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"In this Maple file, we \+ check that the las for the Lax matrices are correct provided that the \+ Hamiltonian are given as the linear combination of the spectral invari ants (Theorem 8.1). We check the formulas both in the oper or in the g eometric gauge using also the symmetric Darboux coordinates." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 212 54 "Loarding some procedures for the \+ symmetric polynomials" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "res tart:\n" }{MPLTEXT 1 0 21 "with(LinearAlgebra):\n" }{MPLTEXT 1 0 17 "w ith(ListTools):\n" }{MPLTEXT 1 0 16 "with(combinat):\n" }{MPLTEXT 1 0 23 "with(PolynomialTools):\n" }{MPLTEXT 1 0 16 "with(Groebner):\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 12 "chk:=proc()\n" }{MPLTEXT 1 0 26 "local VV,AA,pp,LL,K,N,KK:\n" }{MPLTEXT 1 0 31 "VV:=[seq(args[i],i=2.. nargs)];\n" }{MPLTEXT 1 0 34 "AA:=[seq(sigma[i],i=1..nargs-1)];\n" } {MPLTEXT 1 0 53 "pp:=simplify(expand(mul(x_-args[i],i=2..nargs)),x_); \n" }{MPLTEXT 1 0 62 "LL := Reverse([seq((-1)^(r+nargs-1)*coeff(pp, x_ , r), r = 0..\n" }{MPLTEXT 1 0 15 "nargs-2)])-AA;\n" }{MPLTEXT 1 0 25 "K:=Basis(LL,tdeg(VV[])):\n" }{MPLTEXT 1 0 37 "N:=NormalForm(args[1],K ,tdeg(VV[]));\n" }{MPLTEXT 1 0 26 "KK:=Basis(AA,tdeg(AA[]));\n" } {MPLTEXT 1 0 29 "NormalForm(N,KK,tdeg(AA[]));\n" }{MPLTEXT 1 0 65 "if \+ is(NormalForm(N,KK,tdeg(AA[]))=0) then print(\"symmetric\")else\n" } {MPLTEXT 1 0 26 "print(\"not symmetric\")fi;\n" }{MPLTEXT 1 0 10 "end \+ proc:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 11 "es:=proc()\n" } {MPLTEXT 1 0 25 "local VV, AA, pp, LL, K;\n" }{MPLTEXT 1 0 64 "VV:=[se q(args[i],i=2..nargs)];AA:=[seq(sigma[i],i=1..nargs-1)];\n" }{MPLTEXT 1 0 53 "pp:=simplify(expand(mul(x_-args[i],i=2..nargs)),x_);\n" } {MPLTEXT 1 0 62 "LL := Reverse([seq((-1)^(r+nargs-1)*coeff(pp, x_, r), r = 0..\n" }{MPLTEXT 1 0 15 "nargs-2)])-AA;\n" }{MPLTEXT 1 0 25 "K:=B asis(LL,tdeg(VV[]));\n" }{MPLTEXT 1 0 34 "NormalForm(args[1],K,tdeg(VV []));\n" }{MPLTEXT 1 0 10 "end proc:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 41 "ss:=proc() local LL, LLL, t, LLLL, H, K;\n" }{MPLTEXT 1 0 31 "LL:=[seq(args[i],i=2..nargs)];\n" }{MPLTEXT 1 0 41 "LLL:=[seq(map(x-> x^r,LL),r=1..nargs-1)];\n" }{MPLTEXT 1 0 27 "t:=seq(s[i],i=1..nargs-1) ;\n" }{MPLTEXT 1 0 48 "LLLL:=[seq(add(i,i in LLL[u]),u=1..nops(LLL))]; \n" }{MPLTEXT 1 0 13 "H:=LLLL-[t];\n" }{MPLTEXT 1 0 25 "K:=Basis(H,grl ex(LL[]));\n" }{MPLTEXT 1 0 35 "NormalForm(args[1],K,grlex(LL[]));\n" }{MPLTEXT 1 0 9 "end proc:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "ElementaryS:= proc(k)\n" }{MPLTEXT 1 0 19 "local aux ,i,Coeff:\n" }{MPLTEXT 1 0 50 "aux:=0: for i from 1 to g do aux:=aux+q [i]^k od: \n" }{MPLTEXT 1 0 67 "Coeff:=unapply(es(aux,q[1],q[2],q[3]), sigma[1],sigma[2],sigma[3]):\n" }{MPLTEXT 1 0 31 "return(Coeff(Q[1],Q[ 2],Q[3])):\n" }{MPLTEXT 1 0 10 "end proc:\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 11 "rinfty:=6:\n" }{MPLTEXT 1 0 12 "g:=rinfty-3:" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "S[0]:=ElementaryS(0);\n" } {MPLTEXT 1 0 22 "S[1]:=ElementaryS(1);\n" }{MPLTEXT 1 0 22 "S[2]:=Elem entaryS(2);\n" }{MPLTEXT 1 0 22 "S[3]:=ElementaryS(3);\n" }{MPLTEXT 1 0 22 "S[4]:=ElementaryS(4);\n" }{MPLTEXT 1 0 22 "S[5]:=ElementaryS(5); \n" }{MPLTEXT 1 0 22 "S[6]:=ElementaryS(6);\n" }{MPLTEXT 1 0 22 "S[7]: =ElementaryS(7);\n" }{MPLTEXT 1 0 21 "S[8]:=ElementaryS(8);" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"SG6\"6#\"\"!\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 ">&I\"SG6\"6#\"\"\"&I\"QGF%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"SG6\"6#\"\"#,&*$&I\"QGF%6#\"\"\"F'F-&F+F&! \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"SG6\"6#\"\"$,(*$&I\"QGF%6#\" \"\"F'F-*&F*F-&F+6#\"\"#F-!\"$&F+F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"SG6\"6#\"\"%,**$&I\"QGF%6#\"\"\"F'F-*&F*\"\"#&F+6#F/F-!\"%*&F*F- &F+6#\"\"$F-F'*$F0F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"SG6\"6#\" \"&,,*$&I\"QGF%6#\"\"\"F'F-*&F*\"\"$&F+6#\"\"#F-!\"&*&F*F2&F+6#F/F-F'* &F*F-F0F2F'*&F0F-F5F-F3" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"SG6\"6# \"\"',0*$&I\"QGF%6#\"\"\"F'F-*&F*\"\"%&F+6#\"\"#F-!\"'*&F*\"\"$&F+6#F5 F-F'*&F*F2F0F2\"\"**(F*F-F0F-F6F-!#7*$F0F5!\"#*$F6F2F5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"SG6\"6#\"\"(,2*$&I\"QGF%6#\"\"\"F'F-*&F*\"\"&&F +6#\"\"#F-!\"(*&F*\"\"%&F+6#\"\"$F-F'*&F*F8F0F2\"#9*(F*F2F0F-F6F-!#@*& F*F-F0F8F3*&F*F-F6F2F'*&F0F2F6F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">& I\"SG6\"6#\"\"),6*$&I\"QGF%6#\"\"\"F'F-*&F*\"\"'&F+6#\"\"#F-!\")*&F*\" \"&&F+6#\"\"$F-F'*&F*\"\"%F0F2\"#?*(F*F8F0F-F6F-!#K*&F*F2F0F8!#;*&F*F2 F6F2\"#7*(F*F-F0F2F6F-\"#C*$F0F:F2*&F0F-F6F2F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "res:=-lambda^(2*rinfty-5):\n" }{MPLTEXT 1 0 67 "for k from (rinfty-2) to (2*rinfty-7) do aux:=2*tau[2*rinfty-k-6]:\n" }{MPLTEXT 1 0 89 "for m from (k-rinfty+6) to (rinfty-3) do aux:=aux+t au[rinfty-m-2]*tau[rinfty-k+m-5]: od:\n" }{MPLTEXT 1 0 27 "res:=res-au x*lambda^k: od:\n" }{MPLTEXT 1 0 23 "aux2:=2*tau[rinfty-3]:\n" } {MPLTEXT 1 0 71 "for m from 3 to (rinfty-3) do aux2:=aux2+tau[rinfty-m -2]*tau[m-2]: od:\n" }{MPLTEXT 1 0 33 "res:=res-aux2*lambda^(rinfty-3) :\n" }{MPLTEXT 1 0 4 "res;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**$I'lamb daG6\"\"\"(!\"\"*&&I$tauGF%6#\"\"\"F,F$\"\"&!\"#*&&F*6#\"\"#F,F$\"\"%F .*&,&*$F)F2F,&F*6#\"\"$F2F,F$F9F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "tdP2:=unapply(-lambda^7-2*tau[1]*lambda^5-2*tau[2]*la mbda^4-(tau[1]^2+2*tau[3])*lambda^3,lambda);\n" }{MPLTEXT 1 0 101 "for k from rinfty-3 to 2*rinfty-5 do P2[k]:=-residue(tdP2(lambda)/lambda^ (k+1), lambda=infinity): od:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 7 "P2 [3];\n" }{MPLTEXT 1 0 7 "P2[4];\n" }{MPLTEXT 1 0 7 "P2[5];\n" } {MPLTEXT 1 0 7 "P2[6];\n" }{MPLTEXT 1 0 6 "P2[7];" }{MPLTEXT 1 0 1 "\n " }{MPLTEXT 1 0 10 "q1:=q[1]:\n" }{MPLTEXT 1 0 10 "q2:=q[2]:\n" } {MPLTEXT 1 0 9 "q3:=q[3]:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 10 "p1:= p[1]:\n" }{MPLTEXT 1 0 9 "p2:=p[2]:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 10 "p3:=p[3]:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "Elementaryh: = proc(k)\n" }{MPLTEXT 1 0 19 "local aux,i,Coeff:\n" }{MPLTEXT 1 0 55 "aux:=1: for i from 1 to g do aux:=aux/(1-t*q[i]): od: \n" }{MPLTEXT 1 0 76 "Coeff:=unapply(es(residue(aux/t^(k+1),t=0),q[1],q[2],q[3]),sig ma[1],sigma[2]" }{MPLTEXT 1 0 9 ",sigma[3]" }{MPLTEXT 1 0 3 "):\n" } {MPLTEXT 1 0 31 "return(Coeff(Q[1],Q[2],Q[3])):\n" }{MPLTEXT 1 0 10 "e nd proc:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "h[0]:=simplify(Elem entaryh(0));\n" }{MPLTEXT 1 0 32 "h[1]:=simplify(Elementaryh(1));\n" } {MPLTEXT 1 0 32 "h[2]:=simplify(Elementaryh(2));\n" }{MPLTEXT 1 0 32 " h[3]:=simplify(Elementaryh(3));\n" }{MPLTEXT 1 0 32 "h[4]:=simplify(El ementaryh(4));\n" }{MPLTEXT 1 0 32 "h[5]:=simplify(Elementaryh(5));\n" }{MPLTEXT 1 0 32 "h[6]:=simplify(Elementaryh(6));\n" }{MPLTEXT 1 0 32 "h[7]:=simplify(Elementaryh(7));\n" }{MPLTEXT 1 0 15 "h[8]:=simplif y(" }{MPLTEXT 1 0 16 "Elementaryh(8));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "Q[0]:=1:\n" }{MPLTEXT 1 0 22 "Q[1]:=q[1]+ q[2]+q[3]:\n" }{MPLTEXT 1 0 16 "Q[2]:=q[1]*q[2]+" }{MPLTEXT 1 0 10 "q[ 1]*q[3]+" }{MPLTEXT 1 0 9 "q[2]*q[3]" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 20 "Q[3]:=q[1]*q[2]*q[3]" }{MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 26 "SymMatrix:=Matrix(g,g,0):\n" }{MPLTEXT 1 0 83 "for \+ i from 1 to g do for j from 1 to g do SymMatrix[i,j]:=diff(Q[j],q[i]): od: od:\n" }{MPLTEXT 1 0 11 "SymMatrix:\n" }{MPLTEXT 1 0 24 "Vectorp: =Matrix(g,1,0):\n" }{MPLTEXT 1 0 45 "for i from 1 to g do Vectorp[i,1] :=p[i]: od:\n" }{MPLTEXT 1 0 9 "Vectorp:\n" }{MPLTEXT 1 0 43 "VectorP: =Multiply(SymMatrix^(-1),Vectorp);\n" }{MPLTEXT 1 0 44 "for i from 1 t o g do P[i]:=VectorP[i,1]: od:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%tdP2G6\"f*6# I'lambdaGF$F$6$I)operatorGF$I&arrowGF$F$,**$9$\"\"(!\"\"*&&I$tauGF$6# \"\"\"F4F-\"\"&!\"#*&&F26#\"\"#F4F-\"\"%F6*&,&*$F1F:F4&F26#\"\"$F:F4F- FAF/F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$&I$tauG6\"6#\"\"\"\"\" #!\"\"&F%6#\"\"$!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$&I$tauG6\"6#\" \"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$&I$tauG6\"6#\"\"\"!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"hG6\"6#\"\"!\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ">&I\"hG6\"6#\"\"\"&I\"QGF%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"hG6\"6#\"\"#,&*$&I\"QGF%6#\"\"\"F'F-&F+F&! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"hG6\"6#\"\"$,(*$&I\"QGF%6# \"\"\"F'F-*&F*F-&F+6#\"\"#F-!\"#&F+F&F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"hG6\"6#\"\"%,**$&I\"QGF%6#\"\"\"F'F-*&F*\"\"#&F+6#F/F-!\"$*&F *F-&F+6#\"\"$F-F/*$F0F/F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"hG6\"6 #\"\"&,,*$&I\"QGF%6#\"\"\"F'F-*&F*\"\"$&F+6#\"\"#F-!\"%*&F*F2&F+6#F/F- F/*&F*F-F0F2F/*&F0F-F5F-!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"hG6 \"6#\"\"',0*$&I\"QGF%6#\"\"\"F'F-*&F*\"\"%&F+6#\"\"#F-!\"&*&F*\"\"$&F+ 6#F5F-F/*&F*F2F0F2F'*(F*F-F0F-F6F-!\"'*$F0F5!\"\"*$F6F2F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"hG6\"6#\"\"(,2*$&I\"QGF%6#\"\"\"F'F-*&F*\"\" &&F+6#\"\"#F-!\"'*&F*\"\"%&F+6#\"\"$F-F/*&F*F8F0F2\"#5*(F*F2F0F-F6F-!# 7*&F*F-F0F8!\"%*&F*F-F6F2F8*&F0F2F6F-F8" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"hG6\"6#\"\"),6*$&I\"QGF%6#\"\"\"F'F-*&F*\"\"'&F+6#\"\"#F-!\"( *&F*\"\"&&F+6#\"\"$F-F/*&F*\"\"%F0F2\"#:*(F*F8F0F-F6F-!#?*&F*F2F0F8!#5 *&F*F2F6F2F/*(F*F-F0F2F6F-\"#7*$F0F:F-*&F0F-F6F2!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibG F'6'-I%mrowGF$6#-I'mtableGF$67-I$mtrGF$6&-I$mtdGF$6(-F,6'-I&mfracGF$6) -F,6%-I(msubsupGF$6'-I#miGF$6&Q\"qF'/%'italicGQ%trueF'/%+foregroundGQ( [0,0,0]F'/%,mathvariantGQ'italicF'-F,6#-I#mnGF$6%Q\"1F'FH/FLQ'normalF' -FQ6%Q\"2F'FHFT/%1superscriptshiftGQ\"0F'/%/subscriptshiftGFen-I#moGF$ 6-Q1⁢F'FT/%&fenceGQ&falseF'/%*separatorGF^o/%)stretchyG F^o/%*symmetricGF^o/%(largeopGF^o/%.movablelimitsGF^o/%'accentGF^o/%'l spaceGQ&0.0emF'/%'rspaceGF]p-I%msubGF$6%-FB6&Q\"pF'FEFHFKFNFfn-F,6#-F, 6)-F,6#F>-Fin6.Q(−F'FHFTF\\oF_oFaoFcoFeoFgoFio/F\\pQ,0.2222222em F'/F_pF`q-F,6%-Fap6%FAFNFfnFhn-Fap6%FA-F,6#FVFfnF\\q-F,6%FdqFhn-Fap6%F A-F,6#-FQ6%Q\"3F'FHFTFfn-Fin6.Q\"+F'FHFTF\\oF_oFaoFcoFeoFgoFioF_qFaq-F ,6%FfqFhnF\\r/%.linethicknessGFS/%+denomalignGQ'centerF'/%)numalignGF \\s/%)bevelledGF^oFHF\\q-F:6)-F,6%-F?6'FAFhqFVFYFfnFhn-Fap6%FcpFhqFfn- F,6#-F,6)FbqF\\qFjqF\\q-F,6#FesFcrFfrFhrFjrF]sF_sFHFcr-F:6)-F,6%-F?6'F AF^rFVFYFfnFhn-Fap6%FcpF^rFfn-F,6#-F,6)FbqF\\qFjqF\\qFfrFcr-F,6#FctFhr FjrF]sF_sFH/%)rowalignGQ!F'/%,columnalignGF_u/%+groupalignGF_u/%(rowsp anGFS/%+columnspanGFSF]uF`uFbu-F26&-F56(-F,6(-Fin6.Q*&uminus0;F'FHFTF \\oF_oFaoFcoFeoFgoFioF_qFaq-F:6)-F,6%FdqFhnF`pFfpFhrFjrF]sF_sFHFcr-F:6 )-F,6%FfqFhnFgsFisFhrFjrF]sF_sFHF\\q-F:6)-F,6%F\\rFhnFetFgtFhrFjrF]sF_ sFHF]uF`uFbuFduFfuF]uF`uFbu-F26&-F56(-F,6'-F:6)-F,6#F`pFfpFhrFjrF]sF_s FHF\\q-F:6)-F,6#FgsFisFhrFjrF]sF_sFHFcr-F:6)-F,6#FetFgtFhrFjrF]sF_sFHF ]uF`uFbuFduFfuF]uF`uFbu/%&alignGQ%axisF'/F^uQ)baselineF'/FauF\\s/FcuQ' |frleft|hrF'/%/alignmentscopeGFG/%,columnwidthGQ%autoF'/%&widthGF[y/%+ rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,c olumnlinesGFfy/%&frameGFfy/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsG F^o/%-equalcolumnsGF^o/%-displaystyleGF^o/%%sideGQ&rightF'/%0minlabels pacingGFcyFHFT/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 64 "Computing the spectral invariants using the theoretical formulas" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "V:=Matrix(3,3, 0):\n" }{MPLTEXT 1 0 11 "V[1,1]:=1:\n" }{MPLTEXT 1 0 12 "V[1,2]:=q1:\n " }{MPLTEXT 1 0 14 "V[1,3]:=q1^2:\n" }{MPLTEXT 1 0 11 "V[2,1]:=1:\n" } {MPLTEXT 1 0 12 "V[2,2]:=q2:\n" }{MPLTEXT 1 0 14 "V[2,3]:=q2^2:\n" } {MPLTEXT 1 0 11 "V[3,1]:=1:\n" }{MPLTEXT 1 0 12 "V[3,2]:=q3:\n" } {MPLTEXT 1 0 14 "V[3,3]:=q3^2:\n" }{MPLTEXT 1 0 3 "V;\n" }{MPLTEXT 1 0 24 "HVector:=Matrix(3,1,0):\n" }{MPLTEXT 1 0 18 "HVector[1,1]:=H0:\n " }{MPLTEXT 1 0 18 "HVector[2,1]:=H1:\n" }{MPLTEXT 1 0 18 "HVector[3,1 ]:=H2:\n" }{MPLTEXT 1 0 9 "HVector;\n" }{MPLTEXT 1 0 21 "RHSH:=Matrix( 3,1,0):\n" }{MPLTEXT 1 0 62 "RHSH[1,1]:=p1^2+tdP2(q1)+h*(p2-p1)/(q1-q2 )+h*(p3-p1)/(q1-q3):\n" }{MPLTEXT 1 0 62 "RHSH[2,1]:=p2^2+tdP2(q2)+h*( p1-p2)/(q2-q1)+h*(p3-p2)/(q2-q3):\n" }{MPLTEXT 1 0 62 "RHSH[3,1]:=p3^2 +tdP2(q3)+h*(p1-p3)/(q3-q1)+h*(p2-p3)/(q3-q2):\n" }{MPLTEXT 1 0 6 "RHS H;\n" }{MPLTEXT 1 0 41 "HVector:=simplify(Multiply(V^(-1),RHSH));" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modu lenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$67-I$mtr GF$6(-I$mtdGF$6(-I#mnGF$6%Q\"1F'/%+foregroundGQ([0,0,0]F'/%,mathvarian tGQ'normalF'/%)rowalignGQ!F'/%,columnalignGFC/%+groupalignGFC/%(rowspa nGF:/%+columnspanGF:-F56(-I%msubGF$6%-I#miGF$6&Q\"qF'/%'italicGQ%trueF 'F;/F?Q'italicF'-F,6#F7/%/subscriptshiftGQ\"0F'FAFDFFFHFJ-F56(-F,6#-I( msubsupGF$6'FQFZ-F86%Q\"2F'F;F>/%1superscriptshiftGFhnFfnFAFDFFFHFJFAF DFF-F26(F4-F56(-FO6%FQ-F,6#F`oFfnFAFDFFFHFJ-F56(-F,6#-F^o6'FQF[pF`oFco FfnFAFDFFFHFJFAFDFF-F26(F4-F56(-FO6%FQ-F,6#-F86%Q\"3F'F;F>FfnFAFDFFFHF J-F56(-F,6#-F^o6'FQFipF`oFcoFfnFAFDFFFHFJFAFDFF/%&alignGQ%axisF'/FBQ)b aselineF'/FEQ'centerF'/FGQ'|frleft|hrF'/%/alignmentscopeGFW/%,columnwi dthGQ%autoF'/%&widthGFar/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8e mF'/%)rowlinesGQ%noneF'/%,columnlinesGF\\s/%&frameGF\\s/%-framespacing GQ,0.4em~0.5exF'/%*equalrowsGQ&falseF'/%-equalcolumnsGFfs/%-displaysty leGFfs/%%sideGQ&rightF'/%0minlabelspacingGFirF;F>/%%openGQ\"[F'/%&clos eGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6 \"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$67-I$mtrGF$6&-I$ mtdGF$6(-I#miGF$6&Q#H0F'/%'italicGQ%trueF'/%+foregroundGQ([0,0,0]F'/%, mathvariantGQ'italicF'/%)rowalignGQ!F'/%,columnalignGFF/%+groupalignGF F/%(rowspanGQ\"1F'/%+columnspanGFMFDFGFI-F26&-F56(-F86&Q#H1F'F;F>FAFDF GFIFKFNFDFGFI-F26&-F56(-F86&Q#H2F'F;F>FAFDFGFIFKFNFDFGFI/%&alignGQ%axi sF'/FEQ)baselineF'/FHQ'centerF'/FJQ'|frleft|hrF'/%/alignmentscopeGF=/% ,columnwidthGQ%autoF'/%&widthGFeo/%+rowspacingGQ&1.0exF'/%.columnspaci ngGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF`p/%&frameGF`p/%-frame spacingGQ,0.4em~0.5exF'/%*equalrowsGQ&falseF'/%-equalcolumnsGFjp/%-dis playstyleGFjp/%%sideGQ&rightF'/%0minlabelspacingGF]pF>/FBQ'normalF'/%% openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfenced G6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableG F$67-I$mtrGF$6&-I$mtdGF$6(-F,6/-F,6#-I(msubsupGF$6'-I#miGF$6&Q\"pF'/%' italicGQ%trueF'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'italicF'-F,6# -I#mnGF$6%Q\"1F'FE/FIQ'normalF'-FN6%Q\"2F'FEFQ/%1superscriptshiftGQ\"0 F'/%/subscriptshiftGFX-I#moGF$6.Q(−F'FEFQ/%&fenceGQ&falseF'/%*se paratorGF[o/%)stretchyGF[o/%*symmetricGF[o/%(largeopGF[o/%.movablelimi tsGF[o/%'accentGF[o/%'lspaceGQ,0.2222222emF'/%'rspaceGFjo-F,6#-F<6'-F? 6&Q\"qF'FBFEFHFK-FN6%Q\"7F'FEFQFVFYFen-F,6'FS-Ffn6-Q1⁢F 'FQFinF\\oF^oF`oFboFdoFfo/FioQ&0.0emF'/F\\pF]q-I%msubGF$6%-F?6&Q&τ F'/FCF[oFEFQFKFYFip-F<6'FapFK-FN6%Q\"5F'FEFQFVFYFen-F,6'FSFip-F`q6%Fbq -F,6#FSFYFip-F<6'FapFK-FN6%Q\"4F'FEFQFVFYFen-F,6%-F#6%-F,6%-F,6#-F<6'F 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}{MPLTEXT 1 0 12 "HVector[2,1]" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 17 "H2:=HVector[3,1]:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 24 "TMatrix:=Matrix(3,3,0):\n" }{MPLTEXT 1 0 17 "TMatrix[ 1,1]:=1:\n" }{MPLTEXT 1 0 17 "TMatrix[2,2]:=1:\n" }{MPLTEXT 1 0 17 "TM atrix[3,3]:=1:\n" }{MPLTEXT 1 0 22 "TMatrix[3,1]:=tau[1]:\n" }{MPLTEXT 1 0 9 "TMatrix;\n" }{MPLTEXT 1 0 24 "RHStau1:=Matrix(3,1,0):\n" } {MPLTEXT 1 0 14 "RHStau1[1,1]:=" }{MPLTEXT 1 0 22 "1/(2*rinfty-2*1-5)* 1:\n" }{MPLTEXT 1 0 24 "RHStau2:=Matrix(3,1,0):\n" }{MPLTEXT 1 0 14 "R HStau2[2,1]:=" }{MPLTEXT 1 0 21 "1/(2*rinfty-2*2-5)*1:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 24 "RHStau3:=Matrix(3,1,0):\n" }{MPLTEXT 1 0 35 " RHStau3[3,1]:=1/(2*rinfty-2*3-5)*1:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "RHStau1;\n" }{MPLTEXT 1 0 9 "RHStau2;\n" }{MPLTEXT 1 0 9 "RHStau3 ;\n" }{MPLTEXT 1 0 45 "nutau1Vector:=Multiply(TMatrix^(-1),RHStau1);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 46 "nutau2Vector:=Multiply(TMatrix^ (-1),RHStau2);\n" }{MPLTEXT 1 0 45 "nutau3Vector:=Multiply(TMatrix^(-1 ),RHStau3);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "nu1tau1:=" }{MPLTEXT 1 0 17 "nutau1Vector[1,1]" }{MPLTEXT 1 0 2 "; \n" }{MPLTEXT 1 0 9 "nu2tau1:=" }{MPLTEXT 1 0 17 "nutau1Vector[2,1]" } {MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 9 "nu3tau1:=" }{MPLTEXT 1 0 17 "nut au1Vector[3,1]" }{MPLTEXT 1 0 1 ";" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 28 "nu1tau2:=nutau2Vector[1,1];\n" }{MPLTEXT 1 0 28 "nu2tau2:=nutau2Vector[2,1];\n" }{MPLTEXT 1 0 27 "nu3tau2:=nutau2 Vector[3,1];" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 28 "nu1tau3:=nutau3Vector[1,1];\n" }{MPLTEXT 1 0 28 "nu2tau3:=nutau3Ve ctor[2,1];\n" }{MPLTEXT 1 0 27 "nu3tau3:=nutau3Vector[3,1];" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 24 "mutau1Vector:=Multiply((" }{MPLTEXT 1 0 27 "LinearAlgebra[Transpos e](V)" }{MPLTEXT 1 0 7 ")^(-1)," }{MPLTEXT 1 0 13 "nutau1Vector)" } {MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 9 "mu1tau1:=" }{MPLTEXT 1 0 19 "mut au1Vector[1,1]:\n" }{MPLTEXT 1 0 9 "mu2tau1:=" }{MPLTEXT 1 0 19 "mutau 1Vector[2,1]:\n" }{MPLTEXT 1 0 9 "mu3tau1:=" }{MPLTEXT 1 0 18 "mutau1V ector[3,1]:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 13 "mutau1Vector;" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 24 "mutau2Vecto r:=Multiply((" }{MPLTEXT 1 0 27 "LinearAlgebra[Transpose](V)" } {MPLTEXT 1 0 22 ")^(-1),nutau2Vector);\n" }{MPLTEXT 1 0 28 "mu1tau2:=m utau2Vector[1,1]:\n" }{MPLTEXT 1 0 28 "mu2tau2:=mutau2Vector[2,1]:\n" }{MPLTEXT 1 0 28 "mu3tau2:=mutau2Vector[3,1]:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 24 "mutau3Vector:=Multiply((" }{MPLTEXT 1 0 27 "LinearAl gebra[Transpose](V)" }{MPLTEXT 1 0 22 ")^(-1),nutau3Vector);\n" } {MPLTEXT 1 0 28 "mu1tau3:=mutau3Vector[1,1]:\n" }{MPLTEXT 1 0 28 "mu2t au3:=mutau3Vector[2,1]:\n" }{MPLTEXT 1 0 28 "mu3tau3:=mutau3Vector[3,1 ]:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "Hamtau1:= nu1tau1*H0+nu2t au1*H1+" }{MPLTEXT 1 0 10 "nu3tau1*H2" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 31 "Hamtau2:= nu1tau2*H0+nu2tau2*H1" }{MPLTEXT 1 0 1 "+" } {MPLTEXT 1 0 10 "nu3tau2*H2" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 42 "H amtau3:= nu1tau3*H0+nu2tau3*H1+nu3tau3*H2" }{MPLTEXT 1 0 1 ";" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 5 "QQ:=u" } {MPLTEXT 1 0 35 "napply(-p1*(lambda-q2)*(lambda-q3)/" }{MPLTEXT 1 0 59 "(q1-q2)/(q1-q3)-p2*(lambda-q1)*(lambda-q3)/(q2-q1)/(q2-q3)\n" } {MPLTEXT 1 0 43 "-p3*(lambda-q1)*(lambda-q2)/(q3-q1)/(q3-q2)" } {MPLTEXT 1 0 9 ",lambda);" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "J:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 11 "J[1,1]:=1:\n" }{MPLTEXT 1 0 11 "J[1,2]:=0:\n" }{MPLTEXT 1 0 42 "J[2,1]:=QQ(lambda)/ (lambda-q1)/(lambda-q2)" }{MPLTEXT 1 0 12 "/(lambda-q3)" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 33 "J[2,2]:=1/(lambda-q1)/(lambda-q2)" } {MPLTEXT 1 0 12 "/(lambda-q3)" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 3 " J;\n" }{MPLTEXT 1 0 26 "dJdlambda:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 42 "for i from 1 to 2 do for j from 1 to 2 do\n" }{MPLTEXT 1 0 45 "dJdlam bda[i,j]:=diff(J[i,j],lambda): od: od:\n" }{MPLTEXT 1 0 10 "dJdlambda; " }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Type settingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$67-I$mtrGF$6(-I$mtdGF$6( -I#mnGF$6%Q\"1F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%)r owalignGQ!F'/%,columnalignGFC/%+groupalignGFC/%(rowspanGF:/%+columnspa nGF:-F56(-F86%Q\"0F'F;F>FAFDFFFHFJFLFAFDFF-F26(FLF4FLFAFDFF-F26(-F56(- I%msubGF$6%-I#miGF$6&Q&τF'/%'italicGQ&falseF'F;F>-F,6#F7/%/subscri ptshiftGFPFAFDFFFHFJFLF4FAFDFF/%&alignGQ%axisF'/FBQ)baselineF'/FEQ'cen terF'/FGQ'|frleft|hrF'/%/alignmentscopeGQ%trueF'/%,columnwidthGQ%autoF '/%&widthGF]p/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowli nesGQ%noneF'/%,columnlinesGFhp/%&frameGFhp/%-framespacingGQ,0.4em~0.5e xF'/%*equalrowsGFjn/%-equalcolumnsGFjn/%-displaystyleGFjn/%%sideGQ&rig htF'/%0minlabelspacingGFepF;F>/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 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6.Q*&uminus0;F'F;F>/%&fenceGQ&falseF'/%*separatorGFjn/%)stretchyGFjn/% *symmetricGFjn/%(largeopGFjn/%.movablelimitsGFjn/%'accentGFjn/%'lspace GQ,0.2222222emF'/%'rspaceGFio-FT6)-F,6%-I%msubGF$6%-I#miGF$6&Q\"pF'/%' italicGQ%trueF'F;/F?Q'italicF'-F,6#-F86%FJF;F>/%/subscriptshiftGF:-Fen 6-Q1⁢F'F>FhnF[oF]oF_oFaoFcoFeo/FhoQ&0.0emF'/F[pFfq-F#6% -F,6%-Fdp6&Q)λF'/FhpFjnF;F>-Fen6.Q(−F'F;F>FhnF[oF]oF_oFao FcoFeoFgoFjo-Fap6%-Fdp6&Q\"qF'FgpF;Fjp-F,6#-F86%Q\"3F'F;F>F`qF;F>-F,6% -F#6%-F,6%-Fap6%FerF\\qF`qF`r-Fap6%Fer-F,6#-F86%Q\"2F'F;F>F`qF;F>Fbq-F #6%-F,6%FcsF`rFcrF;F>/%.linethicknessGFJ/%+denomalignGQ'centerF'/%)num alignGFdt/%)bevelledGFjnF;F`r-FT6)-F,6%F`pFbq-F#6%-F,6%F\\rF`rFesF;F>F ]sF`tFbtFetFgtF;F`r-FT6)-F,6%-Fap6%FcpFgsF`qFbqFhq-F,6%-F#6%-F,6%FesF` rFcsF;F>Fbq-F#6%-F,6%FesF`rFcrF;F>F`tFbtFetFgtF;F`r-FT6)-F,6%FeuFbq-F# 6%-F,6%F\\rF`rFcsF;F>FguF`tFbtFetFgtF;F`r-FT6)-F,6%-Fap6%FcpFhrF`qFbqF ]u-F,6%-F#6%-F,6%FcrF`rFcsF;F>Fbq-F#6%-F,6%FcrF`rFesF;F>F`tFbtFetFgtF; F`r-FT6)-F,6%F]wFbqFevF_wF`tFbtFetFgtF;-F,6'FevFbqF]uFbqFhqF`tFbtFetFg tF;F`r-FT6)-F,6#-F,6(FZ-FT6)-F,6'F`pFbqF]uFbqFhqF]sF`tFbtFetFgtF;F`r-F T6)-F,6'FeuFbqFevFbqFhqFguF`tFbtFetFgtF;F`r-FT6)-F,6'F]wFbqFevFbqF]uF_ wF`tFbtFetFgtF;-F,6'-I%msupGF$6%FevFis/%1superscriptshiftGF:FbqF]uFbqF hqF`tFbtFetFgtF;F`r-FT6)Fax-F,6'FevFbq-Fdy6%F]uFisFfyFbqFhqF`tFbtFetFg tF;F`r-FT6)Fax-F,6'FevFbqF]uFbq-Fdy6%FhqFisFfyF`tFbtFetFgtF;FAFDFFFHFK -F56(-F,6(FZ-FT6)F^qFayF`tFbtFetFgtF;F`r-FT6)F^qFjyF`tFbtFetFgtF;F`r-F T6)F^qF`zF`tFbtFetFgtF;FAFDFFFHFKFAFDFF/%&alignGQ%axisF'/FBQ)baselineF '/FEFdt/FGQ'|frleft|hrF'/%/alignmentscopeGFip/%,columnwidthGQ%autoF'/% &widthGFj[l/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowline sGQ%noneF'/%,columnlinesGFe\\l/%&frameGFe\\l/%-framespacingGQ,0.4em~0. 5exF'/%*equalrowsGFjn/%-equalcolumnsGFjn/%-displaystyleGFjn/%%sideGQ&r ightF'/%0minlabelspacingGFb\\lF;F>/%%openGQ\"[F'/%&closeGQ\"]F'" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 212 150 "Defining the Lax matrix in the o per gauge using the general formula. Then defining the normalized geom etric Lax matrix using the gauge transformation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "L:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 11 "L[1,1] :=0:\n" }{MPLTEXT 1 0 11 "L[1,2]:=1:\n" }{MPLTEXT 1 0 21 "L[2,1]:=-tdP 2(lambda)" }{MPLTEXT 1 0 12 "+H2*lambda^2" }{MPLTEXT 1 0 47 "+H1*lambd a+H0-h*p1/(lambda-q1)-h*p2/(lambda-q2)" }{MPLTEXT 1 0 17 "-h*p3/(lambd a-q3)" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 8 "L[2,2]:=" }{MPLTEXT 1 0 27 "h/(lambda-q1)+h/(lambda-q2)" }{MPLTEXT 1 0 14 "+h/(lambda-q3)" } {MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 2 "L:" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 78 "CheckL:=simplify(Multiply(Multip ly(J,L),J^(-1))+h*Multiply(dJdlambda,J^(-1))):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 35 "Defining the Hamiltonian evolutions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "dq1dtau1:=factor(1/h*diff(Hamtau1,p 1));\n" }{MPLTEXT 1 0 41 "dp1dtau1:=factor(-1/h*diff(Hamtau1,q1));\n" }{MPLTEXT 1 0 40 "dq2dtau1:=factor(1/h*diff(Hamtau1,p2));\n" }{MPLTEXT 1 0 41 "dp2dtau1:=factor(-1/h*diff(Hamtau1,q2));\n" }{MPLTEXT 1 0 40 "dq3dtau1:=factor(1/h*diff(Hamtau1,p3));\n" }{MPLTEXT 1 0 41 "dp3dtau1 :=factor(-1/h*diff(Hamtau1,q3));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 40 "dq1dtau2:=factor(1/h*diff(Hamtau2,p1));\n" }{MPLTEXT 1 0 41 "dp1 dtau2:=factor(-1/h*diff(Hamtau2,q1));\n" }{MPLTEXT 1 0 40 "dq2dtau2:=f actor(1/h*diff(Hamtau2,p2));\n" }{MPLTEXT 1 0 41 "dp2dtau2:=factor(-1/ h*diff(Hamtau2,q2));\n" }{MPLTEXT 1 0 40 "dq3dtau2:=factor(1/h*diff(Ha mtau2,p3));\n" }{MPLTEXT 1 0 41 "dp3dtau2:=factor(-1/h*diff(Hamtau2,q3 ));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 40 "dq1dtau3:=factor(1/h*dif f(Hamtau3,p1));\n" }{MPLTEXT 1 0 41 "dp1dtau3:=factor(-1/h*diff(Hamtau 3,q1));\n" }{MPLTEXT 1 0 40 "dq2dtau3:=factor(1/h*diff(Hamtau3,p2));\n " }{MPLTEXT 1 0 41 "dp2dtau3:=factor(-1/h*diff(Hamtau3,q2));\n" } {MPLTEXT 1 0 40 "dq3dtau3:=factor(1/h*diff(Hamtau3,p3));\n" }{MPLTEXT 1 0 41 "dp3dtau3:=factor(-1/h*diff(Hamtau3,q3));\n" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 22 "Atau1:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 83 "Atau1 [1,1]:=-p1*mu1tau1/(lambda-q1)-p2*mu2tau1/(lambda-q2)-p3*mu3tau1/(lamb da-q3):\n" }{MPLTEXT 1 0 74 "Atau1[1,2]:= mu1tau1/(lambda-q1)+mu2tau1/ (lambda-q2)+mu3tau1/(lambda-q3):\n" }{MPLTEXT 1 0 57 "Atau1[2,1]:=h*di ff(Atau1[1,1],lambda)+Atau1[1,2]*L[2,1]:\n" }{MPLTEXT 1 0 68 "Atau1[2, 2]:=h*diff(Atau1[1,2],lambda)+Atau1[1,1]+Atau1[1,2]*L[2,2]:\n" } {MPLTEXT 1 0 7 "Atau1:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "Atau2 :=Matrix(2,2,0):\n" }{MPLTEXT 1 0 83 "Atau2[1,1]:=-p1*mu1tau2/(lambda- q1)-p2*mu2tau2/(lambda-q2)-p3*mu3tau2/(lambda-q3):\n" }{MPLTEXT 1 0 73 "Atau2[1,2]:=mu1tau2/(lambda-q1)+mu2tau2/(lambda-q2)+mu3tau2/(lambd a-q3):\n" }{MPLTEXT 1 0 57 "Atau2[2,1]:=h*diff(Atau2[1,1],lambda)+Atau 2[1,2]*L[2,1]:\n" }{MPLTEXT 1 0 68 "Atau2[2,2]:=h*diff(Atau2[1,2],lamb da)+Atau2[1,1]+Atau2[1,2]*L[2,2]:\n" }{MPLTEXT 1 0 7 "Atau2:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "Atau3:=Matrix(2,2,0):\n" } {MPLTEXT 1 0 83 "Atau3[1,1]:=-p1*mu1tau3/(lambda-q1)-p2*mu2tau3/(lambd a-q2)-p3*mu3tau3/(lambda-q3):\n" }{MPLTEXT 1 0 73 "Atau3[1,2]:=mu1tau3 /(lambda-q1)+mu2tau3/(lambda-q2)+mu3tau3/(lambda-q3):\n" }{MPLTEXT 1 0 57 "Atau3[2,1]:=h*diff(Atau3[1,1],lambda)+Atau3[1,2]*L[2,1]:\n" } {MPLTEXT 1 0 68 "Atau3[2,2]:=h*diff(Atau3[1,2],lambda)+Atau3[1,1]+Atau 3[1,2]*L[2,2]:\n" }{MPLTEXT 1 0 7 "Atau3:\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 24 "dJdtau1:=Matrix(2,2,0):\n" } {MPLTEXT 1 0 42 "for i from 1 to 2 do for j from 1 to 2 do\n" } {MPLTEXT 1 0 191 "dJdtau1[i,j]:=diff(J[i,j],tau1)+diff(J[i,j],q1)*dq1d tau1+diff(J[i,j],p1)*dp1dtau1+diff(J[i,j],q2)*dq2dtau1+diff(J[i,j],p2) *dp2dtau1+diff(J[i,j],q3)*dq3dtau1+diff(J[i,j],p3)*dp3dtau1: od: od:\n " }{MPLTEXT 1 0 9 "dJdtau1:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 24 " dJdtau2:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 42 "for i from 1 to 2 do for \+ j from 1 to 2 do\n" }{MPLTEXT 1 0 191 "dJdtau2[i,j]:=diff(J[i,j],tau2) +diff(J[i,j],q1)*dq1dtau2+diff(J[i,j],p1)*dp1dtau2+diff(J[i,j],q2)*dq2 dtau2+diff(J[i,j],p2)*dp2dtau2+diff(J[i,j],q3)*dq3dtau2+diff(J[i,j],p3 )*dp3dtau2: od: od:\n" }{MPLTEXT 1 0 9 "dJdtau2:\n" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 24 "dJdtau3:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 42 "for i from 1 to 2 do for j from 1 to 2 do\n" }{MPLTEXT 1 0 191 "dJdtau3[i ,j]:=diff(J[i,j],tau3)+diff(J[i,j],q1)*dq1dtau3+diff(J[i,j],p1)*dp1dta u3+diff(J[i,j],q2)*dq2dtau3+diff(J[i,j],p2)*dp2dtau3+diff(J[i,j],q3)*d q3dtau3+diff(J[i,j],p3)*dp3dtau3: od: od:\n" }{MPLTEXT 1 0 9 "dJdtau3: \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 85 "CheckAtau1:=simplify(Multip ly(Multiply(J,Atau1),J^(-1))+h*Multiply(dJdtau1,J^(-1)));\n" }{MPLTEXT 1 0 85 "CheckAtau2:=simplify(Multiply(Multiply(J,Atau2),J^(-1))+h*Mul tiply(dJdtau2,J^(-1)));\n" }{MPLTEXT 1 0 84 "CheckAtau3:=simplify(Mult iply(Multiply(J,Atau3),J^(-1))+h*Multiply(dJdtau3,J^(-1)));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)dq1dtau1G6\",$**,,*(&I\"p 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F,F0**FduF,F`uF,F)F,F.F0F4*(FftF0F.F0FHF,F4**FftF0F.F,F1F,FHF,F0*(FftF 0F1F0FHF,F4*(F[uF0F)F0FHF,F4**F[uF0F)F,F.F,FHF,F0**F[uF0F.F,F1F,FHF,FA *(F[uF0F1F0FHF,F,*(F`uF0F)F0FHF,F,**F`uF0F)F,F.F,FHF,FA*(F`uF0F.F0FHF, F,F,FduF4,&F)F,F.F4FA,&F)F,F1F4F4,&F.F,F1F4FA#F4FC" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)dq3dtau1G6\",$**,,*(&I\"pGF$6#\"\"$\"\"\"&I\"qGF$6#F- F-&F/6#\"\"#F-F3*&I\"hGF$F-F.F-F-*&F5F-F1F-F-*&F5F-&F/F+F-!\"\"*&F)F-& I$tauGF$F0F-!\"#F-,&F1F-F8F9F9,&F.F-F8F9F9F5F9#F-\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)dp3dtau1G6\",$*,,d]l*&&I\"qGF$6#\"\"\"\"\")&F*6# \"\"#\"\"$!\"\"*(F)F-F.F0&F*6#F1F,F0*(F)F-F.F,F4F0F2*&F)F1F.F-F,*(F)F1 F.F0F4\"\"'!\"(*(F)F1F.F,F4\"\"(F9*(F)F0F.F-F4F,!\"#*(F)F0F.F1F4F9F<*( F)F0F.F,F4F-!\"&*(F)F,F.F-F4F0F,*(F)F,F.F1F4F*(F)F**F)F9F.F0F4F,FHF,\"\"%**F)F9F.F,F4F0FHF,F>*(F)F1F.F9FHF,F0**F)F 1F.F0F4FNFHF,!#5**F)F1F.F,F4FFFHF,F-*(F)F0F.F**F)FFF.F0F4F,F_oF,FN*(F)FFF.F0FHF0F0**F)FFF.F,F4F0F_ oF,F>**F)FFF.F,F4F,FHF0FV*(F)FFF4F0FHF0F0*(F)FNF.F1FHF0F2**F)FNF.F0F4F ,FHF0F0**F)FNF.F,F4F0FHF0F2*(F)F1F.FFF_oF,F0*(F)F1F.FNFHF0F,**F)F1F.F0 F4F1F_oF,Fhn**F)F1F.F0F4F0FHF0!\"$**F)F1F.F,F4FNF_oF,F9**F)F1F.F,F4F1F HF0F0**F)F0F.FFF4F,F_oF,FV*(F)F0F.FFFHF0F>**F)F0F.FNF4F,FHF0F>**F)F0F. 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FNFeoF,F0**F)F0F.FSF3F,FeoF,Fhn*(F)F0F.FSFPF0F,**F)F0F.F5F3F,FPF0FB**F )F0F.F,F3FSFeoF,F=**F)F0F.F,F3F5FPF0F0*(F)F0F3FNFeoF,Fhn*(F)F0F3FSFPF0 F1**F)F,F.FNF3F,FeoF,Fhn**F)F,F.FSF3F0FeoF,F0**F)F,F.FSF3F,FPF0FB**F)F ,F.F5F3F5FeoF,F-**F)F,F.F5F3F0FPF0FS**F)F,F.F0F3FSFeoF,FJ**F)F,F.F0F3F 5FPF0FB*(F.FNF3F0FeoF,F0*(F.FSF3F0FPF0F,*(F.F5F3FSFeoF,FJ*(F.F5F3F5FPF 0FB*(F.F0F3FNFeoF,FS*(F.F0F3FSFPF0F,*(F)FSF.F0&FQF4F,FB**F)FSF.F,F3F,F jqF,FS*(F)FSF3F0FjqF,FB**F)F5F.F0F3F,FjqF,FS**F)F5F.F,F3F0FjqF,F`o*(F) F5F3F5FjqF,FS*(F)F0F.FSFjqF,F0**F)F0F.F5F3F,FjqF,Fhn**F)F0F.F,F3F5FjqF ,FS*(F)F0F3FSFjqF,FB**F)F,F.FSF3F,FjqF,Fhn**F)F,F.F5F3F0FjqF,F-**F)F,F .F0F3F5FjqF,Fhn*(F.FSF3F0FjqF,F0*(F.F5F3F5FjqF,Fhn*(F.F0F3FSFjqF,F0*(& I\"pGF$F+F0F)F,F.F0F,**F[sF0F)F,F.F,F3F,FB*(F[sF0F)F,F3F0F,*&F[sF0F.F5 F,*(F[sF0F.F0F3F,FB*(F[sF0F.F,F3F0F,*&&F\\sF/F0F)F5F1*(FcsF0F)F0F.F,F1 *(FcsF0F)F0F3F,F0**FcsF0F)F,F.F,F3F,F0*(FcsF0F)F,F3F0F1*(FcsF0F.F,F3F0 F1*&&F\\sF4F0F)F5F,*(FjsF0F)F0F.F,F,*(FjsF0F)F0F3F,FB*(FjsF0F)F,F.F0F1 *&FjsF0F.F5F1*(FjsF0F.F0F3F,F0*(I\"hGF$F,F[sF,F.F0F,**FatF,F[sF,F.F,F3 F,FB*(FatF,F[sF,F3F0F,*(FatF,FcsF,F)F0F1**FatF,FcsF,F)F,F3F,F0*(FatF,F csF,F3F0F1*(FatF,FjsF,F)F0F,**FatF,FjsF,F)F,F3F,FB*(FatF,FjsF,F.F0F1** FatF,FjsF,F.F,F3F,F0F,FatF1,&F)F,F.F1F1,&F)F,F3F1FB,&F.F,F3F1FB#F,F5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)dq1dtau3G6\",$**&I\"pGF$6#\"\"\"F*, &&I\"qGF$F)F*&F-6#\"\"$!\"\"F1,&F,F*&F-6#\"\"#F1F1I\"hGF$F1F5" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I)dp1dtau3G6\"*,,hs*&&I\"qGF$6#\"\"\"\" \")&F)6#\"\"#F+\"\"&*&F(F,&F)6#\"\"$F+!\"&*&F(\"\"(F-F/!\"'*&F(F7F2F/ \"\"'*(F(F:F-F/F2F+F7*(F(F:F-F+F2F/!\"(*&F(F/F-F7F+*&F(F/F2F7!\"\"*(F( F+F-F7F2F+!\"#*(F(F+F-F+F2F7F/*&F-F7F2F/F+*&F-F/F2F7F@*(F(F:F-F+&I$tau GF$F*F+F:*(F(F:F2F+FGF+F8*(F(F0F-F/FGF+!\")*(F(F0F2F/FGF+F,**F(\"\"%F- F/F2F+FGF+\"#5**F(FNF-F+F2F/FGF+!#5*(F(F/F-F0FGF+F/*(F(F/F2F0FGF+FB**F (F+F-F0F2F+FGF+!\"%**F(F+F-F+F2F0FGF+FN*(F-F0F2F/FGF+F/*(F-F/F2F0FGF+F B*(F(F0F-F+&FHF.F+FN*(F(F0F2F+FZF+FU*(F(FNF-F/FZF+F8*(F(FNF-F+FGF/F+*( F(FNF2F/FZF+F:*(F(FNF2F+FGF/F@**F(F4F-F/F2F+FZF+F,*(F(F4F-F/FGF/FB**F( F4F-F+F2F/FZF+FK*(F(F4F2F/FGF/F/*(F(F/F-FNFZF+F/*(F(F/F-F4FGF/F+**F(F/ F-F/F2F+FGF/F4**F(F/F-F+F2F/FGF/!\"$*(F(F/F2FNFZF+FB*(F(F/F2F4FGF/F@** F(F+F-FNF2F+FZF+FU**F(F+F-F4F2F+FGF/FB**F(F+F-F+F2FNFZF+FN**F(F+F-F+F2 F4FGF/F/*(F-FNF2F/FZF+F/*(F-F4F2F/FGF/F+*(F-F/F2FNFZF+FB*(F-F/F2F4FGF/ F@*(F(FNF-F+&FHF3F+F/*(F(FNF2F+F^pF+FB*(F(F4F-F/F^pF+FU*(F(F4F2F/F^pF+ FN*(F(F/F-F4F^pF+F/**F(F/F-F/F2F+F^pF+F:**F(F/F-F+F2F/F^pF+F8*(F(F/F2F 4F^pF+FB**F(F+F-F4F2F+F^pF+FU**F(F+F-F+F2F4F^pF+FN*(F-F4F2F/F^pF+F/*(F -F/F2F4F^pF+FB*(&I\"pGF$F*F/F(F+F-F+F/*(F[qF/F(F+F2F+FB*&F[qF/F-F/F@*& F[qF/F2F/F+*&&F\\qF.F/F(F/F@*(FaqF/F(F+F2F+F/*&FaqF/F2F/F@*&&F\\qF3F/F (F/F+*(FeqF/F(F+F-F+FB*&FeqF/F-F/F+F+I\"hGF$F@,&F(F+F-F@FB,&F(F+F2F@FB ,&F-F+F2F@F@" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)dq2dtau3G6\",$**&I\"p GF$6#\"\"#\"\"\",&&I\"qGF$F)F+&F.6#\"\"$!\"\"F2,&&F.6#F+F+F-F2F2I\"hGF $F2!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)dp2dtau3G6\",$*,,hs*&&I\"q GF$6#\"\"\"\"\"(&F*6#\"\"#F0!\"\"*(F)F-F.F,&F*6#\"\"$F,F0*&F)F-F3F0F1* &F)F0F.F-\"\"'*(F)F0F.F8F3F,!\"(*&F)F0F3F-F,*&F)F,F.\"\")!\"&*(F)F,F.F 8F3F0F-*(F)F,F.F,F3F-!\"#*&F.F=F3F,\"\"&*&F.F-F3F0!\"'*&F.F0F3F-F,*(F) FCF.F0&I$tauGF$F+F,FA**F)FCF.F,F3F,FHF,\"\"%*(F)FCF3F0FHF,FA*(F)F0F.FC FHF,F=**F)F0F.FKF3F,FHF,!#5*(F)F0F3FCFHF,F0*(F)F,F.F8FHF,FE**F)F,F.FKF 3F0FHF,\"#5**F)F,F.F,F3FCFHF,!\"%*(F.F8F3F,FHF,F8*(F.FCF3F0FHF,!\")*(F .F0F3FCFHF,F0*(F)FKF.F0&FIF/F,FA**F)FKF.F,F3F,FenF,FK*(F)FKF3F0FenF,FA *(F)F5F.F0FHF0F1**F)F5F.F,F3F,FHF0F0*(F)F5F3F0FHF0F1*(F)F0F.FKFenF,F8* *F)F0F.F5F3F,FenF,FX*(F)F0F.F5FHF0F0**F)F0F.F0F3F,FHF0!\"$*(F)F0F3FKFe nF,F0*(F)F0F3F5FHF0F,*(F)F,F.FCFenF,FU*(F)F,F.FKFHF0F1**F)F,F.F5F3F0Fe nF,F=**F)F,F.F0F3F0FHF0F5**F)F,F.F,F3FKFenF,FU**F)F,F.F,F3F5FHF0FA*(F. FCF3F,FenF,FK*(F.FKF3F0FenF,FE*(F.FKF3F,FHF0F,*(F.F5F3F0FHF0FA*(F.F0F3 FKFenF,F0*(F.F0F3F5FHF0F,*(F)F5F.F0&FIF4F,FA**F)F5F.F,F3F,F_pF,FK*(F)F 5F3F0F_pF,FA*(F)F0F.F5F_pF,FK**F)F0F.F0F3F,F_pF,FE*(F)F0F3F5F_pF,F0*(F )F,F.FKF_pF,FA**F)F,F.F0F3F0F_pF,F8**F)F,F.F,F3F5F_pF,FU*(F.FKF3F,F_pF ,F0*(F.F5F3F0F_pF,FU*(F.F0F3F5F_pF,F0*&&I\"pGF$F+F0F.F0F,*(F\\qF0F.F,F 3F,FA*&F\\qF0F3F0F,*&&F]qF/F0F)F0F,*(FaqF0F)F,F.F,FA*(FaqF0F.F,F3F,F0* &FaqF0F3F0F1*&&F]qF4F0F)F0F1*(FfqF0F)F,F.F,F0*&FfqF0F.F0F1F,I\"hGF$F1, &F)F,F.F1FA,&F)F,F3F1F1,&F.F,F3F1FAF1" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)dq3dtau3G6\",$**&I\"pGF$6#\"\"$\"\"\",&&I\"qGF$6#\"\"#F+&F.F)!\"\" F2,&&F.6#F+F+F1F2F2I\"hGF$F2F0" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)dp3 dtau3G6\",$*,,hs*&&I\"qGF$6#\"\"\"\"\"(&F*6#\"\"#F0!\"\"*(F)F-F.F,&F*6 #\"\"$F,F0*&F)F-F3F0F1*&F)F0F.F-F,*(F)F0F.F,F3\"\"'!\"(*&F)F0F3F-F9*(F )F,F.F-F3F,!\"#*(F)F,F.F0F3F9F-*&F)F,F3\"\")!\"&*&F.F-F3F0F,*&F.F0F3F- !\"'*&F.F,F3F@\"\"&*(F)FFF.F0&I$tauGF$F+F,F=**F)FFF.F,F3F,FHF,\"\"%*(F )FFF3F0FHF,F=*(F)F0F.FFFHF,F0**F)F0F.F,F3FKFHF,!#5*(F)F0F3FFFHF,F@**F) F,F.FFF3F,FHF,!\"%**F)F,F.F0F3FKFHF,\"#5*(F)F,F3F9FHF,FD*(F.FFF3F0FHF, F0*(F.F0F3FFFHF,!\")*(F.F,F3F9FHF,F9*(F)FKF.F0&FIF/F,F=**F)FKF.F,F3F,F enF,FK*(F)FKF3F0FenF,F=*(F)F5F.F0FHF0F1**F)F5F.F,F3F,FHF0F0*(F)F5F3F0F HF0F1*(F)F0F.FKFenF,F0*(F)F0F.F5FHF0F,**F)F0F.F,F3F5FenF,FX**F)F0F.F,F 3F0FHF0!\"$*(F)F0F3FKFenF,F9*(F)F0F3F5FHF0F0**F)F,F.FKF3F,FenF,FR**F)F ,F.F5F3F,FHF0F=**F)F,F.F0F3F5FenF,F@**F)F,F.F0F3F0FHF0F5*(F)F,F3FFFenF ,FR*(F)F,F3FKFHF0F1*(F.FKF3F0FenF,F0*(F.F5F3F0FHF0F,*(F.F0F3FKFenF,FD* (F.F0F3F5FHF0F=*(F.F,F3FFFenF,FK*(F.F,F3FKFHF0F,*(F)F5F.F0&FIF4F,F=**F )F5F.F,F3F,F_pF,FK*(F)F5F3F0F_pF,F=*(F)F0F.F5F_pF,F0**F)F0F.F,F3F0F_pF ,FD*(F)F0F3F5F_pF,FK**F)F,F.F5F3F,F_pF,FR**F)F,F.F0F3F0F_pF,F9*(F)F,F3 FKF_pF,F=*(F.F5F3F0F_pF,F0*(F.F0F3F5F_pF,FR*(F.F,F3FKF_pF,F0*&&I\"pGF$ F+F0F.F0F,*(F\\qF0F.F,F3F,F=*&F\\qF0F3F0F,*&&F]qF/F0F)F0F1*(FaqF0F)F,F 3F,F0*&FaqF0F3F0F1*&&F]qF4F0F)F0F,*(FeqF0F)F,F3F,F=*&FeqF0F.F0F1*(FeqF 0F.F,F3F,F0F,I\"hGF$F1,&F)F,F.F1F1,&F)F,F3F1F=,&F.F,F3F1F=F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI( _syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-F,6%-I&mf racGF$6)-I#mnGF$6%Q\"1F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'norm alF'-F=6%Q\"5F'F@FC/%.linethicknessGF?/%+denomalignGQ'centerF'/%)numal ignGFM/%)bevelledGQ&falseF'F@-I#moGF$6-Q1⁢F'FC/%&fenceG FR/%*separatorGFR/%)stretchyGFR/%*symmetricGFR/%(largeopGFR/%.movablel imitsGFR/%'accentGFR/%'lspaceGQ&0.0emF'/%'rspaceGFao-F:6)-F,6#-F,6+-F, 6%-F#6%-F,6%-I%msubGF$6%-I#miGF$6&Q\"pF'/%'italicGQ%trueF'F@/FDQ'itali cF'-F,6#-F=6%Q\"2F'F@FC/%/subscriptshiftGQ\"0F'-FT6.Q(−F'F@FCFWF YFenFgnFinF[oF]o/F`oQ,0.2222222emF'/FcoFhq-Fap6%Fcp-F,6#-F=6%Q\"3F'F@F CFaqF@FCFS-I(msubsupGF$6'-Fdp6&Q\"qF'FgpF@Fjp-F,6#FFAFDFFFHFKFA FDFF-F26'-F56(-F,6)-F,6%-F86%Q\"2F'F;F>-I#moGF$6-Q1⁢F'F >/%&fenceGQ&falseF'/%*separatorGF\\o/%)stretchyGF\\o/%*symmetricGF\\o/ %(largeopGF\\o/%.movablelimitsGF\\o/%'accentGF\\o/%'lspaceGQ&0.0emF'/% 'rspaceGF[p-I%msubGF$6%-I#miGF$6&Q\"qF'/%'italicGQ%trueF'F;/F?Q'italic F'-F,6#FY/%/subscriptshiftGF:-Fgn6.Q\"+F'F;F>FjnF]oF_oFaoFcoFeoFgo/Fjo Q,0.2222222emF'/F]pFbq-Fbp6&Q)λF'/FfpF\\oF;F>F^q-F,6%FYFfn-F_p6 %Fap-F,6#FOF\\qF^q-F,6%FYFfn-F_p6%Fap-F,6#-F86%Q\"3F'F;F>F\\qFAFDFFFHF KF4FAFDFF/%&alignGQ%axisF'/FBQ)baselineF'/FEQ'centerF'/FGQ'|frleft|hrF '/%/alignmentscopeGFgp/%,columnwidthGQ%autoF'/%&widthGFds/%+rowspacing GQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlines GF_t/%&frameGF_t/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGF\\o/%-equ alcolumnsGF\\o/%-displaystyleGF\\o/%%sideGQ&rightF'/%0minlabelspacingG F\\tF;F>/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 54 "Check of the theoretical formulas for the Lax matrix L" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "tdL11theo:=0:\n" }{MPLTEXT 1 0 132 "for j from 0 to g-1 do aux:=0: for i from j+1 to g do aux:=au x+P[i]*Q[i-j-1]: od: tdL11theo:=tdL11theo-(-1)^(j-1)*aux*lambda^j: od: \n" }{MPLTEXT 1 0 32 "tdL11theo:=simplify(tdL11theo);\n" }{MPLTEXT 1 0 32 "simplify(tdL11theo-CheckL[1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*tdL11theoG6\"**,,*&,(*&,&&I\"pGF$6#\"\"$\"\"\"&F,6#\"\"#!\"\"F/&I \"qGF$6#F/F/F/*&,&&F,F6F/F+F3F/&F5F1F/F/*&&F5F-F/,&F9F/F0F3F/F3F/I'lam bdaGF$F2F/*&,(*&,&F0F/F+F3F/F4F2F/*&,&F+F/F9F3F/F:F2F/*&F F/F/*&,&*&F0F/F " 0 "" {MPLTEXT 1 0 14 "tdL12theo:=0:\n" }{MPLTEXT 1 0 74 "for m from 0 to g \+ do tdL12theo:=tdL12theo+(-1)^(g-m)*Q[g-m]*lambda^m: od:\n" }{MPLTEXT 1 0 32 "tdL12theo:=simplify(tdL12theo);\n" }{MPLTEXT 1 0 32 "simplify( tdL12theo-CheckL[1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*tdL12theoG 6\"*(,&I'lambdaGF$\"\"\"&I\"qGF$6#F(!\"\"F(,&F'F(&F*6#\"\"#F,F(,&F'F(& F*6#\"\"$F,F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Term1:=0:\n" }{MPLTEXT 1 0 107 "for i fro m 0 to rinfty-2 do for j from g+i to 2*rinfty-5 do Term1:=Term1- P2[j] *h[j-g-i]*lambda^i: od: od:\n" }{MPLTEXT 1 0 7 "Term1:\n" }{MPLTEXT 1 0 10 "Term2:=0:\n" }{MPLTEXT 1 0 212 "for i from 0 to g-2 do for j1 fr om i+1 to g-1 do for j2 from g+i-j1 to g-1 do for i1 from j1+1 to g do for i2 from j2+1 to g do Term2:=Term2- (-1)^(j1+j2)*P[i1]*Q[i1-j1-1]* P[i2]*Q[i2-j2-1]*h[j1+j2-g-i]*lambda^i:\n" }{MPLTEXT 1 0 20 "od: od: o d: od: od:\n" }{MPLTEXT 1 0 7 "Term2:\n" }{MPLTEXT 1 0 34 "tdL21theo:= simplify(Term1+Term2):\n" }{MPLTEXT 1 0 32 "simplify(tdL21theo-CheckL[ 2,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 43 "Inverting the symmetric Darboux coordinates" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "factor(P[1] -( q[1]^2*(q[2] -q[3])*p[1]- q[2]^2*(q[1]-q[3])*p[2]+q[3]^2*(q[1]-q[2])*p[3])/(q[1]-q[ 2])/(q[1]-q[3])/(q[2]-q[3]));\n" }{MPLTEXT 1 0 15 "factor(P[2]-(-(" } {MPLTEXT 1 0 66 "q[1]*(q[2]-q[3])*p[1]- q[2]*(q[1]-q[3])*p[2]+q[3]*(q[ 1]-q[2])*p[3]" }{MPLTEXT 1 0 2 ")/" }{MPLTEXT 1 0 35 "(q[1]-q[2])/(q[1 ]-q[3])/(q[2]-q[3])" }{MPLTEXT 1 0 4 " ));" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 106 "factor(P[3]-(((q[2]-q[3])*p[1]- (q[1]-q[3])*p[2]+(q[ 1]-q[2])*p[3])/(q[1]-q[2])/(q[1]-q[3])/(q[2]-q[3]) ));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 359 "solve(\{PP1 -( q[1]^2*( q[2]-q[3])*p[1]- q[2]^2*(q[1]-q[3])*p[2]+q[3]^2*(q[1]-q[2])*p[3])/(q[1 ]-q[2])/(q[1]-q[3])/(q[2]-q[3])=0, PP2-(-(q[1]*(q[2]-q[3])*p[1]- q[2]* (q[1]-q[3])*p[2]+q[3]*(q[1]-q[2])*p[3])/(q[1]-q[2])/(q[1]-q[3])/(q[2]- q[3]) ), PP3-(((q[2]-q[3])*p[1]- (q[1]-q[3])*p[2]+(q[1]-q[2])*p[3])/(q [1]-q[2])/(q[1]-q[3])/(q[2]-q[3]) )\},\{p[1],p[2],p[3]\});\n\n" } {MPLTEXT 1 0 48 "QQfunction:=unapply(QQ(lambda),p[1],p[2],p[3]):\n" } {MPLTEXT 1 0 176 "simplify(series(es(simplify(QQfunction(PP3*q[2]*q[3] +PP2*q[2]+PP2*q[3]+PP1,PP3*q[1]*q[3]+PP2*q[1]+PP2*q[3]+PP1,PP3*q[1]*q[ 2]+PP2*q[1]+PP2*q[2]+PP1)),q[1],q[2],q[3]),lambda=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "< %/&I\"pG6\"6#\"\"\",**(I$PP3GF&F(&I\"qGF&6#\"\"#F(&F-6#\"\"$F(F(*&I$PP 2GF&F(F,F(F(*&F4F(F0F(F(I$PP1GF&F(/&F%F.,**(F+F(&F-F'F(F0F(F(*&F4F(F;F (F(F5F(F6F(/&F%F1,**(F+F(F;F(F,F(F(F " 0 "" {MPLTEXT 1 0 20 "H0function:=unapply(" }{MPLTEXT 1 0 19 "H0,p[1],p [2],p[3]):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 40 "H1function:=unapply (H1,p[1],p[2],p[3]):\n" }{MPLTEXT 1 0 40 "H2function:=unapply(H2,p[1], p[2],p[3]):\n" }{MPLTEXT 1 0 172 "simplify(series(es(simplify(H0functi on(PP3*q[2]*q[3]+PP2*q[2]+PP2*q[3]+PP1,PP3*q[1]*q[3]+PP2*q[1]+PP2*q[3] +PP1,PP3*q[1]*q[2]+PP2*q[1]+PP2*q[2]+PP1)),q[1],q[2],q[3]),h=0));\n" } {MPLTEXT 1 0 172 "simplify(series(es(simplify(H1function(PP3*q[2]*q[3] +PP2*q[2]+PP2*q[3]+PP1,PP3*q[1]*q[3]+PP2*q[1]+PP2*q[3]+PP1,PP3*q[1]*q[ 2]+PP2*q[1]+PP2*q[2]+PP1)),q[1],q[2],q[3]),h=0));\n" }{MPLTEXT 1 0 171 "simplify(series(es(simplify(H2function(PP3*q[2]*q[3]+PP2*q[2]+PP2 *q[3]+PP1,PP3*q[1]*q[3]+PP2*q[1]+PP2*q[3]+PP1,PP3*q[1]*q[2]+PP2*q[1]+P P2*q[2]+PP1)),q[1],q[2],q[3]),h=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 " +'I\"hG6\",D*&&I&sigmaGF$6#\"\"\"\"\"%&F(6#\"\"$F*!\"\"*&I$PP2GF$\"\"# F'F2F***F1F*I$PP3GF$F*F'F*&F(6#F2F*F2*(F4F2F'F*F,F*F/*&F4F2F5F2F**(F'F 2F5F*F,F*F.*(F'F2F,F*&I$tauGF$F)F*!\"#*(I$PP1GF$F*F1F*F'F*F2*(F?F*F4F* F5F*F2*(F1F*F4F*F,F*F=*&F'F*F,F2F=*(F'F*F,F*&F*(F'F*F/F*F4F*F2*&F'F*F4F1F,*&I$PP 1GF$F*F9F*F1*$F8F1F**&F'F*&F5F=F*F6*&F/F*F " 0 "" {MPLTEXT 1 0 27 "CheckL11function:= unapply(" }{MPLTEXT 1 0 28 "CheckL [1,1],p[1],p[2],p[3]):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 27 "CheckL1 2function:= unapply(" }{MPLTEXT 1 0 29 "CheckL[1,2],p[1],p[2],p[3]):\n " }{MPLTEXT 1 0 55 "CheckL21function:= unapply(CheckL[2,1],p[1],p[2],p [3]):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 183 "simplify(series(es(simp lify(CheckL11function(PP3*q[2]*q[3]+PP2*q[2]+PP2*q[3]+PP1,PP3*q[1]*q[3 ]+PP2*q[1]+PP2*q[3]+PP1,PP3*q[1]*q[2]+PP2*q[1]+PP2*q[2]+PP1)),q[1],q[2 ],q[3]),lambda=0));\n" }{MPLTEXT 1 0 183 "simplify(series(es(simplify( CheckL12function(PP3*q[2]*q[3]+PP2*q[2]+PP2*q[3]+PP1,PP3*q[1]*q[3]+PP2 *q[1]+PP2*q[3]+PP1,PP3*q[1]*q[2]+PP2*q[1]+PP2*q[2]+PP1)),q[1],q[2],q[3 ]),lambda=0));\n" }{MPLTEXT 1 0 182 "simplify(series(es(simplify(Check L21function(PP3*q[2]*q[3]+PP2*q[2]+PP2*q[3]+PP1,PP3*q[1]*q[3]+PP2*q[1] +PP2*q[3]+PP1,PP3*q[1]*q[2]+PP2*q[1]+PP2*q[2]+PP1)),q[1],q[2],q[3]),la mbda=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "+)I'lambdaG6\",(*&I$PP2GF$ \"\"\"&I&sigmaGF$6#F(F(F(*&I$PP3GF$F(&F*6#\"\"#F(F(I$PP1GF$F(\"\"!,&*& F-F(F)F(!\"\"F'F5F(F-F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "++I'lambdaG6 \",$&I&sigmaGF$6#\"\"$!\"\"\"\"!&F'6#\"\"#\"\"\",$&F'6#F/F*F.F/F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "+-I'lambdaG6\",2*$&I&sigmaGF$6#\"\"\"\" \"%F**&,&&I$tauGF$F)\"\"#&F(6#F0!\"$F*F'F0F**&,(*$I$PP3GF$F0F*&F(6#\" \"$F0&F/F2F0F*F'F*F**&I$PP2GF$F*F7F*F0*$F.F0F**&F1F*F.F*!\"#*$F1F0F*&F /F9F0\"\"!,,*$F'F:F**&,&F.F0F1F@F*F'F*F*F6!\"\"F;F0F8F*F*,(*$F'F0F*F1F HF.F0F0F'F:F*F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 69 "Expressing th e Hamiltonians in terms of symmetric Darboux coordinates" }{TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Hamtau1function:=una pply(Hamtau1,p[1],p[2],p[3]):\n" }{MPLTEXT 1 0 7 "Hamtau2" }{MPLTEXT 1 0 18 "function:=unapply(" }{MPLTEXT 1 0 7 "Hamtau2" }{MPLTEXT 1 0 18 ",p[1],p[2],p[3]):\n" }{MPLTEXT 1 0 7 "Hamtau3" }{MPLTEXT 1 0 18 "f unction:=unapply(" }{MPLTEXT 1 0 7 "Hamtau3" }{MPLTEXT 1 0 18 ",p[1],p [2],p[3]):\n" }{MPLTEXT 1 0 10 "Hamtau1R:=" }{MPLTEXT 1 0 28 "simplify (series(es(simplify(" }{MPLTEXT 1 0 7 "Hamtau1" }{MPLTEXT 1 0 142 "fun ction(PP3*q[2]*q[3]+PP2*q[2]+PP2*q[3]+PP1,PP3*q[1]*q[3]+PP2*q[1]+PP2*q [3]+PP1,PP3*q[1]*q[2]+PP2*q[1]+PP2*q[2]+PP1)),q[1],q[2],q[3]),h=0));\n " }{MPLTEXT 1 0 10 "Hamtau2R:=" }{MPLTEXT 1 0 28 "simplify(series(es(s implify(" }{MPLTEXT 1 0 7 "Hamtau2" }{MPLTEXT 1 0 142 "function(PP3*q[ 2]*q[3]+PP2*q[2]+PP2*q[3]+PP1,PP3*q[1]*q[3]+PP2*q[1]+PP2*q[3]+PP1,PP3* q[1]*q[2]+PP2*q[1]+PP2*q[2]+PP1)),q[1],q[2],q[3]),h=0));\n" }{MPLTEXT 1 0 10 "Hamtau3R:=" }{MPLTEXT 1 0 28 "simplify(series(es(simplify(" } {MPLTEXT 1 0 7 "Hamtau3" }{MPLTEXT 1 0 141 "function(PP3*q[2]*q[3]+PP2 *q[2]+PP2*q[3]+PP1,PP3*q[1]*q[3]+PP2*q[1]+PP2*q[3]+PP1,PP3*q[1]*q[2]+P P2*q[1]+PP2*q[2]+PP1)),q[1],q[2],q[3]),h=0));" }{MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I)Hamtau1RG6\"+'I\"hGF$,fn*(I$PP3GF$\" \"#&I&sigmaGF$6#\"\"\"F.&F,6#\"\"$F.#!\"\"\"\"&*(F+F*&F,6#F*F.F/F.#F1F 4*(F+F*F/F.&I$tauGF$F-F.#F.F4*(I$PP2GF$F.F)F.F/F.#!\"#F4*(F+F.F/F.&F;F 7F.F?*(I$PP1GF$F.F)F.F6F.#F*F4*(FDF.F>F.F+F.FE*&F>F*F+F*F<*&F+\"\"%F/F .F2*&F+F.F/F*F?*&F6F*F/F.F2*&F/F.F:F*F<*&F/F.&F;F0F.F?*&F)F*F6F*F<**F> F.F)F.F+F.F:F.F?**F>F.F)F.F+F.F6F.FE*$FDF*F<*(FDF.F)F.F:F.F?*&F>F*F:F. F2*(F+F1F6F.F:F.#!\"%F4*(F)F*F6F.F:F.F2*(F+F.F6F*F:F.F8*(F+F.F6F.F:F*F V*(F6F.F:F.FBF.F?*(F+F*F:F.FBF.FE*(F+F.F:F.FNF.FE*&F+F4F:F.F<*&F+F1F:F *FE*&F+F.F:F1F<\"\"!,&F>FE*&F)F.F+F.FI)Hamtau2RG6\"+'I\"hGF$,H*&&I&sigmaGF$6#\"\"\"\"\"%&F*6#\"\"#F,#F ,\"\"$*(I$PP2GF$F,I$PP3GF$F,F)F0#!\"#F2*(F5F0F)F,F.F,#!\"\"F2*&F)F2&F* 6#F2F,F9*&F)F0F.F0F:*(F)F0F.F,&I$tauGF$F+F,#F0F2*(I$PP1GF$F,F5F,F)F,F6 *&F4F0F)F,F6*&F5F0FI)Ham tau3RG6\"+%I\"hGF$,B*$&I&sigmaGF$6#\"\"\"\"\"&!\"\"*&F)\"\"$&F*6#\"\"# F,\"\"%*&F)F0&I$tauGF$F+F,!\"#*(I$PP2GF$F,I$PP3GF$F,F)F,F3*&F;F3F1F,F, *&F)F3&F*6#F0F,!\"$*&F)F3&F7F2F,F8*&F)F,F1F3F@*(F)F,F1F,F6F,F4*&F)F,F6 F3F.*&I$PP1GF$F,F;F,F3*$F:F3F,*&F)F,&F7F?F,F8*&F1F,F>F,F3*&F1F,FBF,F3* &F>F,F6F,F8\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 828 "Hamtau 1bis:=((1/5)*PP3*sigma[1]+2*PP2*(1/5))*h-(1/5)*PP3^2*sigma[2]*tau[1]-( 1/5)*PP3^2*sigma[1]*sigma[3]+(1/5)*sigma[1]^2*sigma[3]*tau[1]+(1/5)*si gma[1]^5*tau[1]+(1/5)*PP2^2*sigma[1]^2+(1/5)*sigma[1]*tau[1]^3-(1/5)*s igma[1]^4*sigma[3]-(1/5)*sigma[2]^2*sigma[3]+(1/5)*sigma[3]*tau[1]^2+( 1/5)*PP3^2*sigma[2]^2-(2/5*PP1)*PP3*tau[1]-(2/5*sigma[3])*tau[3]-(2/5* sigma[1])*sigma[3]^2+(2/5*(sigma[1]^3))*tau[1]^2+(1/5)*PP1^2-(2/5*sigm a[1])*sigma[3]*tau[2]-(2/5*PP2)*PP3*sigma[3]+(2/5*(sigma[1]^2))*tau[1] *tau[2]+(2/5*sigma[1])*tau[1]*tau[3]+(2/5*PP1)*PP2*sigma[1]+(3/5*sigma [1])*sigma[2]^2*tau[1]-(4/5*sigma[1])*sigma[2]*tau[1]^2+(3/5*(sigma[1] ^2))*sigma[2]*sigma[3]+(2/5*PP1)*PP3*sigma[2]-(2/5*sigma[2])*tau[1]*ta u[2]-(1/5)*PP2^2*tau[1]-(4/5*(sigma[1]^3))*sigma[2]*tau[1]+(2/5*PP2)*P P3*sigma[1]*sigma[2]-(2/5*PP2)*PP3*sigma[1]*tau[1]:\n" }{MPLTEXT 1 0 31 "simplify(Hamtau1R-Hamtau1bis);\n" }{MPLTEXT 1 0 263 "Hamtau1ter:=1 /5*((sigma[2]^2-sigma[1]*sigma[3])*PP3^2+2*(sigma[1]+sigma[2])*PP1*PP3 +sigma[1]^2*PP2^2+2*sigma[1]*PP1*PP2+PP1^2+(2*sigma[1]*sigma[2]-2*sig ma[3])*PP2*PP3 -2*sigma[1]*PP1*PP3-sigma[3]*(sigma[1]^4-3*sigma[1]^2*s igma[2]+2*sigma[1]*sigma[3]+sigma[2]^2)\n" }{MPLTEXT 1 0 79 "+ 2*sigma [1]^2*tau[1]*tau[2]-2*sigma[2]*tau[1]*tau[2]+ 2*sigma[1]*tau[1]*tau[3] " }{MPLTEXT 1 0 18 "+sigma[1]*tau[1]^3" }{MPLTEXT 1 0 54 "+(sigma[3]+2 *sigma[1]^3-4*sigma[1]*sigma[2])*tau[1]^2\n" }{MPLTEXT 1 0 209 "-(2*PP 1*PP3+2*sigma[1]*PP2*PP3+PP2^2+sigma[2]*PP3^2-sigma[1]^5-sigma[1]^2*si gma[3]-3*sigma[1]*sigma[2]^2+4*sigma[1]^3*sigma[2])*tau[1]-2*sigma[1]* sigma[3]*tau[2]- 2*sigma[3]*tau[3]+ (PP3*sigma[1]+2*PP2)*h ):\n" } {MPLTEXT 1 0 55 "factor(simplify(series(Hamtau1bis-Hamtau1ter,PP3=0))) ;\n" }{MPLTEXT 1 0 11 "Hamtau1ter;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",D*&,&*&&I&sigmaG6\"6#\"\"\"F*&F'6#\"\"$F*!\"\"*$&F'6#\" \"#F2F*F*I$PP3GF(F2#F*\"\"&*(,&F&F*F0F*F*I$PP1GF(F*F3F*#F2F5*&I$PP2GF( F2F&F2F4*(F8F*F;F*F&F*F9*$F8F2F4*(,&*&F&F*F0F*F2F+!\"#F*F;F*F3F*F4*(F8 F*F3F*F&F*#FAF5*&F+F*,**$F&\"\"%F**&F&F2F0F*!\"$F%F2F/F*F*#F.F5*(F&F2& I$tauGF(F)F*&FMF1F*F9*(F0F*FLF*FNF*FC*(F&F*FLF*&FMF,F*F9*&F&F*FLF-F4*& ,(*$F&F-F2F@!\"%F+F*F*FLF2F4*&,2*$F&F5F.*&F&F-F0F*FG*(F;F*F3F*F&F*F2*& F3F2F0F*F**&F&F2F+F*F.*&F&F*F0F2FI*&F8F*F3F*F2*$F;F2F*F*FLF*FJ*(F&F*F+ F*FNF*FC*&F+F*FQF*FC*&,&*&F3F*F&F*F*F;F2F*I\"hGF(F*F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 493 "Hamtau2bis:=(1/3)*sigma[1]^4*sigma [2]-(2/3*PP2)*PP3*sigma[1]^2-(1/3)*PP3^2*sigma[1]*sigma[2]-(1/3)*sigma [1]^3*sigma[3]-sigma[1]^2*sigma[2]^2+(2/3*(sigma[1]^2))*sigma[2]*tau[1 ]-(2/3*PP1)*PP3*sigma[1]-(2/3*(PP2^2))*sigma[1]+(1/3)*PP3^2*sigma[3]+( 4/3*sigma[1])*sigma[2]*sigma[3]+(2/3*sigma[1])*sigma[2]*tau[2]-(2/3*si gma[1])*sigma[3]*tau[1]+(1/3)*sigma[2]^3-(2/3*(sigma[2]^2))*tau[1]+(1/ 3)*sigma[2]*tau[1]^2-(2/3*PP1)*PP2+(2/3*sigma[2])*tau[3]-(1/3)*sigma[3 ]^2-(2/3*sigma[3])*tau[2]-h*(1/3)*PP3:\n" }{MPLTEXT 1 0 31 "simplify(H amtau2R-Hamtau2bis);\n" }{MPLTEXT 1 0 382 "Hamtau2ter:=1/3*(-2*sigma[1 ]^2*PP2*PP3-2*sigma[1]*PP1*PP3-2*PP1*PP2+(sigma[3]-sigma[1]*sigma[2])* PP3^2-2*sigma[1]*PP2^2+4*sigma[1]*sigma[2]*sigma[3]+sigma[1]^4*sigma[2 ]-3*sigma[1]^2*sigma[2]^2-sigma[1]^3*sigma[3]+sigma[2]^3-sigma[3]^2+ \+ sigma[2]*tau[1]^2+2*(sigma[1]^2*sigma[2]-sigma[1]*sigma[3]-sigma[2]^2) *tau[1]+ 2*(sigma[1]*sigma[2]-sigma[3])*tau[2]+2*sigma[2]*tau[3]-PP3 *h):\n" }{MPLTEXT 1 0 57 "factor(simplify(series(Hamtau2bis-Hamtau2ter ,tau[1]=0)));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 11 "Hamtau2ter;" } {MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",B*(I$PP2G 6\"\"\"\"I$PP3GF%F&&I&sigmaGF%6#F&\"\"##!\"#\"\"$*(I$PP1GF%F&F'F&F(F&F ,*&F0F&F$F&F,*&,&*&F(F&&F)6#F+F&!\"\"&F)6#F.F&F&F'F+#F&F.*&F$F+F(F&F,* (F(F&F5F&F8F&#\"\"%F.*&F(F>F5F&F:*&F(F+F5F+F7*&F(F.F8F&#F7F.*$F5F.F:*$ F8F+FB*&F5F&&I$tauGF%F*F+F:*&,(*&F(F+F5F&F&*&F(F&F8F&F7*$F5F+F7F&FFF&# F+F.*&,&F4F&F8F7F&&FGF6F&FM*&F5F&&FGF9F&FM*&F'F&I\"hGF%F&FB" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Hamtau3bis:=" }{MPLTEXT 1 0 288 "-sigma[1]^5+4*sigma[1]^3*sigma[2]-2*sigma[1]^3*tau[1]+2*PP2*PP3*s igma[1]+PP3^2*sigma[2]-3*sigma[1]^2*sigma[3]-2*sigma[1]^2*tau[2]-3*sig ma[1]*sigma[2]^2+4*sigma[1]*sigma[2]*tau[1]-sigma[1]*tau[1]^2+2*PP1*PP 3+PP2^2-2*sigma[1]*tau[3]+2*sigma[2]*sigma[3]+2*sigma[2]*tau[2]-2*sigm a[3]*tau[1]:\n" }{MPLTEXT 1 0 31 "simplify(Hamtau3R-Hamtau3bis);\n" } {MPLTEXT 1 0 279 "Hamtau3ter:=2*PP1*PP3+PP2^2+sigma[2]*PP3^2+2*sigma[1 ]*PP2*PP3-sigma[1]^5+4*sigma[1]^3*sigma[2]-3*sigma[1]^2*sigma[3]-3*sig ma[1]*sigma[2]^2+2*sigma[2]*sigma[3]-sigma[1]*tau[1]^2+2*(2*sigma[1]*s igma[2]-sigma[1]^3-sigma[3])*tau[1]+ 2*(sigma[2]-sigma[1]^2)*tau[2]-2* sigma[1]*tau[3]:\n" }{MPLTEXT 1 0 58 "factor(simplify(series(Hamtau3bi s-Hamtau3ter,tau[1]=0)));\n" }{MPLTEXT 1 0 11 "Hamtau3ter;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ",<*&I$PP1G6\"\"\"\"I$PP3GF%F&\"\"#*$I$PP 2GF%F(F&*&F'F(&I&sigmaGF%6#F(F&F&*(F*F&F'F&&F-6#F&F&F(*$F0\"\"&!\"\"*& F0\"\"$F,F&\"\"%*&F0F(&F-6#F6F&!\"$*&F0F&F,F(F;*&F,F&F9F&F(*&F0F&&I$ta uGF%F1F(F4*&,(*$F0F6F4*&F0F&F,F&F(F9F4F&F?F&F(*&,&*$F0F(F4F,F&F&&F@F.F &F(*&F0F&&F@F:F&!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 53 "Verifica tion of the formulas for the matrix A^\{tau1\}." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "nu[1]:=nu1tau1;\n" }{MPLTEXT 1 0 16 "nu[2]:=nu 2tau1;\n" }{MPLTEXT 1 0 15 "nu[3]:=nu3tau1;" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "nu[rinfty-2]:=0:\n" }{MPLTEXT 1 0 79 "for k from 1 to g do nu[rinfty-2]:=nu[rinfty-2]+(-1)^(g-k)*nu[k] *Q[g+1-k]: od:\n" }{MPLTEXT 1 0 13 "nu[rinfty-2];" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#nuG6\"6#\"\"\"#F'\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#nuG6\"6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#nuG6\" 6#\"\"$,$&I$tauGF%6#\"\"\"#!\"\"\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*(&I\"qG6\"6#\"\"\"F(&F%6#\"\"#F(&F%6#\"\"$F(#F(\"\"&*&&I$tauGF&F'F (,(F$F(F)F(F,F(F(#!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "tdA11theo:=0:\n" }{MPLTEXT 1 0 150 "for i from 0 to g-2 do for m f rom 1 to g-1-i do for r from i+m+1 to g do tdA11theo:=tdA11theo-(-1)^( i+m-1)*nu[m]*P[r]*Q[r-i-m-1]*lambda^i od: od: od:\n" }{MPLTEXT 1 0 32 "tdA11theo:=simplify(tdA11theo);\n" }{MPLTEXT 1 0 44 "factor(simplify( tdA11theo-CheckAtau1[1,1]));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*tdA11theoG6\",$**,,*&,&&I\"pGF$6#\"\"#\"\"\"&F+6#\"\"$ !\"\"F.&I\"qGF$6#F.F-F.*(I'lambdaGF$F.F)F.F3F.F2*&,&F/F.&F+F5F2F.&F4F, F-F.*(F7F.,&F:F.F/F2F.F;F.F.*(&F4F0F.,&F:F.F*F2F.,&F7F.F?F2F.F2F.,&F3F .F;F2F2,&F3F.F?F2F2,&F;F.F?F2F2#F.\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "tdA12theo:=0: \n" }{MPLTEXT 1 0 113 "for j from 0 to g-1 do for m from 1 to g-j do t dA12theo:=tdA12theo+(-1)^(g-j-m)*nu[m]*Q[g-j-m]*lambda^j: od: od:\n" } {MPLTEXT 1 0 32 "tdA12theo:=simplify(tdA12theo);\n" }{MPLTEXT 1 0 44 " factor(simplify(tdA12theo-CheckAtau1[1,2]));" }{MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I*tdA12theoG6\",,*$I'lambdaGF$\"\"##\" \"\"\"\"&*&,(&I\"qGF$6#F*!\"\"&F/6#F(F1&F/6#\"\"$F1F*F'F*F)*&,&F2F*F4F *F*F.F*F)*&F4F*F2F*F)&I$tauGF$F0#F1F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "tdA21theoTerm1:=0 :\n" }{MPLTEXT 1 0 111 "for i from 0 to g do lowerpoint:=max(g,g+i-1): for j from lowerpoint to 2*rinfty-5 do for m from 1 to j-g-i do\n" } {MPLTEXT 1 0 66 " tdA21theoTerm1:=tdA21theoTerm1-nu[m]*h[j-g-m-i]*P2[j ]*lambda^i: \n" }{MPLTEXT 1 0 12 "od: od: od:\n" }{MPLTEXT 1 0 16 "tdA 21theoTerm1;\n" }{MPLTEXT 1 0 19 "tdA21theoTerm2:=0:\n" }{MPLTEXT 1 0 149 "for i from 0 to g do for j1 from 0 to g-1 do for j2 from 1 to g-1 do for m from 1 to j1+j2-g-i do for r1 from j1+1 to g do for r2 from \+ j2+1 to g do \n" }{MPLTEXT 1 0 16 "tdA21theoTerm2:=" }{MPLTEXT 1 0 14 "tdA21theoTerm2" }{MPLTEXT 1 0 79 "-(-1)^(j1+j2)*nu[m]*h[j1+j2-g-i-m]* P[r1]*P[r2]*Q[r1-j1-1]*Q[r2-j2-1]*lambda^i:\n" }{MPLTEXT 1 0 25 "od: o d: od: od: od: od: \n" }{MPLTEXT 1 0 16 "tdA21theoTerm2:\n" }{MPLTEXT 1 0 52 "tdA21theo:=simplify(tdA21theoTerm1+tdA21theoTerm2):\n" } {MPLTEXT 1 0 44 "factor(simplify(tdA21theo-CheckAtau1[2,1]));" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",2&I$tauG6\"6#\"\"## F'\"\"&*$,(&I\"qGF%6#\"\"\"F/&F-F&F/&F-6#\"\"$F/F3#F/F)*&F+F/,(*&F,F/F 0F/F/*&F,F/F1F/F/*&F0F/F1F/F/F/#!\"#F)*(F,F/F0F/F1F/F(*&&F$F.F/I'lambd aGF%F/F4*&,*F7!\"\"*&F1F/F,F/FB*&F1F/F0F/FB*$F+F'F/F/F?F/F4*&F+F/F?F'F 4*$F?F3F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 49 "Verification of the formulas for the Hamiltonians" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Term0Hamtheo:=0:\n" } {MPLTEXT 1 0 115 "for i from 1 to g do for k from i+1 to g do Term0Ham theo:=Term0Hamtheo-h*nu[i]*(-1)^i*(g-i)*P[k]*Q[k-1-i]: od: od:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 16 "Term1Hamtheo:=0:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 145 "for i from 1 to g do for k from i+1 to g do \+ for m from i+1 to k-1 do Term1Hamtheo:=Term1Hamtheo-h*nu[i]*(-1)^m*P[k ]*Q[k-1-m]*S[m-i]: od: od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "Term2Hamtheo:=0:\n" }{MPLTEXT 1 0 109 "for i from 1 to g do for k1 from 1 to g do for k2 from 1 to g do for r1 from max(0,i-k2) to min(k 1-1,i-1) do\n" }{MPLTEXT 1 0 95 "Term2Hamtheo:=Term2Hamtheo+nu[i]*P[k1 ]*P[k2]*(-1)^(i-1)*Q[k1-1-r1]*Q[k2-i+r1]: od: od: od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 298 "for i from 1 to g do for k1 from 1 to g do for k2 from 1 to g do for r1 from 0 to k1-1 do for r2 from 0 to k2 -1 do for m from i to g do if r1+r2>=g then Term2Hamtheo:=Term2Hamthe o+nu[i]*P[k1]*P[k2]*(-1)^(r1+r2)*Q[k1-1-r1]*Q[k2-1-r2]*(-1)^(g-m)*Q[g- m]*h[r1+r2+m-i-g+1]: fi: od: od: od: od: od: od:\n" }{MPLTEXT 1 0 2 " \+ \n" }{MPLTEXT 1 0 17 "Term3Hamtheo:=0:\n" }{MPLTEXT 1 0 155 "for i fro m 1 to g do for r from g to 2*rinfty-5 do for m from i to g do Term3Ha mtheo:=Term3Hamtheo+nu[i]*(-1)^(g-m)*P2[r]*Q[g-m]*h[r+m-i-g+1]: od: od : od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "Hamilton:=simplify(" } {MPLTEXT 1 0 13 "Term0Hamtheo+" }{MPLTEXT 1 0 41 "Term1Hamtheo+Term2Ha mtheo+Term3Hamtheo):\n" }{MPLTEXT 1 0 18 "simplify(Hamilton-" } {MPLTEXT 1 0 9 "Hamtau1);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Term1 Hamtheo:=0:\n" }{MPLTEXT 1 0 115 "for i from 1 to g do for k from i+1 \+ to g do Term1Hamtheo:=Term1Hamtheo-h*nu[i]*(-1)^i*(g-i)*P[k]*Q[k-1-i]: od: od:\n" }{MPLTEXT 1 0 145 "for i from 1 to g do for k from i+1 to \+ g do for m from i+1 to k-1 do Term1Hamtheo:=Term1Hamtheo-h*nu[i]*(-1)^ m*P[k]*Q[k-1-m]*S[m-i]: od: od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "Term2Hamtheo:=0:\n" }{MPLTEXT 1 0 109 "for i from 1 to g do fo r k1 from 1 to g do for k2 from 1 to g do for r1 from max(0,i-k2) to m in(k1-1,i-1) do\n" }{MPLTEXT 1 0 95 "Term2Hamtheo:=Term2Hamtheo+nu[i]* P[k1]*P[k2]*(-1)^(i-1)*Q[k1-1-r1]*Q[k2-i+r1]: od: od: od: od:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 298 "for i from 1 to g do for k1 fro m 1 to g do for k2 from 1 to g do for r1 from 0 to k1-1 do for r2 from 0 to k2-1 do for m from i to g do if r1+r2>=g then Term2Hamtheo:=Ter m2Hamtheo+nu[i]*P[k1]*P[k2]*(-1)^(r1+r2)*Q[k1-1-r1]*Q[k2-1-r2]*(-1)^(g -m)*Q[g-m]*h[r1+r2+m-i-g+1]: fi: od: od: od: od: od: od:\n" }{MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 17 "Term3Hamtheo:=0:\n" }{MPLTEXT 1 0 155 "f or i from 1 to g do for r from g to 2*rinfty-5 do for m from i to g do Term3Hamtheo:=Term3Hamtheo+nu[i]*(-1)^(g-m)*P2[r]*Q[g-m]*h[r+m-i-g+1] : od: od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 60 "Hamilton:=simp lify(Term1Hamtheo+Term2Hamtheo+Term3Hamtheo):\n" }{MPLTEXT 1 0 27 "sim plify(Hamilton-Hamtau1);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 66 "Expression of \\td\{A\} in terms of the symmetric Darboux coordinates" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "CheckA11tau1function:= unapp ly(" }{MPLTEXT 1 0 15 "CheckAtau1[1,1]" }{MPLTEXT 1 0 18 ",p[1],p[2],p [3]):\n" }{MPLTEXT 1 0 31 "CheckA12tau1function:= unapply(" }{MPLTEXT 1 0 10 "CheckAtau1" }{MPLTEXT 1 0 23 "[1,2],p[1],p[2],p[3]):\n" } {MPLTEXT 1 0 31 "CheckA21tau1function:= unapply(" }{MPLTEXT 1 0 10 "Ch eckAtau1" }{MPLTEXT 1 0 23 "[2,1],p[1],p[2],p[3]):\n" }{MPLTEXT 1 0 187 "simplify(series(es(simplify(CheckA11tau1function(PP3*q[2]*q[3]+PP 2*q[2]+PP2*q[3]+PP1,PP3*q[1]*q[3]+PP2*q[1]+PP2*q[3]+PP1,PP3*q[1]*q[2]+ PP2*q[1]+PP2*q[2]+PP1)),q[1],q[2],q[3]),lambda=0));\n" }{MPLTEXT 1 0 28 "simplify(series(es(simplify(" }{MPLTEXT 1 0 12 "CheckA12tau1" } {MPLTEXT 1 0 147 "function(PP3*q[2]*q[3]+PP2*q[2]+PP2*q[3]+PP1,PP3*q[1 ]*q[3]+PP2*q[1]+PP2*q[3]+PP1,PP3*q[1]*q[2]+PP2*q[1]+PP2*q[2]+PP1)),q[1 ],q[2],q[3]),lambda=0));\n" }{MPLTEXT 1 0 28 "simplify(series(es(simpl ify(" }{MPLTEXT 1 0 12 "CheckA21tau1" }{MPLTEXT 1 0 146 "function(PP3* q[2]*q[3]+PP2*q[2]+PP2*q[3]+PP1,PP3*q[1]*q[3]+PP2*q[1]+PP2*q[3]+PP1,PP 3*q[1]*q[2]+PP2*q[1]+PP2*q[2]+PP1)),q[1],q[2],q[3]),lambda=0));" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "+'I'lambdaG6\",&*&I$ PP3GF$\"\"\"&I&sigmaGF$6#F(F(#!\"\"\"\"&I$PP2GF$F,\"\"!,$F'#F(F.F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "+)I'lambdaG6\",&&I&sigmaGF$6#\"\"##\"\" \"\"\"&&I$tauGF$6#F+#!\"\"F,\"\"!,$&F'F/F0F+F*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "++I'lambdaG6\",,*$&I&sigmaGF$6#\"\"\"\"\"$#F*\"\"&*$I$PP3 GF$\"\"##!\"\"F-*&F'F*&F(6#F0F*#!\"#F-&F(6#F+#F0F-&I$tauGF$F5F:\"\"!,( *$F'F0F,F4F1&F " 0 "" {MPLTEXT 1 0 16 "nu[1]:=nu1tau2;\n" } {MPLTEXT 1 0 16 "nu[2]:=nu2tau2;\n" }{MPLTEXT 1 0 16 "nu[3]:=nu3tau2; \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "nu[rinfty-2]:=0:\n" } {MPLTEXT 1 0 79 "for k from 1 to g do nu[rinfty-2]:=nu[rinfty-2]+(-1)^ (g-k)*nu[k]*Q[g+1-k]: od:\n" }{MPLTEXT 1 0 13 "nu[rinfty-2];" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#nuG6\"6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#nuG6\"6#\"\"##\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#nuG6\"6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ", (*&&I\"qG6\"6#\"\"\"F(&F%6#\"\"#F(#!\"\"\"\"$*&&F%6#F.F(F$F(F,*&F0F(F) F(F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "tdA11theo:=0:\n" } {MPLTEXT 1 0 150 "for i from 0 to g-2 do for m from 1 to g-1-i do for \+ r from i+m+1 to g do tdA11theo:=tdA11theo-(-1)^(i+m-1)*nu[m]*P[r]*Q[r- i-m-1]*lambda^i od: od: od:\n" }{MPLTEXT 1 0 32 "tdA11theo:=simplify(t dA11theo);\n" }{MPLTEXT 1 0 44 "factor(simplify(tdA11theo-CheckAtau2[1 ,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*tdA11theoG6\",$**,(*&,&&I\" pGF$6#\"\"$\"\"\"&F+6#\"\"#!\"\"F.&I\"qGF$6#F.F.F.*&,&&F+F5F.F*F2F.&F4 F0F.F.*&&F4F,F.,&F8F.F/F2F.F2F.,&F9F.F;F2F2,&F3F.F;F2F2,&F3F.F9F2F2#F. F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "tdA12theo:=0:\n" }{MPLTEXT 1 0 113 "for j from 0 to \+ g-1 do for m from 1 to g-j do tdA12theo:=tdA12theo+(-1)^(g-j-m)*nu[m]* Q[g-j-m]*lambda^j: od: od:\n" }{MPLTEXT 1 0 32 "tdA12theo:=simplify(td A12theo);\n" }{MPLTEXT 1 0 44 "factor(simplify(tdA12theo-CheckAtau2[1, 2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*tdA12theoG6\",*&I\"qGF$6#\" \"##!\"\"\"\"$I'lambdaGF$#\"\"\"F,&F'6#F/F*&F'6#F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "td A21theoTerm1:=0:\n" }{MPLTEXT 1 0 111 "for i from 0 to g do lowerpoint :=max(g,g+i-1): for j from lowerpoint to 2*rinfty-5 do for m from 1 to j-g-i do\n" }{MPLTEXT 1 0 66 " tdA21theoTerm1:=tdA21theoTerm1-nu[m]*h [j-g-m-i]*P2[j]*lambda^i: \n" }{MPLTEXT 1 0 12 "od: od: od:\n" } {MPLTEXT 1 0 16 "tdA21theoTerm1;\n" }{MPLTEXT 1 0 19 "tdA21theoTerm2:= 0:\n" }{MPLTEXT 1 0 149 "for i from 0 to g do for j1 from 0 to g-1 do \+ for j2 from 1 to g-1 do for m from 1 to j1+j2-g-i do for r1 from j1+1 \+ to g do for r2 from j2+1 to g do \n" }{MPLTEXT 1 0 109 "tdA21theoTerm2 :=tdA21theoTerm2-(-1)^(j1+j2)*nu[m]*h[j1+j2-g-i-m]*P[r1]*P[r2]*Q[r1-j1 -1]*Q[r2-j2-1]*lambda^i:\n" }{MPLTEXT 1 0 25 "od: od: od: od: od: od: \+ \n" }{MPLTEXT 1 0 16 "tdA21theoTerm2:\n" }{MPLTEXT 1 0 52 "tdA21theo:= simplify(tdA21theoTerm1+tdA21theoTerm2):\n" }{MPLTEXT 1 0 44 "factor(s implify(tdA21theo-CheckAtau2[2,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 " ,0&I$tauG6\"6#\"\"\"#\"\"#\"\"$*&&I\"qGF%F&F'&F-6#F)F'#!\"#F**&&F-6#F* F'F,F'F0*&F3F'F.F'F0*$,(F,F'F.F'F3F'F)#F'F**&F7F'I'lambdaGF%F'F8*$F:F) F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Term1Hamtheo:=0:\n" }{MPLTEXT 1 0 115 "for i from 1 \+ to g do for k from i+1 to g do Term1Hamtheo:=Term1Hamtheo-h*nu[i]*(-1) ^i*(g-i)*P[k]*Q[k-1-i]: od: od:\n" }{MPLTEXT 1 0 145 "for i from 1 to \+ g do for k from i+1 to g do for m from i+1 to k-1 do Term1Hamtheo:=Ter m1Hamtheo-h*nu[i]*(-1)^m*P[k]*Q[k-1-m]*S[m-i]: od: od: od:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "Term2Hamtheo:=0:\n" }{MPLTEXT 1 0 109 "for i from 1 to g do for k1 from 1 to g do for k2 from 1 to g d o for r1 from max(0,i-k2) to min(k1-1,i-1) do\n" }{MPLTEXT 1 0 95 "Ter m2Hamtheo:=Term2Hamtheo+nu[i]*P[k1]*P[k2]*(-1)^(i-1)*Q[k1-1-r1]*Q[k2-i +r1]: od: od: od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 298 "for i from 1 to g do for k1 from 1 to g do for k2 from 1 to g do for r1 fro m 0 to k1-1 do for r2 from 0 to k2-1 do for m from i to g do if r1+r2> =g then Term2Hamtheo:=Term2Hamtheo+nu[i]*P[k1]*P[k2]*(-1)^(r1+r2)*Q[k 1-1-r1]*Q[k2-1-r2]*(-1)^(g-m)*Q[g-m]*h[r1+r2+m-i-g+1]: fi: od: od: od: od: od: od:\n" }{MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 17 "Term3Hamtheo:= 0:\n" }{MPLTEXT 1 0 155 "for i from 1 to g do for r from g to 2*rinfty -5 do for m from i to g do Term3Hamtheo:=Term3Hamtheo+nu[i]*(-1)^(g-m) *P2[r]*Q[g-m]*h[r+m-i-g+1]: od: od: od:\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 60 "Hamilton:=simplify(Term1Hamtheo+Term2Hamtheo+Term3Ham theo):\n" }{MPLTEXT 1 0 27 "simplify(Hamilton-Hamtau2);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "CheckA11tau2function:= unapply(CheckAtau2[1,1],p[ 1],p[2],p[3]):\n" }{MPLTEXT 1 0 64 "CheckA12tau2function:= unapply(Che ckAtau2[1,2],p[1],p[2],p[3]):\n" }{MPLTEXT 1 0 64 "CheckA21tau2functio n:= unapply(CheckAtau2[2,1],p[1],p[2],p[3]):\n" }{MPLTEXT 1 0 187 "sim plify(series(es(simplify(CheckA11tau2function(PP3*q[2]*q[3]+PP2*q[2]+P P2*q[3]+PP1,PP3*q[1]*q[3]+PP2*q[1]+PP2*q[3]+PP1,PP3*q[1]*q[2]+PP2*q[1] +PP2*q[2]+PP1)),q[1],q[2],q[3]),lambda=0));\n" }{MPLTEXT 1 0 187 "simp lify(series(es(simplify(CheckA12tau2function(PP3*q[2]*q[3]+PP2*q[2]+PP 2*q[3]+PP1,PP3*q[1]*q[3]+PP2*q[1]+PP2*q[3]+PP1,PP3*q[1]*q[2]+PP2*q[1]+ PP2*q[2]+PP1)),q[1],q[2],q[3]),lambda=0));\n" }{MPLTEXT 1 0 186 "simpl ify(series(es(simplify(CheckA21tau2function(PP3*q[2]*q[3]+PP2*q[2]+PP2 *q[3]+PP1,PP3*q[1]*q[3]+PP2*q[1]+PP2*q[3]+PP1,PP3*q[1]*q[2]+PP2*q[1]+P P2*q[2]+PP1)),q[1],q[2],q[3]),lambda=0));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "+%I'lambdaG6\",$I$PP3GF$#\"\"\"\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "+'I'lambdaG6\",$&I&sigmaGF$6#\"\"\"#!\" \"\"\"$\"\"!#F)F,F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "+)I'lambdaG6\",(* $&I&sigmaGF$6#\"\"\"\"\"##F*\"\"$&F(6#F+#!\"#F-&I$tauGF$F)#F+F-\"\"!,$ F'F,F*F,F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 58 "We have verified t he \\td\{A\}^2 formula. Let us do A^\{tau3\}." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "nu[1]:=nu1tau3;\n" }{MPLTEXT 1 0 16 "nu[2]:=nu2t au3;\n" }{MPLTEXT 1 0 16 "nu[3]:=nu3tau3;\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 17 "nu[rinfty-2]:=0:\n" }{MPLTEXT 1 0 79 "for k from 1 to g do nu[rinfty-2]:=nu[rinfty-2]+(-1)^(g-k)*nu[k]*Q[g+1-k]: od:\n" } {MPLTEXT 1 0 13 "nu[rinfty-2];" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#nu G6\"6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#nuG6\"6#\"\"#\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#nuG6\"6#\"\"$\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(&I\"qG6\"6#\"\"\"F'&F$6#\"\"#F'&F$6#\"\"$F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "tdA11theo:=0:\n" } {MPLTEXT 1 0 150 "for i from 0 to g-2 do for m from 1 to g-1-i do for \+ r from i+m+1 to g do tdA11theo:=tdA11theo-(-1)^(i+m-1)*nu[m]*P[r]*Q[r- i-m-1]*lambda^i od: od: od:\n" }{MPLTEXT 1 0 32 "tdA11theo:=simplify(t dA11theo);\n" }{MPLTEXT 1 0 44 "factor(simplify(tdA11theo-CheckAtau3[1 ,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*tdA11theoG6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "tdA12theo:=0:\n" }{MPLTEXT 1 0 113 "for j from 0 to g -1 do for m from 1 to g-j do tdA12theo:=tdA12theo+(-1)^(g-j-m)*nu[m]*Q [g-j-m]*lambda^j: od: od:\n" }{MPLTEXT 1 0 32 "tdA12theo:=simplify(tdA 12theo);\n" }{MPLTEXT 1 0 44 "factor(simplify(tdA12theo-CheckAtau3[1,2 ]));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*tdA12theoG6\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "tdA21theoTerm1:=0:\n" }{MPLTEXT 1 0 111 "for i from 0 to g do lo werpoint:=max(g,g+i-1): for j from lowerpoint to 2*rinfty-5 do for m f rom 1 to j-g-i do\n" }{MPLTEXT 1 0 66 " tdA21theoTerm1:=tdA21theoTerm1 -nu[m]*h[j-g-m-i]*P2[j]*lambda^i: \n" }{MPLTEXT 1 0 12 "od: od: od:\n" }{MPLTEXT 1 0 16 "tdA21theoTerm1;\n" }{MPLTEXT 1 0 19 "tdA21theoTerm2 :=0:\n" }{MPLTEXT 1 0 149 "for i from 0 to g do for j1 from 0 to g-1 d o for j2 from 1 to g-1 do for m from 1 to j1+j2-g-i do for r1 from j1+ 1 to g do for r2 from j2+1 to g do \n" }{MPLTEXT 1 0 109 "tdA21theoTer m2:=tdA21theoTerm2-(-1)^(j1+j2)*nu[m]*h[j1+j2-g-i-m]*P[r1]*P[r2]*Q[r1- j1-1]*Q[r2-j2-1]*lambda^i:\n" }{MPLTEXT 1 0 25 "od: od: od: od: od: od : \n" }{MPLTEXT 1 0 16 "tdA21theoTerm2:\n" }{MPLTEXT 1 0 52 "tdA21theo :=simplify(tdA21theoTerm1+tdA21theoTerm2):\n" }{MPLTEXT 1 0 44 "factor (simplify(tdA21theo-CheckAtau3[2,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",*&I\"qG6\"6#\"\"#F'I'lambdaGF%\"\"\"&F$6#F)F'&F$6#\"\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Term1Hamtheo:=0:\n" }{MPLTEXT 1 0 115 "for i from 1 to g do for \+ k from i+1 to g do Term1Hamtheo:=Term1Hamtheo-h*nu[i]*(-1)^i*(g-i)*P[k ]*Q[k-1-i]: od: od:\n" }{MPLTEXT 1 0 145 "for i from 1 to g do for k f rom i+1 to g do for m from i+1 to k-1 do Term1Hamtheo:=Term1Hamtheo-h* nu[i]*(-1)^m*P[k]*Q[k-1-m]*S[m-i]: od: od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "Term2Hamtheo:=0:\n" }{MPLTEXT 1 0 109 "for i from 1 to g do for k1 from 1 to g do for k2 from 1 to g do for r1 from max(0 ,i-k2) to min(k1-1,i-1) do\n" }{MPLTEXT 1 0 95 "Term2Hamtheo:=Term2Ham theo+nu[i]*P[k1]*P[k2]*(-1)^(i-1)*Q[k1-1-r1]*Q[k2-i+r1]: od: od: od: o d:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 298 "for i from 1 to g do for k1 from 1 to g do for k2 from 1 to g do for r1 from 0 to k1-1 do for \+ r2 from 0 to k2-1 do for m from i to g do if r1+r2>=g then Term2Hamth eo:=Term2Hamtheo+nu[i]*P[k1]*P[k2]*(-1)^(r1+r2)*Q[k1-1-r1]*Q[k2-1-r2]* (-1)^(g-m)*Q[g-m]*h[r1+r2+m-i-g+1]: fi: od: od: od: od: od: od:\n" } {MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 17 "Term3Hamtheo:=0:\n" }{MPLTEXT 1 0 155 "for i from 1 to g do for r from g to 2*rinfty-5 do for m from i to g do Term3Hamtheo:=Term3Hamtheo+nu[i]*(-1)^(g-m)*P2[r]*Q[g-m]*h[ r+m-i-g+1]: od: od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 60 "Hami lton:=simplify(Term1Hamtheo+Term2Hamtheo+Term3Hamtheo):\n" }{MPLTEXT 1 0 27 "simplify(Hamilton-Hamtau3);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 " CheckA11tau3function:= unapply(CheckAtau3[1,1],p[1],p[2],p[3]):\n" } {MPLTEXT 1 0 64 "CheckA12tau3function:= unapply(CheckAtau3[1,2],p[1],p [2],p[3]):\n" }{MPLTEXT 1 0 64 "CheckA21tau3function:= unapply(CheckAt au3[2,1],p[1],p[2],p[3]):\n" }{MPLTEXT 1 0 187 "simplify(series(es(sim plify(CheckA11tau3function(PP3*q[2]*q[3]+PP2*q[2]+PP2*q[3]+PP1,PP3*q[1 ]*q[3]+PP2*q[1]+PP2*q[3]+PP1,PP3*q[1]*q[2]+PP2*q[1]+PP2*q[2]+PP1)),q[1 ],q[2],q[3]),lambda=0));\n" }{MPLTEXT 1 0 187 "simplify(series(es(simp lify(CheckA12tau3function(PP3*q[2]*q[3]+PP2*q[2]+PP2*q[3]+PP1,PP3*q[1] *q[3]+PP2*q[1]+PP2*q[3]+PP1,PP3*q[1]*q[2]+PP2*q[1]+PP2*q[2]+PP1)),q[1] ,q[2],q[3]),lambda=0));\n" }{MPLTEXT 1 0 186 "simplify(series(es(simpl ify(CheckA21tau3function(PP3*q[2]*q[3]+PP2*q[2]+PP2*q[3]+PP1,PP3*q[1]* q[3]+PP2*q[1]+PP2*q[3]+PP1,PP3*q[1]*q[2]+PP2*q[1]+PP2*q[2]+PP1)),q[1], q[2],q[3]),lambda=0));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "+%I'lambdaG6\"\"\" \"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "+'I'lambdaG6\",$&I&sigmaGF$6# \"\"\"\"\"#\"\"!F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }