
   \end{array}
$$
${\bf s} \in {\bf S}$$B$*$h$S(B${\bf x}^T{\bf s} = 0$$B$KCm0U$9$k!%$3$&$7$F!"1?F0NL(B${\bf p}$$B$rF07B@.J,(B$y$$B$H@\@~@.J,(B${\bf x}$$B$KJ,$1$?!%$3$3$G(B
$$
  {\bf T} = \{ ({\bf q}, {\bf p}) \in {\bf Q} \times {\bf P}: 
{\bf q} \in {\bf S}, {\bf p}^T{\bf q} = 0\},
$$
$B$H$*$/!%$3$l$O(B${\bf S}$$B$N@\%P%s%I%k$H$_$J$9$3$H$,$G$-$k!%$3$N$H$-(B$r \in (0, \infty), y \in R^1$$B$*$h$S(B$({\bf s},{\bf x}) \in {\bf T}$$B$G$"$k!%8E$$JQ?t$O?7$7$$JQ?t$K$h$C$F<!$N$h$&$KI=$o$5$l$k$3$H$KCm0U$9$k!%(B
$$ \begin{array}{l}
 {\bf q} = r{\bf s}, \\
 {\bf p} = {\bf x} + yM{\bf s}.
     \end{array}
$$
$B$3$&$7$F<B2r@OHyJ,F1Aj(B
$$
 (0,\infty) \times R^1 \times {\bf T} \rightarrow ({\bf Q} - \{ 0 \}) \times {\bf P}: 
(r,y,({\bf s},{\bf x})) \mapsto (r{\bf s},{\bf x}+yM{\bf s}),
  \eqno(3.1)
$$
$B$rDj5A$7$?$3$H$K$J$k!%(B

$B$3$N?7$7$$:BI8$G1?F0%(%M%k%.!<$O(B
$$
 T({\bf p}) = \frac{1}{2} ({\bf x}^T M^{-1} {\bf x} + y^2),
$$
$B$H=q$1!"%]%F%s%7%c%k%(%M%k%.!<$O(B
$$
 U({\bf q}) = r^{-1} U({\bf s}),
$$
$B$H=q$1$k!%$@$+$i%(%M%k%.!<4X78<0(B$H({\bf q},{\bf p}) = h$$B$O<!$N$h$&$K=q$1$k!%(B
$$
 \frac{1}{2} ({\bf x}^T M^{-1} {\bf x} + y^2) - r^{-1} U({\bf s}) = h.       \eqno(3.2)
$$
$B1?F0J}Dx<0(B(2.3)$B$O<!$N$h$&$K$J$k!%(B
$$ \begin{array}{l}
 \dot{r} = y, \\
 \dot{y} = r^{-1} {\bf x}^T M^{-1} {\bf x} - r^{-2} U({\bf s}), \\
 \dot{\bf s} = r^{-1} M^{-1} {\bf x}, \\ 
 \dot{\bf x} = - r^{-1} y {\bf x} - r^{-1} ({\bf x}^T M^{-1} {\bf x}) 
M {\bf s} + r^{-2} U({\bf s}) M {\bf s} + r^{-2} \nabla U({\bf s}).
   \end{array}              \eqno(3.3)
$$
$U$$B$,(B$-1$$B<!$NF1<!<0$G$"$k$3$H$KCm0U$9$l$P$3$l$i$NJ}Dx<0$r4JC1$KF3=P$G$-$k!%(B
$B$f$($K(B$\nabla U$$B$O(B$-2$$B<!$NF1<!<0$G$"$j!"(BEuler$B$N8x<0$h$j!"(B
$$
 {\bf q}^T \nabla U ({\bf q}) = - U ({\bf q}).
$$

$BJ}Dx<0(B(3.3)$B$O(B$[0,\infty) \times R^1 \times {\bf T}$$B>e$KFC0[@-$r;}$D%Y%/%H%k>l$rDj5A$9$k!%;0BN>WFM$K$h$kFC0[@-$rKD$i$^$;$?$3$H$K$J$k!%J}Dx<0(B(2.3)$B$G$O!";0BN>WFME@$N=89g$O(B$\{ 0 \} \times {\bf P}$$B$G$"$C$?$,!"J}Dx<0(B(3.3)$B$G$O!"(B$\{ 0 \} \times R^1 \times {\bf T}$$B!"$9$J$o$A(B$r = 0$$B$N=89g$K$J$C$?!%FsBN>WFME@$N=89g$O(B$(0, \infty) \times R^1 \times ({\bf T} \cap (\Delta \times {\bf P}))$$B!"$9$J$o$A(B${\bf s} = (s_1, s_2, s_3)$$B$,(B$s_1 = s_2, s_2 = s_3$$B$^$?$O(B$s_3 = s_1$$B$rK~$?$9=89g$K$J$C$?!%FsBN>WFM$O(B5$B@a$G07$&!%$^$:(B$r=0$$B$NFC0[@-$r<h$j=|$3$&!%(B

$B$U$?$?$S?7$7$$JQ?t(B
$$ \begin{array}{l}
 {\bf u} = r^{1/2} {\bf x}, \\
 v = r^{1/2} y,
   \end{array}
$$
$B$rF3F~$7$h$&!%$9$J$o$A!"<B2r@OE*HyJ,F1Aj(B
$$
 (0,\infty) \times R^1 \times {\bf T} \rightarrow (0, \infty) \times R^1 \times {\bf T}: (r,v,({\bf s},{\bf u})) \mapsto 
(r,r^{-1/2}v,({\bf s},r^{-1/2}{\bf u})),      \eqno(3.4)
$$
$B$rF3F~$9$k!%$3$N$H$-%(%M%k%.!<4X78<0(B(3.2)$B$O<!$N$h$&$K=q$1$k!%(B
$$
 \frac{1}{2} ({\bf u}^T M^{-1} {\bf u} + v^2)
 - U({\bf s}) = rh.       \eqno(3.5)
$$
$B$^$?1?F0J}Dx<0(B(3.3)$B$O<!$N$h$&$K$J$k!%(B
$$ \begin{array}{l}
 \dot{r} = r^{-1/2} v, \\
 \dot{v} = r^{-3/2} 
[ (1/2)v^2 + {\bf u}^T M^{-1} {\bf u} - U({\bf s})], \\
 \dot{\bf s} = r^{-3/2} M^{-1} {\bf u}, \\ 
 \dot{\bf u} = - r^{-3/2} [ -(1/2)v {\bf u} - ({\bf u}^T M^{-1} {\bf u})M {\bf s} + U({\bf s}) M {\bf s} + \nabla U({\bf s}) ].
   \end{array}                                          \eqno(3.6)
$$
$BA0$HF1MM!"$3$l$i$NJ}Dx<0$O(B$[0, \infty) \times R^1 \times {\bf T}$$B>e$KFC0[@-$r;}$D%Y%/%H%k>l$rDj5A$9$k!%FsBN$*$h$S;0BN>WFME@$OJ}Dx<0(B(3.3)$B$N>l9g$H$^$C$?$/F1$8$G$"$k!%$1$l$I$b$3$s$I$O!";~4VJQ49(B
$$
  dt = r^{3/2}dt',$B!!!!!!!!!!!!!!!!!!!!!!!!!!(B\eqno(3.7)
$$
$B$K$h$C$F%Y%/%H%k>l$N<\EYJQ49$r9T$J$($P!"(B$r=0$$B$NFC0[@-$r<h$j=|$1$k!%J}Dx<0(B(3.6)$B$O<!$N$h$&$K$J$k!%(B
$$ \begin{array}{l}
 \frac{\displaystyle dr}{\displaystyle dt'} = rv, \\
 \frac{\displaystyle dv}{\displaystyle dt'} = 
\frac{\displaystyle 1}{\displaystyle 2}v^2 + 
{\bf u}^T M^{-1} {\bf u} - U({\bf s}), \\
 \frac{\displaystyle d{\bf s}}{\displaystyle dt'} = M^{-1} {\bf u}, \\ 
 \frac{\displaystyle d{\bf u}}{\displaystyle dt'} = 
- \frac{\displaystyle 1}{\displaystyle 2} v {\bf u} - 
({\bf u}^T M^{-1} {\bf u}) M {\bf s} + 
U({\bf s}) M {\bf s} + \nabla U({\bf s}).
   \end{array}                                    \eqno(3.8)
$$

$B>e$NJ}Dx<0$O(B$r=0$$B$KFC0[@-$r;}$?$J$$$+$i!";0BN>WFM$r4^$`$h$&$K1?F0J}Dx<0$r3HD%$7$?$3$H$K$J$k!%(B$\{ r = 0 \}$$B$OJ}Dx<0(B(3.8)$B$K$*$$$FITJQ$G$"$k$3$H$KCm0U$7$h$&!%;~4VJQ49(B(3.7)$B$K$h$j!">.$5$J(B$r$$B$KBP$7$F50F;$,8:B.$5$l$k$N$G!";0BN>WFM$K=*$o$k50F;$OL58B;~4V$+$+$C$F$=$3$KE~C#$9$k!%;0BN>WFM$K=*$o$k50F;$N=89g$O$$$^$dITJQ=89g(B$\{ r=0 \}$$B$KA26a$9$k50F;$N=89g$G$"$k!%$^$?!"ITJQ=89g(B$\{ r=0 \}$$B>e$N50F;$r;H$($P(B$r$$B$,>.$5$$$H$-$N(B(3.8)$B$N50F;!"$D$^$j;0BN>WFM$N6a$/$rDL2a$9$k50F;$r5-=R$G$-$k!%(B
