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

\noindent
{\it Inventiones mathematicae} (1980), {\bf 60}, 249-267

\vspace{0.5cm}
\begin{center}
{\Large
{\bf $BJ?LLFsEyJU;0BNLdBj$N;0BN>WFM(B \\
Triple collision in the planar isosceles \\
three body problem} 
}

\vspace{0.3cm}
Robert L. Devaney
\end{center}

\vspace{0.5cm}
{\bf $BMWLs(B.} McGehee$B$NAuCV$r;H$C$F!"J?LL;0BNLdBj$NFCJL$J>l9g$K$*$$$F!";0BN>WFM$KC#$7$?$j!"$=$N6a$/$KMh$k50F;$NDj@-E*$U$k$^$$$K$D$$$F5DO@$9$k!%;0BN>WFM$K;O$^$j$+$D=*$k50F;$,L58B8DB8:_$9$k$3$H$r<($9!%6aK5$N50F;$OD>@~Cf?4?^7A$K6a$E$/$+@5;03Q7ACf?4?^7A$K6a$E$/$+$K$h$C$F0[$J$k$U$k$^$$$r$9$k!%:G8e$K!"!V%S%j%"!<%I(Bshot$B!W$H8F$V!";0BNLdBj$K$*$1$k?7$7$$7?$N50F;$K$D$$$F5DO@$9$k!%(B

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

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\vspace{0.5cm}
\noindent
{\bf 1. $B;0BN>WFMB?MMBN(B}

$B$3$N@a$NL\I8$O(BMcGehee$B$NF3F~$7$?AuCV$rMxMQ$7$F;0BN>WFM$K$*$1$k7O$NFC0[E@$rKD$i$^$;$k$3$H$K$"$k!%$3$3$G;H$&:`NA$N$[$H$s$I$O(BMcGehee$B$NO@J8$N$b$N$H$^$C$?$/F1$8$G$"$k!%$+$l$NO@J8$r;2>H$7$FM_$7$$!%0J2<$G$OJQ?t$r$$$/$i$+JQ$($k$,$=$l$K$D$$$F$O(B[2]$B$r;2>H$7$FM_$7$$!%$7$+$740A4$r4|$9$?$a!"5;=QE*>\:Y$N$9$Y$F$r$3$3$G=R$Y$k!%(B

$BJ?LLFsEyJU;0BNLdBj$N@_Dj$r;W$$5/$3$=$&!%J?LLFb$K#3$D$N<AE@(B$m_1 = m_2, m_3$$B$,M?$($i$l$k!%Bh;0BN$r$O$8$a$K(B$y$$B<4$K$*$-!"B.EY$O(B$y$$B$K1h$o$;$k!%<ANL$NEy$7$$#2<AE@$N0LCV$HB.EY$O(B$y$$B<4$K4X$7BP>N$G$"$k!%$@$+$iN3;R$O$O$8$a$KFsEyJU;03Q7A$ND:E@$K0LCV$9$k!%<AE@$O%K%e!<%H%s$NK!B'$K$7$?$,$C$F8_$$$r0z$-9g$&$H$9$k!%LdBj$NBP>N@-$+$i!"N3;R$O$$$D$b$"$kFsEyJU;03Q7A$ND:E@>e$K$"$k!%(B

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$B$3$l$i$N:BI8$G!"1?F0J}Dx<0$O<!$N$h$&$K=q$1$k!%(B
$$  \begin{array}{l}
\displaystyle 
  \ddot{x}_1 = \displaystyle \frac{-2m_1}{x_1^2} - 
\frac {8m_3x_1}{(x_1^2+4x_2^2)^{3/2}}, \\
  \ddot{x}_2 = \displaystyle 
\frac{-8(2m_1 +m_3)x_2}{(x_1^2+4x_2^2)^{3/2}}, 
    \end{array}           \eqno(1.1)
$$
$B$3$N<0$NF3$-J}$K$D$$$F$O(BPollard[4]$B$r;2>H$7$FM_$7$$!%(B

\vspace{6cm}
$p_1$$B$*$h$S(B$p_2$$B$O(B$x_1$$B$*$h$S(B$x_2$$B$K6&Lr$J1?F0NL$rI=$o$9$H$7!"<!<0$GDj5A$9$k!%(B
$$  \begin{array}{l}
  p_1 = m_1 \dot{x}_1/2, \\
  p_2 = 2m_1 m_3 \dot{x}_2 / (2m_1 + m_3).
    \end{array} 
$$
$B$3$N$H$-1?F0J}Dx<0$O#13,$NHyJ,J}Dx<07O$H$7$F%O%_%k%H%s7A<0$G(B
$$  \begin{array}{cc}
\displaystyle  \frac{d x_i}{dt} = \displaystyle 
\frac{\partial H}{\partial p_i}, &
\displaystyle  \frac{d p_i}{dt} = \displaystyle 
- \frac{\partial H}{\partial x_i}, 
    \end{array}           \eqno(1.2)
$$
$B$H=q$1$k!%$3$3$G%O%_%k%H%s4X?t$9$J$o$AA4%(%M%k%.!<$O(B
$$
  H(x_1, x_2, p_1, p_2) = \displaystyle 
\frac{p_1^2}{m_1} + \frac{(2m_1+m_3)p_2^2}{4m_1m_3} 
- \frac{m_1^2}{x_1} - \frac{4m_1m_3}{(x_1^2+4x_2^2)^{1/2}},  \eqno(1.3)
$$
$B$GM?$($i$l$k!%$h$/CN$i$l$F$$$k$h$&$K!"(B$H$$B$O(B(1.2)$B$KBP$9$k1?F0$NDj?t$G$"$k!%(B

$B0J2<$N5-K!$r;H$($P$3$N7O$r$b$C$H4J7i$K=q$1$k!%(B$x=(x_1,x_2)$$B$*$h$S(B$p = (p_1, p_2)$$B$H$*$/!%(B$M = 2m_1 + m_3$$B$r7O$NA4<ANL$H$9$k!%$3$N$H$-<!$N$h$&$K=q$1$k!%(B
$$
   H (x, p) = (1/2) p^t A^{-1} p + V(x),     \eqno(1.4)
$$
$B$3$3$G(B$A^{-1}$$B$O(B$2 \times 2$$B9TNs(B
$$
  A^{-1} = \left( 
\begin{array}{cc}
    2/m_1     &     0 \\
     0        &  M/2m_1m_3
\end{array}                \right),           \eqno(1.5)
$$
$B$G$"$j!"%]%F%s%7%c%k%(%M%k%.!<(B$V(x)$$B$O<!<0$GM?$($i$l$k!%(B
$$
  V(x) = \displaystyle 
- \frac{m_1^2}{x_1} - \frac{4m_1m_3}{(x_1^2 + 4x_2^2)^{1/2}}. 
\eqno(1.6)
$$
$V(x) <0$$B$G$"$k$3$H!"$*$h$S(B$V$$B$O(B$-1$$B<!$NF1<!<0$G$"$k$3$H$KCm0U$7$h$&!%(B

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$$ \begin{array}{l}
  r = (x^t A x)^{1/2}, \\
  s = x / r, \\
  v = r^{1/2} (s \cdot p), \\
  u = r^{1/2} (A^{-1} p - (s \cdot p) s).
   \end{array}                              \eqno(1.7)
$$
$s^tAs=1$$B$*$h$S(B$s^tAu=0$$B$KCm0U$7$h$&!%$^$?!"(B$r^2$$B$O7O$N47@-%b!<%a%s%H$G$"$k!%7O$N;~4VJQ?t$N<\EY$b(B
$$
   dt = r^{3/2} d \tau,
$$
$B$K$h$C$FJQ$($k!%$3$l$i$NJQ?t$r;H$&$H!"7O(B$(1.2)$$B$O<!$N$h$&$K$J$k!%(B
$$ \begin{array}{l}
\displaystyle \frac{dr}{d \tau} = rv, \\
\displaystyle \frac{dv}{d \tau} = u^t A u + (1/2)v^2 + V(s), \\
\displaystyle \frac{ds}{d \tau} = u, \\
\displaystyle 
  \frac{du}{d \tau} = -(1/2)vu - (u^tAu)s - {\rm grad} V(s).
   \end{array}                              \eqno(1.8)
$$

$B$3$3$G(B$V(s)$$B$O(B$V$$B$N5e(B$s^tAs$$B$X$N@)8B$G$"$j!"(B${\rm grad} V(s)$$B$O(B$A$$B$K$h$C$FM6F3$5$l$k7WNL$K$*$1$k$=$N8{G[$G$"$k!%(B

$B$3$3$G(B$-\pi/2 < \theta < \pi /2$$B$H$7$F(B
$$ \begin{array}{l}
 s = (A^{-1})^{1/2} (\cos \theta, \sin \theta), \\
 u = u(A^{-1})^{1/2} ( -\sin \theta, \cos \theta),
   \end{array}
$$
$B$rF3F~$9$k!%J}Dx<0$O<!$N$h$&$K$J$k!%(B
$$ \begin{array}{l}
\displaystyle \frac{dr}{d \tau} = rv, \\
\displaystyle \frac{dv}{d \tau} = u^2 + (1/2)v^2 + V(\theta), \\
\displaystyle \frac{d\theta}{d \tau} = u, \\
\displaystyle \frac{du}{d \tau} = -(1/2)vu - V'(\theta).
   \end{array}                              \eqno(1.9)
$$
$B%(%M%k%.!<$N<0(B$(1.4)$$B$O<!$N$h$&$K$J$k!%(B
$$
   re = (1/2)v^2 + (1/2) u^2 + V(\theta).      \eqno(1.10)
$$
$B$3$3$G(B$e$$B$O%O%_%k%H%s4X?t$N0lDjCM$G$"$j!"$^$?(B
$$
\displaystyle 
  V(\theta) = \displaystyle - \frac{m_1^2}{\sqrt{2/m_1}\cos \theta} - 
\frac{4m_1m_3}{\sqrt{(2/m_1)\cos^2 \theta + 
\frac{2M}{m_1m_3} \sin^2 \theta}} .                                      \eqno(1.11)
$$

$x_1=0$$B$+$D(B$x_2>0$$B$K$*$1$kFsBN>WFM$O(B$\theta=\pi/2$$B$KBP1~$7!"(B$x_1=0$$B$+$D(B$x_2<0$$B$K$*$1$kFsBN>WFM$O(B$\theta=-\pi/2$$B$KBP1~$9$k$3$H$KCm0U$7$h$&!%:G8e$K!";0BN>WFM(B$x_1= x_2 = 0$$B$O(B$r=0$$B$KBP1~$9$k!%(B

$B$5$F!"7O(B(1.9)$B$O(B$r=0$$B$G2r@OE*$G$"$k!%;v<B!"(B$r=0$$B$N$H$-(B$dr/d \tau=0$$B$G$"$k!%$f$($K(B$r=0$$B$ON.$l$NITJQB?MMBN$G$"$k!%$@$+$i;0BN>WFM$KBP1~$9$kFC0[@-$r<h$j=|$$$?$3$H$K$J$k!%FC0[E@$N>l=j$K;0BN>WFMB?MMBN$H8F$P$l$k3j$i$+$JB?MMBN$rEI$j9~$a$?$N$G$"$k!%$$$^$^$GM-8B;~4V$G;0BN>WFM$G$O$8$^$C$?$j=*$o$C$?$j$7$?50F;$O$$$^$d8:B.$5$l$F!"(B$t \rightarrow - \infty$$B$d(B$t \rightarrow \infty$$B$G;0BN>WFMB?MMBN$K6a$E$/!%$=$7$F;0BN>WFM$N6a$/$rDL2a$9$k50F;$O!"$$$^$d;0BN>WFMB?MMBN<+?H$N>e$N50F;$HHs>o$K$h$/;w$?$U$k$^$$$r$9$k!%$@$+$i$3$NB?MMBN>e$NN.$l$rM}2r$9$l$P!"$3$l$i6a>WFM50F;$K4X$7$F$?$/$5$s$N>pJs$,F@$i$l$k!%$3$l$+$iD4$Y$k$N$,$3$NN.$l$G$"$k!%(B

$B7O(B(1.9)$B$O(B$0 \leq r < \infty, v \in {\bf R}, -\pi/2 < \theta < \pi/2, u \in {\bf R}$$B$KBP$7$F2r@OE*$J%Y%/%H%k>l$rDj5A$9$k!%$^$:$3$N7O$r6-3&(B$\theta = \pm \pi/2$$B$^$G3HD%$7$h$&!%$3$l$O(BSundman[11,12]$B$N@5B'2=$rMQ$$$F<B9T$G$-$k!%$o$l$o$l$N<h07$$$O(BMcGehee$B$N$b$N$H$d$d0[$J$k!%(B

$W(\theta) = - (\cos \theta) V(\theta)$$B$H$*$/!%(B$W$$B$O(B$[-\pi/2, \pi/2]$$B>e$G@52r@O4X?t$G$"$k$3$H$KCm0U$7$h$&!%?7$7$$JQ?t(B
$$
  w = \displaystyle \frac{\cos \theta}{\sqrt{W(\theta)}} u,
$$
$B$*$h$S$b$&$R$H$D$N;~4V<\EY$NJQ49(B
$$
\displaystyle \frac{d \tau}{dt} = \displaystyle 
\frac{\cos \theta}{\sqrt{W(\theta)}},
$$
$B$rF3F~$9$k!%$3$N?7$7$$;~4VJQ?t(B$t$$B$O$b$H$N;~4VJQ?t$H0[$J$k$3$H$KCm0U$7$h$&!%(B

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$$ \begin{array}{ll}
\displaystyle 
  \frac{dr}{dt} = & 
\displaystyle \frac{\cos \theta}{\sqrt{W(\theta)}} rv, \\
\displaystyle \frac{dv}{dt} = & \displaystyle \sqrt{W(\theta)} 
\left( 1 - \frac{\cos \theta}{2W(\theta)} (v^2 - 4re) \right), \\
\displaystyle \frac{d\theta}{dt} = & w \\
\displaystyle \frac{dw}{dt} = & \displaystyle (\sin \theta) 
\left( - 1 + \frac{\cos \theta}{W(\theta)} (v^2 - 2re) \right) \\
                  & \displaystyle 
- \frac{vw \cos \theta}{2 \sqrt{W(\theta)}} + 
\frac{W'(\theta)}{W(\theta)} (\cos \theta - w^2/2).
   \end{array}                              \eqno(1.12)
$$
$B%(%M%k%.!<4X78<0$O(B
$$
\displaystyle \frac{w^2}{2 \cos \theta} - 1 = 
   \displaystyle \frac{\cos \theta}{W(\theta)} (re - v^2/2),     
                 \eqno(1.13)
$$
$B$H$J$k!%$3$N7O$O(B$[0, \infty) \times {\bf R} \times [- \pi/2, \pi/2] \times {\bf R}$$B>e$G2r@OE*$J%Y%/%H%k>l$G$"$k$+$i!"FsBN>WFM$K$h$kFC0[@-$O<h$j=|$+$l$F$$$k!%<B:]!"(B$\theta = \pm \pi/2$$B$N$H$-(B$dv/dt>0$$B$G$"$k$+$i!"2r$OCF@->WFM$H$7$FFsBN>WFM8e$K$b3HD%$5$l$F$$$k!%(B

$B;0BN>WFMB?MMBN$O$3$NJQ?tJQ49$N$b$H$G$O!"$=$N$^$^<j$D$+$:$G;D$C$F$*$j!"(B$r=0$$B$G$N%(%M%k%.!<4X78<0$H$7$FM?$($i$l$k!'(B
$$
\displaystyle 
   w^2/2 + \frac{v^2 \cos^2 \theta}{2 W(\theta)} = \cos \theta.
\eqno(1.14)
$$
$B$3$N6JLL$NN,?^$O(BFig.2$B$K$"$k!%$3$l$O0LAjE*$K$O#4E@$r<h$j=|$$$?5e$G$"$k$3$H$KCm0U$7$h$&!%$3$l$OD>@~;0BNLdBj$N>l9g$H$^$C$?$/F1$8$G$"$k!%$7$+$7!"(B(1.12)$B$+$i$3$NLL$KM6F3$5$l$kN.$l$OD>@~LdBj$N>l9g$H7`E*$K0[$J$k!%(B

$B$3$NB?MMBN>e$NN.$l$O<!$NHyJ,J}Dx<0$K$h$C$F7h$^$k!%(B
$$ \begin{array}{ll}
\displaystyle 
  \frac{dv}{dt} = & \displaystyle \sqrt{W(\theta)} 
\left( 1 - \frac{\cos \theta}{2W(\theta)} v^2 \right), \\
\displaystyle \frac{d\theta}{dt} = & w, \\
\displaystyle \frac{dw}{dt} = & \displaystyle (\sin \theta)
 \left( - 1 + \frac{v^2 \cos \theta}{W(\theta)} \right) - 
      \frac{vw \cos \theta}{2 \sqrt{W(\theta)}}  \\
               & \displaystyle 
+ \frac{W'(\theta)}{W(\theta)} (\cos \theta - w^2/2).
   \end{array}                              \eqno(1.15)
$$

$B$3$NN.$l$N<g$J@-<A$O<!@a$G5DO@$9$k!%(B

\vspace{6.3cm}
\noindent
\underline{$BCm0U(B.} $W(\theta)$$B!"$7$?$,$C$F>WFMB?MMBN$O!"(B$m_1$$B$H(B$m_3$$B$NA*$SJ}$K0MB8$9$k!%$3$N0MB8@-$O(B$\eta = v/\sqrt{W(\theta)}$$B$rF3F~$9$l$P<h$j=|$1$k!%$3$N$H$-HyJ,J}Dx<0$O(B
$$ \begin{array}{ll}
\displaystyle 
  \frac{d \eta}{dt} = & \displaystyle 1 - (1/2)\eta^2 \cos \theta - 
(1/2)\frac{W'(\theta)}{W(\theta)} \eta w, \\
\displaystyle \frac{d \theta}{dt} = & w, \\
\displaystyle \frac{dw}{dt} = & \displaystyle 
(\sin \theta) ( - 1 + \eta^2 \cos \theta) - 
(1/2) \eta w \cos \theta \\
              & \displaystyle 
+ \frac{W'(\theta)}{W(\theta)} (\cos \theta - w^2/2), 
   \end{array}                              \eqno(1.16)
$$
$B$H$J$j!"%(%M%k%.!<4X78<0$O(B
$$
   w^2/2 + \eta^2 \cos^2 \theta = 2 \cos \theta,         \eqno(1.17)
$$
$B$H$J$j!"<ANL$K0M$i$J$$!%(B

\vspace{0.3cm}
\noindent
\underline{$BCm0U(B.} $B<ANLHf(B$m_3/m_1$$B$r(B$\varepsilon$$B$H$*$1$P!"4JC1$K7W;;$G$-$k$h$&$K!"(B
%\displaystyle 
$$ \begin{array}{l}
  W(\theta) = \displaystyle m_1^{5/2} \left( \frac{1}{\sqrt{2}} + 
\frac{4 \varepsilon^{3/2} \cos \theta}{\sqrt{2 \varepsilon + 4 \sin^2 \theta}} \right), \\
\displaystyle \frac{W'(\theta)}{W(\theta)} = \displaystyle 
\frac{-8 \sqrt{2} \varepsilon^{3/2} (2 + \varepsilon) \sin \theta}
     {(2 \varepsilon + 4 \sin^2 \theta)(\sqrt{2 \varepsilon + 4 \sin^2 \theta} + 4 \sqrt{2} \varepsilon^{3/2} \cos \theta)},
   \end{array}
$$
$B$G$"$k!%(B$W'/W$$B$O(B$\varepsilon$$B$@$1$K$7$+0M$i$J$$$+$i!"(B(1.16)$B$O<ANLHf$K0MB8$9$k%Y%/%H%k>l$N#1%Q%i%a!<%?B2$rI=$o$9!%(B


















