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0 0 0 1 }{CSTYLE "Equati on Label" -1 216 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Hel p Italic" -1 42 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Pro mpt" -1 1 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperli nk" -1 17 "Times" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Capti on Text" -1 217 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 218 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 218 140 "In this Maple file, we \+ check that the formulas found for the second element of the Painlev\35 1 2 hierarchy corresponds to the theoretical ones." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 46 "Loading the results after symplectic reduction" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }{MPLTEXT 1 0 21 "with(LinearAlgebra):\n" }{MPLTEXT 1 0 13 "tinfty14:=1;\n" } {MPLTEXT 1 0 14 "tinfty24:=-1;\n" }{MPLTEXT 1 0 13 "tinfty13:=0;\n" } {MPLTEXT 1 0 13 "tinfty23:=0;\n" }{MPLTEXT 1 0 20 "tinfty12:=1/2*tau2; \n" }{MPLTEXT 1 0 21 "tinfty22:=-tinfty12;\n" }{MPLTEXT 1 0 20 "tinfty 11:=1/2*tau1;\n" }{MPLTEXT 1 0 21 "tinfty21:=-tinfty11;\n" }{MPLTEXT 1 0 21 "tinfty20:=-tinfty10;\n" }{MPLTEXT 1 0 15 "P1:=lambda->0:\n" } {MPLTEXT 1 0 94 "tdP2:=unapply(-lambda^6-tau2*lambda^4-tau1*lambda^3-( 2*tinfty10+1/4*tau2^2)*lambda^2,lambda);\n" }{MPLTEXT 1 0 117 "P2:=una pply(-lambda^6-tau2*lambda^4-tau1*lambda^3-(2*tinfty10+1/4*tau2^2)*lam bda^2+Pinfty02+Pinfty12*lambda,lambda);\n" }{MPLTEXT 1 0 57 "dtdP2dlam bda:=unapply(diff(tdP2(lambda),lambda),lambda);\n" }{MPLTEXT 1 0 50 "t dP2onlambda:=unapply(tdP2(lambda)/lambda,lambda):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)tinfty14G6\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I )tinfty24G6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)tinfty13G6\"\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)tinfty23G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)tinfty12G6\",$I%tau2GF$#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)tinfty22G6\",$I%tau2GF$#!\"\"\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I)tinfty11G6\",$I%tau1GF$#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)tinfty21G6\",$I%tau1GF$#!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)tinfty20G6\",$I)tinfty10GF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%tdP2G6\"f*6#I'lambdaGF$F$6$I)operat orGF$I&arrowGF$F$,**$9$\"\"'!\"\"*&I%tau2GF$\"\"\"F-\"\"%F/*&I%tau1GF$ F2F-\"\"$F/*&,&I)tinfty10GF$\"\"#*$F1F:#F2F3F2F-F:F/F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#P2G6\"f*6#I'lambdaGF$F$6$I)operatorGF$I&arrow GF$F$,.*$9$\"\"'!\"\"*&I%tau2GF$\"\"\"F-\"\"%F/*&I%tau1GF$F2F-\"\"$F/* &,&I)tinfty10GF$\"\"#*$F1F:#F2F3F2F-F:F/I)Pinfty02GF$F2*&I)Pinfty12GF$ F2F-F2F2F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-dtdP2dlambdaG6\"f*6 #I'lambdaGF$F$6$I)operatorGF$I&arrowGF$F$,**$9$\"\"&!\"'*&I%tau2GF$\" \"\"F-\"\"$!\"%*&I%tau1GF$F2F-\"\"#!\"$*&,&I)tinfty10GF$F7*$F1F7#F2\" \"%F2F-F2!\"#F$F$F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 64 "Definitio n of the matrices necessary to define the vector (c_\{\\i" }{TEXT 212 76 "nfty,0\}, c_\{\\infty,1\}) and verification that we recover the co rrect formulas" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Q:=Matrix (2,2,0):\n" }{MPLTEXT 1 0 11 "Q[1,1]:=1:\n" }{MPLTEXT 1 0 12 "Q[1,2]:= q1:\n" }{MPLTEXT 1 0 11 "Q[2,1]:=1:\n" }{MPLTEXT 1 0 12 "Q[2,2]:=q2:\n " }{MPLTEXT 1 0 21 "RHS1:=Matrix(2,1,0):\n" }{MPLTEXT 1 0 51 "RHS1[1,1 ]:=p1^2+tdP2(q1)+h*(p2-p1)/(q1-q2)+h*q1^2:\n" }{MPLTEXT 1 0 51 "RHS1[2 ,1]:=p2^2+tdP2(q2)+h*(p1-p2)/(q2-q1)+h*q2^2:\n" }{MPLTEXT 1 0 3 "Q;\n" }{MPLTEXT 1 0 8 "Q^(-1);\n" }{MPLTEXT 1 0 7 "RHS1;\n\n" }{MPLTEXT 1 0 67 "simplify(1/(q1-q2)*h*(p2-p1)/(q1-q2)-1/(q1-q2)*h*(p1-p2)/(q2-q1) );\n" }{MPLTEXT 1 0 71 "simplify(-q2/(q1-q2)*h*(p2-p1)/(q1-q2)+q1/(q1- q2)*h*(p1-p2)/(q2-q1));\n\n" }{MPLTEXT 1 0 42 "CVector:=simplify(Multi ply(Q^(-1),RHS1)):\n" }{MPLTEXT 1 0 23 "Cinfty0:=CVector[1,1]:\n" } {MPLTEXT 1 0 23 "Cinfty1:=CVector[2,1]:\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 116 "Cinfty1bis:=(p1^2-p2^2)/(q1-q2)-( (q1^2+q2^2+1/2*tau 2)^2-q1^2*q2^2 +(2*tinfty10-h))*(q1+q2)-(q1^2+q1*q2+q2^2)*tau1;\n" } {MPLTEXT 1 0 28 "factor(Cinfty1-Cinfty1bis);\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 168 "Cinfty0bis:=-(p1^2-p2^2)*q1/(q1-q2) -h*(p1-p2)/(q1-q 2)+p1^2 +(q1^4+q2^4+q1^3*q2+q2^3*q1+q1^2*q2^2+1/4*tau2^2+ (q1^2+q1*q2+ q2^2)*tau2+(q1+q2)*tau1+ 2*tinfty10-h)*q1*q2:\n" }{MPLTEXT 1 0 165 "Ci nfty0bis:=(p2^2*q1-q2*p1^2)/(q1-q2) -h*(p1-p2)/(q1-q2) +(q1^4+q2^4+q1^ 3*q2+q2^3*q1+q1^2*q2^2+1/4*tau2^2+ (q1^2+q1*q2+q2^2)*tau2+(q1+q2)*tau1 + 2*tinfty10-h)*q1*q2;\n" }{MPLTEXT 1 0 166 "Cinfty0ter:=-(p1^2-p2^2)* q1/(q1-q2) -h*(p1-p2)/(q1-q2)+p1^2 +((q1^5-q2^5)/(q1-q2)+1/4*tau2^2+ ( q1^3-q2^3)/(q1-q2)*tau2+(q1^2-q2^2)/(q1-q2)*tau1+ 2*tinfty10-h)*q1*q2: \n" }{MPLTEXT 1 0 121 "Cinfty0ter:=-(p1^2-p2^2)*q1/(q1-q2) -h*(p1-p2)/ (q1-q2)+p1^2 -(tdP2onlambda(q1)-tdP2onlambda(q2))/(q1-q2)*q1*q2 -h*q1* q2:\n" }{MPLTEXT 1 0 28 "factor(Cinfty0-Cinfty0bis);\n" }{MPLTEXT 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spacingGFesF=F@/%%openGQ\"[F'/%&closeGQ\"]F'" }}{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/-F,6#-I%msupGF$6 %-I#miGF$6&Q#p1F'/%'italicGQ%trueF'/%+foregroundGQ([0,0,0]F'/%,mathvar iantGQ'italicF'-I#mnGF$6%Q\"2F'FE/FIQ'normalF'/%1superscriptshiftGQ\"0 F'-I#moGF$6.Q(−F'FEFO/%&fenceGQ&falseF'/%*separatorGFZ/%)stretch yGFZ/%*symmetricGFZ/%(largeopGFZ/%.movablelimitsGFZ/%'accentGFZ/%'lspa ceGQ,0.2222222emF'/%'rspaceGFco-F,6#-F<6%-F?6&Q#q1F'FBFEFH-FL6%Q\"6F'F EFOFQFT-F,6%-F?6&Q'τ2F'FBFEFH-FU6-Q1⁢F'FOFXFenFgnFi nF[oF]oF_o/FboQ&0.0emF'/FeoFip-F<6%Fjo-FL6%Q\"4F'FEFOFQFT-F,6%-F?6&Q'& tau;1F'FBFEFHFep-F<6%Fjo-FL6%Q\"3F'FEFOFQFT-F,6%-F#6%-F,6%-F,6%FKFep-F ?6&Q)tinfty10F'FBFEFH-FU6.Q\"+F'FEFOFXFenFgnFinF[oF]oF_oFaoFdo-F,6%-I& mfracGF$6)-FL6%Q\"1F'FEFOF]q/%.linethicknessGF_s/%+denomalignGQ'center F'/%)numalignGFds/%)bevelledGFZFEFep-F,6#-F<6%FbpFKFQFEFOFep-F<6%FjoFK FQFer-F[s6)-F,6%-F?6&Q\"hF'FBFEFHFep-F#6%-F,6%-F?6&Q#p2F'FBFEFHFTF>FEF O-F,6#-F,6%FjoFT-F?6&Q#q2F'FBFEFHF`sFbsFesFgsFEFer-F,6%FctFepF]t/%)row alignGQ!F'/%,columnalignGFhu/%+groupalignGFhu/%(rowspanGF_s/%+columnsp anGF_sFfuFiuF[v-F26&-F56(-F,6/-F,6#-F<6%FjtFKFQFT-F,6#-F<6%FauF]pFQFT- F,6%FbpFep-F<6%FauF]qFQFT-F,6%FbqFep-F<6%FauFgqFQFT-F,6%F\\rFep-F<6%Fa uFKFQFer-F[s6)-F,6%FctFep-F#6%-F,6%F>FTFjtFEFO-F,6#-F,6%FauFTFjoF`sFbs FesFgsFEFer-F,6%FctFepFiwFfuFiuF[vF]vF_vFfuFiuF[v/%&alignGQ%axisF'/Fgu Q)baselineF'/FjuFds/F\\vQ'|frleft|hrF'/%/alignmentscopeGFD/%,columnwid thGQ%autoF'/%&widthGFey/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8em F'/%)rowlinesGQ%noneF'/%,columnlinesGF`z/%&frameGF`z/%-framespacingGQ, 0.4em~0.5exF'/%*equalrowsGFZ/%-equalcolumnsGFZ/%-displaystyleGFZ/%%sid eGQ&rightF'/%0minlabelspacingGF]zFEFO/%%openGQ\"[F'/%&closeGQ\"]F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ", $*(I\"hG6\"\"\"\",&I#p1GF%F&I#p2GF%!\"\"F&,&I#q1GF%F&I#q2GF%F*F*F*" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I+Cinfty1bisG6\",(*&,&*$I#p1GF$\"\"#\" \"\"*$I#p2GF$F*!\"\"F+,&I#q1GF$F+I#q2GF$F.F.F+*&,**$,(*$F0F*F+*$F1F*F+ I%tau2GF$#F+F*F*F+*&F0F*F1F*F.I)tinfty10GF$F*I\"hGF$F.F+,&F0F+F1F+F+F. *&,(F6F+*&F0F+F1F+F+F7F+F+I%tau1GF$F+F." }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+Cinfty0bisG6\",(*&,&*&I# p1GF$\"\"#I#q2GF$\"\"\"!\"\"*&I#p2GF$F*I#q1GF$F,F,F,,&F0F,F+F-F-F,*(I \"hGF$F,,&F)F,F/F-F,F1F-F-*(,6*$F0\"\"%F,*$F+F8F,*&F0\"\"$F+F,F,*&F0F, F+F;F,*&F0F*F+F*F,*$I%tau2GF$F*#F,F8*&,(*$F0F*F,*&F0F,F+F,F,*$F+F*F,F, F?F,F,*&,&F0F,F+F,F,I%tau1GF$F,F,I)tinfty10GF$F*F3F-F,F0F,F+F,F," }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 88 "Definition of the matri x M_\{\\infty\} and of the vectors \\nu_\{\\infty,1\} and \\nu_\{\\inf ty,2\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Minfty:=Matrix(4, 4,0):\n" }{MPLTEXT 1 0 16 "Minfty[1,1]:=2:\n" }{MPLTEXT 1 0 16 "Minfty [2,2]:=2:\n" }{MPLTEXT 1 0 16 "Minfty[3,3]:=2:\n" }{MPLTEXT 1 0 16 "Mi nfty[4,4]:=2:\n" }{MPLTEXT 1 0 19 "Minfty[3,1]:=tau2:\n" }{MPLTEXT 1 0 19 "Minfty[4,1]:=tau1:\n" }{MPLTEXT 1 0 19 "Minfty[4,2]:=tau2:\n" } {MPLTEXT 1 0 8 "Minfty;\n" }{MPLTEXT 1 0 13 "Minfty^(-1);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "E3:=Matrix(4,1,0):\n" }{MPLTEXT 1 0 11 " E3[3,1]:=1:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "E4:=Matrix(4,1,0): \n" }{MPLTEXT 1 0 12 "E4[4,1]:=1:\n" }{MPLTEXT 1 0 4 "E3;\n" }{MPLTEXT 1 0 5 "E4;\n\n" }{MPLTEXT 1 0 26 "Multiply(Minfty^(-1),E3);\n" } {MPLTEXT 1 0 26 "Multiply(Minfty^(-1),E4);\n" }{MPLTEXT 1 0 46 "nu2inf ty1:=1/2*Multiply(Minfty^(-1),E3)[3,1];\n" }{MPLTEXT 1 0 47 "nu2infty2 :=1/2*Multiply(Minfty^(-1),E3)[4,1]; \n" }{MPLTEXT 1 0 42 "nu1infty1:= Multiply(Minfty^(-1),E4)[3,1];\n" }{MPLTEXT 1 0 42 "nu1infty2:=Multipl y(Minfty^(-1),E4)[4,1];\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 31 "nu1i nftyVector:=Matrix(2,1,0):\n" }{MPLTEXT 1 0 32 "nu1inftyVector[1,1]:=n u1infty1:\n" }{MPLTEXT 1 0 32 "nu1inftyVector[2,1]:=nu1infty2:\n" } {MPLTEXT 1 0 16 "nu1inftyVector;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 31 "nu2inftyVector:=Matrix(2,1,0):\n" }{MPLTEXT 1 0 32 "nu2inftyVect or[1,1]:=nu2infty1:\n" }{MPLTEXT 1 0 32 "nu2inftyVector[2,1]:=nu2infty 2:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 15 "nu2inftyVector;" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowG F$6#-I'mtableGF$68-I$mtrGF$6)-I$mtdGF$6(-I#mnGF$6%Q\"2F'/%+foregroundG Q([0,0,0]F'/%,mathvariantGQ'normalF'/%)rowalignGQ!F'/%,columnalignGFC/ %+groupalignGFC/%(rowspanGQ\"1F'/%+columnspanGFJ-F56(-F86%Q\"0F'F;F>FA FDFFFHFKFMFMFAFDFF-F26)FMF4FMFMFAFDFF-F26)-F56(-I#miGF$6&Q'τ2F'/%' italicGQ%trueF'F;/F?Q'italicF'FAFDFFFHFKFMF4FMFAFDFF-F26)-F56(-FY6&Q'& tau;1F'FfnF;FinFAFDFFFHFKFVFMF4FAFDFF/%&alignGQ%axisF'/FBQ)baselineF'/ FEQ'centerF'/FGQ'|frleft|hrF'/%/alignmentscopeGFhn/%,columnwidthGQ%aut oF'/%&widthGF_p/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)row linesGQ%noneF'/%,columnlinesGFjp/%&frameGFjp/%-framespacingGQ,0.4em~0. 5exF'/%*equalrowsGQ&falseF'/%-equalcolumnsGFdq/%-displaystyleGFdq/%%si deGQ&rightF'/%0minlabelspacingGFgpF;F>/%%openGQ\"[F'/%&closeGQ\"]F'" } }{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Typeset tingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$68-I$mtrGF$6)-I$mtdGF$6(-I& mfracGF$6)-I#mnGF$6%Q\"1F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'no rmalF'-F;6%Q\"2F'F>FA/%.linethicknessGF=/%+denomalignGQ'centerF'/%)num alignGFK/%)bevelledGQ&falseF'F>/%)rowalignGQ!F'/%,columnalignGFS/%+gro upalignGFS/%(rowspanGF=/%+columnspanGF=-F56(-F;6%Q\"0F'F>FAFQFTFVFXFZF fnFfnFQFTFV-F26)FfnF4FfnFfnFQFTFV-F26)-F56(-F,6$-I#moGF$6.Q*&uminus0;F 'F>FA/%&fenceGFP/%*separatorGFP/%)stretchyGFP/%*symmetricGFP/%(largeop GFP/%.movablelimitsGFP/%'accentGFP/%'lspaceGQ,0.2222222emF'/%'rspaceGF gp-F,6%-F86)F:-F;6%Q\"4F'F>FAFGFIFLFNF>-Fdo6-Q1⁢F'FAFgo FioF[pF]pF_pFapFcp/FfpQ&0.0emF'/FipFeq-F,6#-I#miGF$6&Q'τ2F'/%'ital icGQ%trueF'F>/FBQ'italicF'FQFTFVFXFZFfnF4FfnFQFTFV-F26)-F56(-F,6$Fco-F ,6%F\\qFaq-F,6#-Fjq6&Q'τ1F'F]rF>F`rFQFTFVFXFZF_oFfnF4FQFTFV/%&alig nGQ%axisF'/FRQ)baselineF'/FUFK/FWQ'|frleft|hrF'/%/alignmentscopeGF_r/% ,columnwidthGQ%autoF'/%&widthGF[t/%+rowspacingGQ&1.0exF'/%.columnspaci ngGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFft/%&frameGFft/%-frame spacingGQ,0.4em~0.5exF'/%*equalrowsGFP/%-equalcolumnsGFP/%-displaystyl eGFP/%%sideGQ&rightF'/%0minlabelspacingGFctF>FA/%%openGQ\"[F'/%&closeG Q\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\" I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$68-I$mtrGF$6&-I$mt dGF$6(-I#mnGF$6%Q\"0F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normal F'/%)rowalignGQ!F'/%,columnalignGFC/%+groupalignGFC/%(rowspanGQ\"1F'/% +columnspanGFJFAFDFFF1-F26&-F56(-F86%FJF;F>FAFDFFFHFKFAFDFFF1/%&alignG Q%axisF'/FBQ)baselineF'/FEQ&rightF'/FGQ'|frleft|hrF'/%/alignmentscopeG Q%trueF'/%,columnwidthGQ%autoF'/%&widthGF[o/%+rowspacingGQ&1.0exF'/%.c olumnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFfo/%&frameGF fo/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGQ&falseF'/%-equalcolumns GF`p/%-displaystyleGF`p/%%sideGFY/%0minlabelspacingGFcoF;F>/%%openGQ\" [F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+mo dulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$68-I$m trGF$6&-I$mtdGF$6(-I#mnGF$6%Q\"0F'/%+foregroundGQ([0,0,0]F'/%,mathvari antGQ'normalF'/%)rowalignGQ!F'/%,columnalignGFC/%+groupalignGFC/%(rows panGQ\"1F'/%+columnspanGFJFAFDFFF1F1-F26&-F56(-F86%FJF;F>FAFDFFFHFKFAF DFF/%&alignGQ%axisF'/FBQ)baselineF'/FEQ&rightF'/FGQ'|frleft|hrF'/%/ali gnmentscopeGQ%trueF'/%,columnwidthGQ%autoF'/%&widthGF[o/%+rowspacingGQ &1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF fo/%&frameGFfo/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGQ&falseF'/%- equalcolumnsGF`p/%-displaystyleGF`p/%%sideGFY/%0minlabelspacingGFcoF;F >/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfe ncedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mta bleGF$68-I$mtrGF$6&-I$mtdGF$6(-I#mnGF$6%Q\"0F'/%+foregroundGQ([0,0,0]F '/%,mathvariantGQ'normalF'/%)rowalignGQ!F'/%,columnalignGFC/%+groupali gnGFC/%(rowspanGQ\"1F'/%+columnspanGFJFAFDFFF1-F26&-F56(-I&mfracGF$6)- F86%FJF;F>-F86%Q\"2F'F;F>/%.linethicknessGFJ/%+denomalignGQ'centerF'/% )numalignGFgn/%)bevelledGQ&falseF'F;FAFDFFFHFKFAFDFFF1/%&alignGQ%axisF '/FBQ)baselineF'/FEFgn/FGQ'|frleft|hrF'/%/alignmentscopeGQ%trueF'/%,co lumnwidthGQ%autoF'/%&widthGFjo/%+rowspacingGQ&1.0exF'/%.columnspacingG Q&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFep/%&frameGFep/%-framespa cingGQ,0.4em~0.5exF'/%*equalrowsGF\\o/%-equalcolumnsGF\\o/%-displaysty leGF\\o/%%sideGQ&rightF'/%0minlabelspacingGFbpF;F>/%%openGQ\"[F'/%&clo seGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG 6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$68-I$mtrGF$6&-I $mtdGF$6(-I#mnGF$6%Q\"0F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'nor malF'/%)rowalignGQ!F'/%,columnalignGFC/%+groupalignGFC/%(rowspanGQ\"1F '/%+columnspanGFJFAFDFFF1F1-F26&-F56(-I&mfracGF$6)-F86%FJF;F>-F86%Q\"2 F'F;F>/%.linethicknessGFJ/%+denomalignGQ'centerF'/%)numalignGFgn/%)bev elledGQ&falseF'F;FAFDFFFHFKFAFDFF/%&alignGQ%axisF'/FBQ)baselineF'/FEFg n/FGQ'|frleft|hrF'/%/alignmentscopeGQ%trueF'/%,columnwidthGQ%autoF'/%& widthGFjo/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesG Q%noneF'/%,columnlinesGFep/%&frameGFep/%-framespacingGQ,0.4em~0.5exF'/ %*equalrowsGF\\o/%-equalcolumnsGF\\o/%-displaystyleGF\\o/%%sideGQ&righ tF'/%0minlabelspacingGFbpF;F>/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*nu2infty1G6\"#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*nu2infty2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "> I*nu1infty1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*nu1infty2G6\" #\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulena meG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6 &-I$mtdGF$6(-I#mnGF$6%Q\"0F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ' normalF'/%)rowalignGQ!F'/%,columnalignGFC/%+groupalignGFC/%(rowspanGQ \"1F'/%+columnspanGFJFAFDFF-F26&-F56(-I&mfracGF$6)-F86%FJF;F>-F86%Q\"2 F'F;F>/%.linethicknessGFJ/%+denomalignGQ'centerF'/%)numalignGFgn/%)bev elledGQ&falseF'F;FAFDFFFHFKFAFDFF/%&alignGQ%axisF'/FBQ)baselineF'/FEFg n/FGQ'|frleft|hrF'/%/alignmentscopeGQ%trueF'/%,columnwidthGQ%autoF'/%& widthGFjo/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesG Q%noneF'/%,columnlinesGFep/%&frameGFep/%-framespacingGQ,0.4em~0.5exF'/ %*equalrowsGF\\o/%-equalcolumnsGF\\o/%-displaystyleGF\\o/%%sideGQ&righ tF'/%0minlabelspacingGFbpF;F>/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_s yslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6&-I$mtdGF$6(-I&mfracGF$6 )-I#mnGF$6%Q\"1F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'-F; 6%Q\"4F'F>FA/%.linethicknessGF=/%+denomalignGQ'centerF'/%)numalignGFK/ %)bevelledGQ&falseF'F>/%)rowalignGQ!F'/%,columnalignGFS/%+groupalignGF S/%(rowspanGF=/%+columnspanGF=FQFTFV-F26&-F56(-F;6%Q\"0F'F>FAFQFTFVFXF ZFQFTFV/%&alignGQ%axisF'/FRQ)baselineF'/FUFK/FWQ'|frleft|hrF'/%/alignm entscopeGQ%trueF'/%,columnwidthGQ%autoF'/%&widthGFjo/%+rowspacingGQ&1. 0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFep/ %&frameGFep/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFP/%-equalcolum nsGFP/%-displaystyleGFP/%%sideGQ&rightF'/%0minlabelspacingGFbpF>FA/%%o penGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 71 "Def inition of the matrix V and the vector \\mu_\{\\tau_1\} and \\mu_\{\\t au_2\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "V:=Matrix(2,2,0): \n" }{MPLTEXT 1 0 11 "V[1,1]:=1:\n" }{MPLTEXT 1 0 11 "V[1,2]:=1:\n" } {MPLTEXT 1 0 12 "V[2,1]:=q1:\n" }{MPLTEXT 1 0 12 "V[2,2]:=q2:\n" } {MPLTEXT 1 0 3 "V;\n" }{MPLTEXT 1 0 8 "V^(-1);\n" }{MPLTEXT 1 0 49 "Mu 1inftyVector:=Multiply(V^(-1),nu1inftyVector);\n" }{MPLTEXT 1 0 49 "Mu 2inftyVector:=Multiply(V^(-1),nu2inftyVector);\n" }{MPLTEXT 1 0 30 "mu tau11:=Mu1inftyVector[1,1];\n" }{MPLTEXT 1 0 30 "mutau12:=Mu1inftyVect or[2,1];\n" }{MPLTEXT 1 0 30 "mutau21:=Mu2inftyVector[1,1];\n" } {MPLTEXT 1 0 29 "mutau22:=Mu2inftyVector[2,1];" }{MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Typesett ingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-I#m nGF$6%Q\"1F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%)rowal ignGQ!F'/%,columnalignGFC/%+groupalignGFC/%(rowspanGF:/%+columnspanGF: F4FAFDFF-F26'-F56(-I#miGF$6&Q#q1F'/%'italicGQ%trueF'F;/F?Q'italicF'FAF DFFFHFJ-F56(-FQ6&Q#q2F'FTF;FWFAFDFFFHFJFAFDFF/%&alignGQ%axisF'/FBQ)bas elineF'/FEQ'centerF'/FGQ'|frleft|hrF'/%/alignmentscopeGFV/%,columnwidt hGQ%autoF'/%&widthGFeo/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF '/%)rowlinesGQ%noneF'/%,columnlinesGF`p/%&frameGF`p/%-framespacingGQ,0 .4em~0.5exF'/%*equalrowsGQ&falseF'/%-equalcolumnsGFjp/%-displaystyleGF jp/%%sideGQ&rightF'/%0minlabelspacingGF]pF;F>/%%openGQ\"[F'/%&closeGQ \"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I ,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtd GF$6(-F,6$-I#moGF$6.Q*&uminus0;F'/%+foregroundGQ([0,0,0]F'/%,mathvaria ntGQ'normalF'/%&fenceGQ&falseF'/%*separatorGFE/%)stretchyGFE/%*symmetr icGFE/%(largeopGFE/%.movablelimitsGFE/%'accentGFE/%'lspaceGQ,0.2222222 emF'/%'rspaceGFT-I&mfracGF$6)-F,6#-I#miGF$6&Q#q2F'/%'italicGQ%trueF'F= /FAQ'italicF'-F,6#-F,6%-Fgn6&Q#q1F'FjnF=F]o-F:6.Q(−F'F=F@FCFFFHF JFLFNFPFRFUFfn/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numali gnGF^p/%)bevelledGFEF=/%)rowalignGQ!F'/%,columnalignGFep/%+groupalignG Fep/%(rowspanGF[p/%+columnspanGF[p-F56(-FX6)-I#mnGF$6%F[pF=F@F_oFioF \\pF_pFapF=FcpFfpFhpFjpF\\qFcpFfpFhp-F26'-F56(-FX6)-F,6#FcoF_oFioF\\pF _pFapF=FcpFfpFhpFjpF\\q-F56(-F,6$F9F`qFcpFfpFhpFjpF\\qFcpFfpFhp/%&alig nGQ%axisF'/FdpQ)baselineF'/FgpF^p/FipQ'|frleft|hrF'/%/alignmentscopeGF \\o/%,columnwidthGQ%autoF'/%&widthGF]s/%+rowspacingGQ&1.0exF'/%.column spacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFhs/%&frameGFhs/%- framespacingGQ,0.4em~0.5exF'/%*equalrowsGFE/%-equalcolumnsGFE/%-displa ystyleGFE/%%sideGQ&rightF'/%0minlabelspacingGFesF=F@/%%openGQ\"[F'/%&c loseGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenam eG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6& -I$mtdGF$6(-I&mfracGF$6)-F,6#-I#mnGF$6%Q\"1F'/%+foregroundGQ([0,0,0]F' /%,mathvariantGQ'normalF'-F,6%-F=6%Q\"2F'F@FC-I#moGF$6-Q1&InvisibleTim es;F'FC/%&fenceGQ&falseF'/%*separatorGFQ/%)stretchyGFQ/%*symmetricGFQ/ %(largeopGFQ/%.movablelimitsGFQ/%'accentGFQ/%'lspaceGQ&0.0emF'/%'rspac eGFjn-F#6%-F,6%-I#miGF$6&Q#q1F'/%'italicGQ%trueF'F@/FDQ'italicF'-FL6.Q (−F'F@FCFOFRFTFVFXFZFfn/FinQ,0.2222222emF'/F\\oF^p-Fbo6&Q#q2F'Fe oF@FhoF@FC/%.linethicknessGF?/%+denomalignGQ'centerF'/%)numalignGFgp/% )bevelledGFQF@/%)rowalignGQ!F'/%,columnalignGF^q/%+groupalignGF^q/%(ro wspanGF?/%+columnspanGF?F\\qF_qFaq-F26&-F56(-F,6$-FL6.Q*&uminus0;F'F@F CFOFRFTFVFXFZFfnF]pF_pF7F\\qF_qFaqFcqFeqF\\qF_qFaq/%&alignGQ%axisF'/F] qQ)baselineF'/F`qFgp/FbqQ'|frleft|hrF'/%/alignmentscopeGFgo/%,columnwi dthGQ%autoF'/%&widthGF\\s/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8 emF'/%)rowlinesGQ%noneF'/%,columnlinesGFgs/%&frameGFgs/%-framespacingG Q,0.4em~0.5exF'/%*equalrowsGFQ/%-equalcolumnsGFQ/%-displaystyleGFQ/%%s ideGQ&rightF'/%0minlabelspacingGFdsF@FC/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Typese ttingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6&-I$mtdGF$6(-F ,6$-I#moGF$6.Q*&uminus0;F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'no rmalF'/%&fenceGQ&falseF'/%*separatorGFE/%)stretchyGFE/%*symmetricGFE/% (largeopGFE/%.movablelimitsGFE/%'accentGFE/%'lspaceGQ,0.2222222emF'/%' rspaceGFT-F,6%-I&mfracGF$6)-I#mnGF$6%Q\"1F'F=F@-Fgn6%Q\"4F'F=F@/%.line thicknessGFin/%+denomalignGQ'centerF'/%)numalignGFao/%)bevelledGFEF=-F :6-Q1⁢F'F@FCFFFHFJFLFNFP/FSQ&0.0emF'/FVFjo-FZ6)-F,6#-I# miGF$6&Q#q2F'/%'italicGQ%trueF'F=/FAQ'italicF'-F,6#-F,6%-Fap6&Q#q1F'Fd pF=Fgp-F:6.Q(−F'F=F@FCFFFHFJFLFNFPFRFUF`pF]oF_oFboFdoF=/%)rowali gnGQ!F'/%,columnalignGFeq/%+groupalignGFeq/%(rowspanGFin/%+columnspanG FinFcqFfqFhq-F26&-F56(-F,6%FYFfo-FZ6)-F,6#F]qFipF]oF_oFboFdoF=FcqFfqFh qFjqF\\rFcqFfqFhq/%&alignGQ%axisF'/FdqQ)baselineF'/FgqFao/FiqQ'|frleft |hrF'/%/alignmentscopeGFfp/%,columnwidthGQ%autoF'/%&widthGFds/%+rowspa cingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnl inesGF_t/%&frameGF_t/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFE/%-e qualcolumnsGFE/%-displaystyleGFE/%%sideGQ&rightF'/%0minlabelspacingGF \\tF=F@/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 " >I(mutau11G6\",$*$,&I#q1GF$\"\"\"I#q2GF$!\"\"F+#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(mutau12G6\",$*$,&I#q1GF$\"\"\"I#q2GF$!\"\"F+#F+ \"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(mutau21G6\",$*&I#q2GF$\"\"\" ,&I#q1GF$F(F'!\"\"F+#F+\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(mutau 22G6\",$*&I#q1GF$\"\"\",&F'F(I#q2GF$!\"\"F+#F(\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 95 "Definition of the theoretical Hamiltonian evolu tions using the formulas proved in the article. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "hdq1dtau1:=2*p1*" }{MPLTEXT 1 0 12 "mutau11- \+ h*(" }{MPLTEXT 1 0 8 "mutau12+" }{MPLTEXT 1 0 18 "mutau11)/(q1-q2);\n" }{MPLTEXT 1 0 54 "hdq2dtau1:=2*p2*mutau12- h*(mutau12+mutau11)/(q2-q1 );\n" }{MPLTEXT 1 0 58 "hdp1dtau1:=simplify(h*(mutau12+mutau11)*(p2-p1 )/(q1-q2)^2+" }{MPLTEXT 1 0 45 "mutau11*(-diff(tdP2(q1),q1)+Cinfty1-2* h*q1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 104 "hdp2dtau1:=simplify(h *(mutau12+mutau11)*(p1-p2)/(q1-q2)^2+mutau12*(-diff(tdP2(q2),q2)+Cinft y1-2*h*q2));\n" }{MPLTEXT 1 0 17 "Ham1:=factor(-h*(" }{MPLTEXT 1 0 33 "mutau12+mutau11)*(p1-p2)/(q1-q2)+" }{MPLTEXT 1 0 7 "mutau11" } {MPLTEXT 1 0 23 "*(p1^2+tdP2(q1)+h*q1^2)" }{MPLTEXT 1 0 1 "+" } {MPLTEXT 1 0 7 "mutau12" }{MPLTEXT 1 0 24 "*(p2^2+tdP2(q2)+h*q2^2))" } {MPLTEXT 1 0 2 " ;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 64 "hdq1dtau2:=simplify(2*p1*mutau21- h*(mutau22+mutau21) /(q1-q2));\n" }{MPLTEXT 1 0 64 "hdq2dtau2:=simplify(2*p2*mutau22- h*(m utau22+mutau21)/(q2-q1));\n" }{MPLTEXT 1 0 104 "hdp1dtau2:=simplify(h* (mutau22+mutau21)*(p2-p1)/(q1-q2)^2+mutau21*(-diff(tdP2(q1),q1)+Cinfty 1-2*h*q1));\n" }{MPLTEXT 1 0 104 "hdp2dtau2:=simplify(h*(mutau22+mutau 21)*(p1-p2)/(q1-q2)^2+mutau22*(-diff(tdP2(q2),q2)+Cinfty1-2*h*q2));\n" }{MPLTEXT 1 0 114 "Ham2:=factor(-h*(mutau22+mutau21)*(p1-p2)/(q1-q2)+ mutau21*(p1^2+tdP2(q1)+h*q1^2)+mutau22*(p2^2+tdP2(q2)+h*q2^2)) ;" } {MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*hdq1dtau1G6\"*& I#p1GF$\"\"\",&I#q1GF$F'I#q2GF$!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*hdq2dtau1G6\",$*&I#p2GF$\"\"\",&I#q1GF$F(I#q2GF$!\"\"F,F," }} {PARA 11 "" 1 "" {XPPMATH 20 ">I*hdp1dtau1G6\",$*&,:*$I#q1GF$\"\"'\"#? *&F)\"\"&I#q2GF$\"\"\"!#C*&I%tau2GF$F/F)\"\"%\"#7*&,&*&F.F/F2F/!#;I%ta u1GF$\"\")F/F)\"\"$F/*&,**&F.F/F9F/!#7*$F2\"\"#F/I\"hGF$!\"%I)tinfty10 GF$F:F/F)FAF/*(F.F/,(F@#!\"\"F3FBF/FD!\"#F/F)F/F:*$F.F*F3*&F2F/F.F3F3* &F9F/F.F;F3*&,(F@F/FBFCFDF:F/F.FAF/*$I#p1GF$FAF3*$I#p2GF$FAFCF/,&F)F/F .FHFI#F/F:" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*hdp2dtau1G6\",$*&,8*$I# q1GF$\"\"'\"\"%*&I%tau2GF$\"\"\"F)F+F+*&I%tau1GF$F.F)\"\"$F+*&,(*$F-\" \"#F.I\"hGF$!\"%I)tinfty10GF$\"\")F.F)F5F.*(I#q2GF$F.,.*$F;F+!\"$*&F-F .F;F5!\"#*&F0F.F;F.#F>F5F4#!\"\"F+F6F.F8F@F.F)F.F9*$F;F*\"#?*&F-F.F;F+ \"#7*&F0F.F;F1F9*&F3F.F;F5F.*$I#p1GF$F5F7*$I#p2GF$F5F+F.,&F)F.F;FDF@#F .F9" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%Ham1G6\",$*&,>*$I#q1GF$\"\"'! \"%*$I#q2GF$F*\"\"%*&F)F.I%tau2GF$\"\"\"F+*&F-F.F0F1F.*&F)\"\"$I%tau1G F$F1F+*&F)\"\"#F0F7!\"\"*&F-F4F5F1F.*&F-F7F0F7F1*&I\"hGF$F1F)F7F.*&FI*hdq1dtau2G6 \"*&,&*&I#p1GF$\"\"\"I#q2GF$F)!\"#I\"hGF$!\"\"F),&I#q1GF$\"\"%F*!\"%F- " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*hdq2dtau2G6\"*&,&*&I#p2GF$\"\"\"I #q1GF$F)\"\"#I\"hGF$F)F),&F*\"\"%I#q2GF$!\"%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*hdp1dtau2G6\",$*&,0*$I#q2GF$\"\"(!\"%*&I%tau2GF$\"\"\" F)\"\"&F+*&I%tau1GF$F.F)\"\"%F+*&,(*$F-\"\"#!\"\"I\"hGF$F2I)tinfty10GF $!\")F.F)\"\"$F.*(I#q1GF$F.,.*$F=F2!\"$*&F=F6F-F.!\"#*&F1F.F=F.#F@F6F5 #F7F2F8F.F9FBF.F)F6F:*&,.*$F=\"\"'!#?*&F-F.F=F2!#7*&F1F.F=F;F:*&F4F.F= F6F.*$I#p1GF$F6F+*$I#p2GF$F6F2F.F)F.F.*&F8F.,&FPF.FRF7F.F+F.,&F=F.F)F7 FB#F.\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*hdp2dtau2G6\",$*&,0*$I#q 1GF$\"\"(!\"%*&I%tau2GF$\"\"\"F)\"\"&F+*&I%tau1GF$F.F)\"\"%F+*&,(*$F- \"\"#!\"\"I\"hGF$F2I)tinfty10GF$!\")F.F)\"\"$F.*(I#q2GF$F.,.*$F=F2!\"$ *&F-F.F=F6!\"#*&F1F.F=F.#F@F6F5#F7F2F8F.F9FBF.F)F6F:*&,.*$F=\"\"'!#?*& F-F.F=F2!#7*&F1F.F=F;F:*&F4F.F=F6F.*$I#p1GF$F6F2*$I#p2GF$F6F+F.F)F.F.* &F8F.,&FPF.FRF7F.F2F.,&F)F.F=F7FB#F.\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%Ham2G6\",$*&,B*&I#q1GF$\"\"'I#q2GF$\"\"\"!\"%*&F)F,F+F*\"\"%*(F )F/F+F,I%tau2GF$F,F-*(F)F,F+F/F1F,F/*(F)\"\"$F+F,I%tau1GF$F,F-*(F)\"\" #F+F,F1F7!\"\"*(F)F,F+F4F5F,F/*(F)F,F+F7F1F7F,*(I\"hGF$F,F)F7F+F,F/*(F " 0 "" {MPLTEXT 1 0 9 "simplify(" } {MPLTEXT 1 0 18 "mutau12+mutau11);\n" }{MPLTEXT 1 0 9 "simplify(" } {MPLTEXT 1 0 19 "mutau22+mutau21);\n\n" }{MPLTEXT 1 0 88 "hdp1dtau1bis :=(p1^2-p2^2)/2/(q1-q2)^2+1/2*(5*q1^4+4*q1^3*q2+3*q1^2*q2^2+2*q1*q2^3+ q2^4)\n" }{MPLTEXT 1 0 70 "+(q1+q2/2)*tau1+1/2*(3*q1^2+2*q1*q2+q2^2)*t au2+tau2^2/8+tinfty10-h/2;\n" }{MPLTEXT 1 0 45 "factor(series(hdp1dtau 1-hdp1dtau1bis,p1=0));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 89 "hdp2d tau1bis:=-(p1^2-p2^2)/2/(q1-q2)^2+1/2*(5*q2^4+4*q2^3*q1+3*q2^2*q1^2+2* q2*q1^3+q1^4)\n" }{MPLTEXT 1 0 70 "+(q2+q1/2)*tau1+1/2*(3*q2^2+2*q2*q1 +q1^2)*tau2+tau2^2/8+tinfty10-h/2;\n" }{MPLTEXT 1 0 48 "factor(series( hdp2dtau1-hdp2dtau1bis,tau2=0));\n\n" }{MPLTEXT 1 0 71 "Ham1bis:=(p1^2 -p2^2)/2/(q1-q2)-(1/2)*(q1+q2)*((q1^2+q2^2)^2-q1^2*q2^2)\n" }{MPLTEXT 1 0 57 "-1/2*(q1^2+q1*q2+q2^2)*tau1-1/2*(q1+q2)*(q1^2+q2^2)*tau2\n" } {MPLTEXT 1 0 49 "-1/8*(q1+q2)*tau2^2 -1/2*(q1+q2)*(2*tinfty10-h);\n" } {MPLTEXT 1 0 36 "factor(series(Ham1-Ham1bis,tau2=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-hdp1dtau1bisG6\",8*&,&*$I#p1GF$\"\" #\"\"\"*$I#p2GF$F*!\"\"F+,&I#q1GF$F+I#q2GF$F.!\"##F+F**$F0\"\"%#\"\"&F **&F0\"\"$F1F+F**&F0F*F1F*#F9F**&F0F+F1F9F+*$F1F5F3*&,&F0F+F1F3F+I%tau 1GF$F+F+*&,(*$F0F*F9*&F0F+F1F+F**$F1F*F+F+I%tau2GF$F+F3*$FFF*#F+\"\")I )tinfty10GF$F+I\"hGF$#F.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I-hdp2dtau1bisG6\",8*&,&*$I#p1GF$\"\"# \"\"\"*$I#p2GF$F*!\"\"F+,&I#q1GF$F+I#q2GF$F.!\"##F.F**$F0\"\"%#F+F**&F 0\"\"$F1F+F+*&F0F*F1F*#F8F**&F0F+F1F8F**$F1F5#\"\"&F**&,&F1F+F0F6F+I%t au1GF$F+F+*&,(*$F0F*F+*&F0F+F1F+F**$F1F*F8F+I%tau2GF$F+F6*$FGF*#F+\"\" )I)tinfty10GF$F+I\"hGF$F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I(Ham1bisG6\",.*&,&*$I#p1GF$\"\"#\"\"\" *$I#p2GF$F*!\"\"F+,&I#q1GF$F+I#q2GF$F.F.#F+F**&,&F0F+F1F+F+,&*$,&*$F0F *F+*$F1F*F+F*F+*&F0F*F1F*F.F+#F.F**&,(F8F+*&F0F+F1F+F+F9F+F+I%tau1GF$F +F;*(F4F+F7F+I%tau2GF$F+F;*&F4F+FAF*#F.\"\")*&F4F+,&I)tinfty10GF$F*I\" hGF$F.F+F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 73 "hdp1dtau2bis:=-(1/4)*q2*(p1^2-p2^2)/(q1-q2)^ 2 -(1/4)*h*(p1-p2)/(q1-q2)^2\n" }{MPLTEXT 1 0 57 "-(1/4)*q2*(5*q1^4+4* q1^3*q2+3*q1^2*q2^2+2*q1*q2^3+q2^4) \n" }{MPLTEXT 1 0 77 "-(1/4)*q2*(2 *q1+q2)*tau1 -1/4*q2*(3*q1^2+2*q1*q2+q2^2)*tau2 -1/16*q2*tau2^2\n" } {MPLTEXT 1 0 24 "-1/4*q2*(2*tinfty10-h);\n" }{MPLTEXT 1 0 50 "factor(s eries(hdp1dtau2-hdp1dtau2bis,tinfty10=0));" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 52 "dp1dtau2ter:=-(p1-p2)*(p1*q2+p2* q2+h)/(4*(q1-q2)^2)\n" }{MPLTEXT 1 0 158 "-(1/4)*q2*(2*q1+q2)*tau1-(1/ 4)*q2*(3*q1^2+2*q1*q2+q2^2)*tau2-(1/16)*q2*tau2^2-(1/4)*q2*(5*q1^4+4*q 1^3*q2+3*q1^2*q2^2+2*q1*q2^3+q2^4)-(1/2)*q2*(tinfty10-h/2);\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 75 "dp2dtau2ter:=(p1-p2)*(p1*q1+p2*q 1+h)/(4*(q1-q2)^2)-(1/4)*q1*(q1+2*q2)*tau1\n" }{MPLTEXT 1 0 109 "-(1/4 )*q1*(q1^2+2*q1*q2+3*q2^2)*tau2-(1/16)*q1*tau2^2-(1/4)*q1*(q1^4+2*q1^3 *q2+3*q1^2*q2^2+4*q1*q2^3+5*q2^4)\n" }{MPLTEXT 1 0 25 "-(1/2)*q1*(tinf ty10-h/2);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 50 "factor(series(hdp1dtau2-dp1dtau2ter,tinfty10=0));\n" }{MPLTEXT 1 0 51 "factor(series(hdp2dtau2-dp2dtau2ter,tinfty10=0));\n\n" }{MPLTEXT 1 0 72 "hdp2dtau2bis:=(1/4)*q1*(p1^2-p2^2)/(q1-q2)^2 +(1/4)*h*(p1-p2) /(q1-q2)^2\n" }{MPLTEXT 1 0 57 "-(1/4)*q1*(5*q2^4+4*q2^3*q1+3*q2^2*q1^ 2+2*q2*q1^3+q1^4) \n" }{MPLTEXT 1 0 77 "-(1/4)*q1*(2*q2+q1)*tau1 -1/4* q1*(3*q2^2+2*q2*q1+q1^2)*tau2 -1/16*q1*tau2^2\n" }{MPLTEXT 1 0 24 "-1 /4*q1*(2*tinfty10-h);\n" }{MPLTEXT 1 0 52 "factor(series(hdp2dtau2-hdp 2dtau2bis,tinfty10=0));\n\n" }{MPLTEXT 1 0 89 "Ham2bis:=(1/4)*(q1*p2^2 -q2*p1^2)/(q1-q2)+1/4*(q1^4+q1^3*q2+q1^2*q2^2+q1*q2^3+q2^4)*q2*q1\n" } {MPLTEXT 1 0 25 "-(1/4)*h*(p1-p2)/(q1-q2)\n" }{MPLTEXT 1 0 79 "+(1/4*( q1+q2))*q2*q1*tau1+(1/4*(q1^2+q1*q2+q2^2))*q2*q1*tau2+1/16*q2*q1*tau2^ 2\n" }{MPLTEXT 1 0 29 "+(1/4)*q2*q1*(2*tinfty10-h);\n" }{MPLTEXT 1 0 34 "factor(series(Ham2-Ham2bis,p2=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-hdp1dtau2bisG6\",0*(I#q2GF$\"\"\",&*$I#p1GF$\"\"#F(*$I#p2GF$F,!\" \"F(,&I#q1GF$F(F'F/!\"##F/\"\"%*(I\"hGF$F(,&F+F(F.F/F(F0F2F3*&F'F(,,*$ F1F4\"\"&*&F1\"\"$F'F(F4*&F1F,F'F,F=*&F1F(F'F=F,*$F'F4F(F(F3*(F'F(,&F1 F,F'F(F(I%tau1GF$F(F3*(F'F(,(*$F1F,F=*&F1F(F'F(F,*$F'F,F(F(I%tau2GF$F( F3*&F'F(FIF,#F/\"#;*&F'F(,&I)tinfty10GF$F,F6F/F(F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,dp1dtau2terG6 \",.*(,&I#p1GF$\"\"\"I#p2GF$!\"\"F),(*&F(F)I#q2GF$F)F)*&F*F)F.F)F)I\"h GF$F)F),&I#q1GF$F)F.F+!\"##F+\"\"%*(F.F),&F2\"\"#F.F)F)I%tau1GF$F)F4*( F.F),(*$F2F8\"\"$*&F2F)F.F)F8*$F.F8F)F)I%tau2GF$F)F4*&F.F)F@F8#F+\"#;* &F.F),,*$F2F5\"\"&*&F2F=F.F)F5*&F2F8F.F8F=*&F2F)F.F=F8*$F.F5F)F)F4*&F. F),&I)tinfty10GF$F)F0#F+F8F)FO" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,dp2 dtau2terG6\",.*(,&I#p1GF$\"\"\"I#p2GF$!\"\"F),(*&F(F)I#q1GF$F)F)*&F*F) F.F)F)I\"hGF$F)F),&F.F)I#q2GF$F+!\"##F)\"\"%*(F.F),&F2\"\"#F.F)F)I%tau 1GF$F)#F+F5*(F.F),(*$F.F8F)*&F.F)F2F)F8*$F2F8\"\"$F)I%tau2GF$F)F:*&F.F )FAF8#F+\"#;*&F.F),,*$F.F5F)*&F.F@F2F)F8*&F.F8F2F8F@*&F.F)F2F@F5*$F2F5 \"\"&F)F:*&F.F),&I)tinfty10GF$F)F0#F+F8F)FP" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-hdp2dtau2bisG6\",0*(I#q1GF$\"\"\",&*$I#p1GF$\" \"#F(*$I#p2GF$F,!\"\"F(,&F'F(I#q2GF$F/!\"##F(\"\"%*(I\"hGF$F(,&F+F(F.F /F(F0F2F3*&F'F(,,*$F'F4F(*&F'\"\"$F1F(F,*&F'F,F1F,F<*&F'F(F1FI(Ham2bisG6\",0*&,&*&I#p1GF$\"\"#I#q2GF$\"\"\"!\"\"*&I#p 2GF$F*I#q1GF$F,F,F,,&F0F,F+F-F-#F,\"\"%*(,,*$F0F3F,*&F0\"\"$F+F,F,*&F0 F*F+F*F,*&F0F,F+F8F,*$F+F3F,F,F+F,F0F,F2*(I\"hGF$F,,&F)F,F/F-F,F1F-#F- F3**,&F0F,F+F,F,F+F,F0F,I%tau1GF$F,F2**,(*$F0F*F,*&F0F,F+F,F,*$F+F*F,F ,F+F,F0F,I%tau2GF$F,F2*(F0F,F+F,FHF*#F,\"#;*(F+F,F0F,,&I)tinfty10GF$F* F=F-F,F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{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 }