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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 17 "Potentials in G_2" }}
{PARA 19 "" 0 "" {TEXT -1 28 "Piotr Kobak and Andrew Swann" }}{PARA 
256 "" 0 "" {TEXT -1 17 "5th October, 1998" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{PARA 3 "" 0 "" {TEXT -1 14 "Initialisation" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 10 "read `g2`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@Def
ining~Lie~algebra~operationsG" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning,
 new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new \+
definition for trace" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Embedding~G2
~in~SO7G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1Our~base~line~isG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG-%'MATRIXG6#7)7)\"\"!%\"sGF**&%
\"tG\"\"\"\"\"##F.F/F*F*F*7)F*F*F*F*F*F*F*F17)F*F*F*F*F*F*,$F,!\"\"7)F
*,$F-F4F*F*F*F*F*7)F*F*F-F*F*F*,$F+F4F1" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%+Defining~JG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>Computing~cha
nge~of~variablesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eta1G,$*&%\"kG
\"\"#,&*$%\"sGF(\"\"\"*$%\"tGF(\"\"$F,F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%eta2G,$*&%\"kG\"\"#,(*$%\"sG\"\"%\"\"\"*&F+F(%\"tGF(\"\"'*$F/
F,\"\"$F-F," }}}{PARA 3 "" 0 "" {TEXT -1 19 "First Main Equation" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "xA := Lb(X,sl3([[0,0,0],[0,0
,1/s],[0,0,0]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xAG-%'MATRIXG6
#7)7)\"\"!F*\"\"\"F*F*F*F*7)F*F*F*F*F*F*F*F,F,7)F*F*F*F*F*F*!\"\"F,F,
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "A1 := J2stP(xA);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G-%'MATRIXG6#7)7)\"\"!F*,(*&%\"kG
!\"%%#fsG\"\"##!\"\"\"\"%*,F-F.%\"tG\"\"\"%\"sGF2%#ftGF6F/F6F1F6F6F*F*
F*F*7)F*F*F*F*F*F*F*F9F97)F*F*F*F*F*F*,(F,#F6F3F4F<F2F6F9F9" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "e1 := -numer(A1[1,3]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G,(*&%#fsG\"\"#%\"sG\"\"\"F**(%\"
tGF*%#ftGF*F'F*F**&%\"kG\"\"%F)F*!\"%" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 103 "This is the first main equation.  Note that xA is X at (
s=1, t=0) and that J^2xA is proportional to xA." }}}{PARA 3 "" 0 "" 
{TEXT -1 15 "Other Equations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
44 "xA := Lb(X,sl3([[1,0,0],[0,0,0],[0,0,-1]]));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#xAG-%'MATRIXG6#7)7)\"\"!,$%\"sG!\"\"F*,$*&%\"tG\"\"
\"\"\"##F1F2F-F*F*F*7)F*F*F*F*F*F*F*F47)F*F*F*F*F*F*F/7)F*F0F*F*F*F*F*
7)F*F*,$F0F-F*F*F*F,F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "A
2 := J2stP(xA);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#A2G-%'MATRIXG6#7
)7)\"\"!,2%\"sG!\"\"**%\"kG!\"%F,\"\"#%#fsG\"\"\"%$fssGF3#F3\"\"%*,F/F
0%\"tGF3F,F3F2F3%$fstGF3#\"\"$\"\")**F/F0F8F1F2F3%$fttGF3#F3F<*(F/F0F,
F3F2F1F5**F/F0F8F3F2F3%#ftGF3F5*,F/F0F8F3F,F3FBF3F4F3F?**F/F0F8F1FBF3F
9F3F?F*,0*&F8F3F1#F3F1F-**F/F0F8F3F2F1F1FGF5*.F/F0F8F3F,F3F2F3F1FGF4F3
F5*,F/F0F8F1F2F3F1FGF9F3F5*.F/F0F8F3F,F3FBF3F1FGF9F3#F3\"#O*,F/F0F8F1F
BF3F1FGF>F3FL**F/F0F8F3FBF1F1FGFLF*,8**F/F0F8F1F,F-F2F1#!\"$\"#;F7#F-
\"#[F=FUFA#F-\"#C**F/F0F8F1F2F3F4F3FR*,F/F0F8F;F,F-F2F3F9F3FR**F/F0F,F
1FBF3F9F3#F3\"$W\"*,F/F0F8F3F,F3FBF3F>F3Ffn*(F/F0F,F3FBF1FfnFCFUFDFUF*
7)F*F*F*F*F*F*,8FQ#F;FTF7#F3FVF=F]oFA#F3FXFYF\\oFZF\\oFen#F-FgnFhnF_oF
inF_oFCF]oFDF]o7)F*F*F*F*F*F*F*7)F*F*,8*,F/F0F8F1F,F-F2F1F1FGF\\o*,F/F
0F8F1F2F3F1FGF4F3F\\o*.F/F0F8F;F,F-F2F3F1FGF9F3F\\o*,F/F0F,F1FBF3F1FGF
9F3F_o*.F/F0F8F3F,F3F2F3F1FGF9F3F]o*,F/F0F8F3FBF3F1FGF2F3F^o*.F/F0F8F3
F,F3FBF3F1FGF4F3F]o*,F/F0F8F1FBF3F1FGF9F3F]o*.F/F0F8F3F,F3FBF3F1FGF>F3
F_o**F/F0F,F3FBF1F1FGF_o*,F/F0F8F1F2F3F1FGF>F3F]oF*F*F*,0FFF3FH#F-F6FI
F_pFJF_pFK#F-FMFNF`pFOF`p7)F*,0**F/F0F8F1F2F3F9F3F_pF8F3*,F/F0F8F3F,F3
F2F3F4F3F_p*(F/F0F8F3F2F1F_p*,F/F0F8F3F,F3FBF3F9F3F`p**F/F0F8F1FBF3F>F
3F`p*(F/F0F8F3FBF1F`pF*,8FcoFRFdoFRFeoFRFfoFfnFgoFUFhoFWFioFUFjoFUF[pF
fnF\\pFfnF]pFUF*F*F*7)F*F*,0FcpF5F8F-FdpF5FepF5FfpFLFgpFLFhpFLF*F*F*,2
F,F3F.F_pF7#FSF<F=#F-F<F@F_pFAF_pFCF^qFDF^qF`o" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 64 "A2e := [numer(A2[1,2]),numer(A2[1,4])/t/sqrt(2
),numer(A2[2,7])];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$A2eG7%,2*&%\"
kG\"\"%%\"sG\"\"\"!\")*(F*\"\"#%#fsGF+%$fssGF+F.**%\"tGF+F*F+F/F+%$fst
GF+\"\"$*(F2F.F/F+%$fttGF+F+*&F/F.F*F+F.*(F2F+%#ftGF+F/F+F.**F2F+F*F+F
9F+F0F+F+*(F2F.F9F+F3F+F+,0*$F(F)!#O*$F/F.\"\"**(F*F+F/F+F0F+F@*(F2F+F
/F+F3F+F@*(F*F+F9F+F3F+F+*(F2F+F9F+F6F+F+*$F9F.F+,8*&F2F.F/F.\"#F**F2F
+F*F.F/F+F3F+F4**F2F.F/F+F6F+F*F+F4**F2F+F9F+F/F+F*F+\"\"'**F2F.F/F+F0
F+F*F+FH*(F2F4F/F+F3F+FH*(F*F4F9F+F3F+!\"\"**F2F+F*F.F9F+F6F+FP*&F*F.F
9F.FP**F2F+F*F.F9F+F0F+F4**F2F.F9F+F3F+F*F+F4" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 54 "Eliminate the second derivatives from these equations.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "A2c := map(collect,A2e,
\{fss,fst,ftt\},distribute,factor);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#>%$A2cG7%,.*(%\"tG\"\"#%#fsG\"\"\"%$fttGF+F+*(%\"sGF+,&*&F*F+F.F+F)*&
F(F+%#ftGF+F+F+%$fssGF+F+*(F(F+,&F1F+F0\"\"$F+%$fstGF+F+*&%\"kG\"\"%F.
F+!\")*(F(F+F2F+F*F+F)*&F*F)F.F+F),.*(F(F+F2F+F,F+F+*(F.F+F*F+F3F+\"\"
**&,&*&F(F+F*F+FA*&F2F+F.F+F+F+F7F+F+*$F9F:!#O*$F*F)FA*$F2F)F+,***F.F+
F(F+,&FEF+FD!\"$F+F,F+!\"\"**F.F+F(F+FCF+F3F+F6*&,**(F.F)F(F+F*F+F6*&F
*F+F(F6\"#F*(F(F)F2F+F.F+F6*&F.F6F2F+FNF+F7F+F+*&,&FEF+FD!\"*F+,&FEF+F
DF6F+FN" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "First fss." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "A2e2 := [ coeff(A2c[2],fss)*A2c[1]
 - coeff(A2c[1],fss)*A2c[2], coeff(A2c[3],fss)*A2c[1] - coeff(A2c[1],f
ss)*A2c[3] ];\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%A2e2G7$,&*(%#fsG
\"\"\"%\"sGF),.*(%\"tG\"\"#F(F)%$fttGF)F)*(F*F),&*&F(F)F*F)F.*&F-F)%#f
tGF)F)F)%$fssGF)F)*(F-F),&F3F)F2\"\"$F)%$fstGF)F)*&%\"kG\"\"%F*F)!\")*
(F-F)F4F)F(F)F.*&F(F.F*F)F.F)\"\"**(F*F)F1F),.*(F-F)F4F)F/F)F)*(F*F)F(
F)F5F)F@*&,&*&F-F)F(F)F@*&F4F)F*F)F)F)F9F)F)*$F;F<!#O*$F(F.F@*$F4F.F)F
)!\"\",&**F*F)F-F)FFF)F+F)F8*(F*F)F1F),***F*F)F-F),&FHF)FG!\"$F)F/F)FM
**F*F)F-F)FFF)F5F)F8*&,**(F*F.F-F)F(F)F8*&F(F)F-F8\"#F*(F-F.F4F)F*F)F8
*&F*F8F4F)FMF)F9F)F)*&,&FHF)FG!\"*F),&FHF)FGF8F)FMF)FM" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "A2c2 := map(collect,A2e2,\{fss,fst,
ftt\},distribute,factor);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%A2c2G7
$,(**%\"sG\"\"\"%\"tGF),(*&F*F)%#fsG\"\"#!\"**&F*F)%#ftGF.F)*(F1F)F-F)
F(F)F.F)%$fttGF)!\"\"*(F(F.F+F)%$fstGF)F4*(F1F)F(F),*F,\"\"*F2!\"#*&F*
F)%\"kG\"\"%\"#OF0F4F)F),(**F(F)F*F),**&F*\"\"$F-F.\"#F*(F-F)F(FCF1F)F
.*(F*F)F(F.F-F.!\"'*(F*F)F1F.F(F.F)F)F3F)F)*(F(F.FAF)F6F)F)*&F(F),.**F
(F)F*F.F-F)F<F=\"$;#*(F*FCF1F)F-F.!#F**F(F.F*F)F1F)F<F=\"#C**F-F.F(F.F
1F)F*F)\"\"'*(F-F)F(FCF1F.F:*(F*F)F1FCF(F.F4F)F4" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 9 "Then fst." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 73 "A2e3 := expand(coeff(A2c2[2],fst)*A2c2[1] - coeff(A2c2[1],fst)*A
2c2[2]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%A2e3G,.**%\"sG\"\"%%\"
tG\"\"$%#fsGF*%\"kGF(\"%W>*,F'F(F)F*%#ftG\"\"#F+\"\"\"F,F(!$;#*,F'\"\"
&F)F0F+F0F/F1F,F(!$K%*,F'\"\"'F+F1F/F0F)F1F,F(\"#C**F'F4F)F0F/F*F,F(\"
#7*,F'F*F)F(F+F0F/F1F,F(\"$s*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "
The surprise is that now there is no ftt term." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "e2 := factor(A2e3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#e2G,$*,%\"kG\"\"%%\"sG\"\"$%\"tG\"\"\",&*&%#fsGF,F)F
,\"\"#*&F+F,%#ftGF,F,F,,&*&F2F,F)F,F,*&F+F,F/F,!\"*F0\"#7" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 53 "This has solutions ft = - 2 s fs/t and ft
 = 9 t fs/s." }}}{PARA 3 "" 0 "" {TEXT -1 9 "Solutions" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "sollist := [ -2*s*fs/t, 9*t*fs/s ];
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(sollistG7$,$*(%\"sG\"\"\"%#fsGF
)%\"tG!\"\"!\"#,$*(F+F)F*F)F(F,\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 43 "Let us verify these are the solutions to e2" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 23 "subs(ft=sollist[1],e2);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "s
ubs(ft=sollist[2],e2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "OK" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "factor(subs(ft=op(1,sollist),e1));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,$*&%\"sG\"\"\",&*$%#fsG\"\"#F&*$%\"kG\"\"%F-F&!\"\"
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "This has no solutions." }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "simplify(su
bs(ft=op(2,sollist),e1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(*&%
#fsG\"\"#%\"sGF(!\"\"*&%\"tGF(F'F(!\"**&%\"kG\"\"%F)F(F0\"\"\"F)F*F*" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "numer(\");" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,(*&%#fsG\"\"#%\"sGF&\"\"\"*&%\"tGF&F%F&\"\"**&%\"
kG\"\"%F'F&!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "collect(
\",fs);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&*$%\"sG\"\"#\"\"\"*$%
\"tGF(\"\"*F)%#fsGF(F)*&%\"kG\"\"%F'F(!\"%" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 67 "So fs=2 k^2s/sqrt(s^2+9t^2) and ft=9fs t/s=18 k^2 t/sqrt(
s^2+9t^2)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "int(2*k^2*s/s
qrt(s^2+9*t^2),s)+c(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&*$%\"
sG\"\"#\"\"\"*$%\"tGF(\"\"*#F)F(%\"kGF(F(-%\"cG6#F+F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "int(18*k^2*t/sqrt(s^2+9*t^2),t)+c(s
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&*$%\"sG\"\"#\"\"\"*$%\"tGF
(\"\"*#F)F(%\"kGF(F(-%\"cG6#F'F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
53 "So both c-terms are zero and f = 2k^2sqrt(s^2+9*t^2)." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "radsimp(k*sqrt(2*(eta1+sqrt(6*(eta1
^2-k^2*eta2)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&*$%\"sG\"\"
#\"\"\"*$%\"tGF(\"\"*#F)F(%\"kGF(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 56 "So this is the expression in terms of the eta variables." }}}
{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{MARK "39 0 0" 56 }{VIEWOPTS 1 1 0 1 
1 1803 }