{VERSION 2018 2 "Windows 10" "2018.2" } {USTYLETAB {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "War ning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Left Justified Maple Outp ut" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Fixed Width" -1 17 1 {CSTYLE " " -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Help" -1 10 1 {CSTYLE "" -1 -1 "Courier" 1 9 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Head ing 4" -1 20 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Line Printed Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "T imes" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Diagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 10 64 128 64 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 1" -1 200 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 2" -1 201 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 36 2 0 2 2 -1 1 }{PSTYLE "Ordered List 3" -1 202 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 72 2 0 2 2 -1 1 }{PSTYLE "Ordered List 4" -1 203 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 108 2 0 2 2 -1 1 } {PSTYLE "Ordered List 5" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 144 2 0 2 2 -1 1 }{PSTYLE "Annota tion Title" -1 205 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output12" -1 206 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "HyperlinkError" -1 207 1 {CSTYLE "" -1 -1 "Courier New" 1 12 255 0 255 1 2 2 1 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "HyperlinkWarning" -1 208 1 {CSTYLE "" -1 -1 "Courier New" 1 12 0 0 255 1 2 2 1 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "M aple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Dash Item" -1 16 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 } {PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{CSTYLE "Help Variable" -1 25 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Itali c Small209212" -1 200 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Comment" -1 21 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "Popup" -1 31 "Times" 1 12 0 128 128 1 1 2 1 2 2 2 0 0 0 1 } {CSTYLE "Atomic Variable" -1 201 "Times" 1 12 175 0 175 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Text" -1 28 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Code" -1 202 "Courier New" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Small" -1 7 "Times" 1 1 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold Small" -1 10 "Times" 1 1 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Bold" -1 39 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Menus" -1 36 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Normal" -1 30 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "Header and Footer" -1 203 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small204" -1 204 "Times " 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small207208 " -1 205 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Ita lic Small206" -1 206 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Output Labels" -1 29 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small205" -1 207 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Notes" -1 37 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small208209215" -1 208 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined" -1 44 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small208" -1 209 "Ti mes" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small207 " -1 210 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Ita lic Small209" -1 211 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 212 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Plot \+ Title" -1 27 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Defaul t" -1 38 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Itali c Bold" -1 40 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input Placeholder" -1 213 "Courier New" 1 12 200 0 200 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "LaTeX" -1 32 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Emphasized" -1 22 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small211" -1 214 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Equation Label" -1 215 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small210" -1 216 "Times " 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic" -1 42 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -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 "Hyperlink" -1 17 "Times" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Caption Text" -1 217 "Times " 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small220" -1 218 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlin ed Bold" -1 41 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "Copy right" -1 34 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Mat h Italic Small222" -1 219 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "2D Math Italic Small221" -1 220 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small223" -1 221 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Dictionary Hyperlink" -1 45 "Times" 1 12 147 0 15 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 1 12 255 0 0 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math Italic Small210220 " -1 222 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Headin g" -1 26 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Inert O utput" -1 223 "Times" 1 12 144 144 144 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Page Number" -1 33 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Fixed" -1 23 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small" -1 224 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small211223" -1 225 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Symbol 2" -1 16 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small211221" -1 226 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Nonterminal" -1 24 "Cour ier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small21 1222" -1 227 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small211220" -1 228 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "Annotation Text" -1 229 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Maple Name" -1 35 "Times" 1 12 104 64 92 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Caption Reference" -1 230 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small208209" -1 231 "Times" 1 1 0 0 0 1 1 2 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 "" -1 232 "Tim es" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 233 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 234 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 235 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 236 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 237 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 238 263 "In this Maple file, we \+ compute the evolution equations for the second element of the Painlev \351 2 hierarchy using the compatibility equation of the Lax system. \+ We also obtain the expression of the Lax matrices in the geometric gau ge without apparent singularities." }{TEXT 238 0 "" }}{PARA 0 "" 0 "" {TEXT 238 28 "The deformation operator is " }}{PARA 0 "" 0 "" {TEXT 238 167 "\\mathcal\{L\}=\\hbar (alpha14\\partial_\{t_\{\\infty^\{(1)\} ,4\} +alpha24\\partial_\{t_\{\\infty^\{(2)\},4\}+alpha13*\\partial_\{t _\{\\infty^\{(1)\},3\} +alpha23*\\partial_\{t_\{\\infty^\{(2)\},3\}\n" }{TEXT 238 150 "+alpha12*\\partial_\{t_\{\\infty^\{(1)\},2\} +alpha22 *\\partial_\{t_\{\\infty^\{(2)\},2\}+alpha11*\\partial_\{t_\{\\infty^ \{(1)\},1\} +alpha21*\\partial_\{t_\{\\infty^\{(2)\},1\}" }{TEXT 238 1 ")" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 239 71 "Solving the compatibili ty equation to obtain the Hamiltonian evolutions" }}{EXCHG {PARA 0 "" 0 "" {TEXT 238 102 "Summary of previous files: We have the expression for some coefficients of the Lax matrix L and of A." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }{MPLTEXT 1 0 21 "with(LinearA lgebra):\n" }{MPLTEXT 1 0 40 "CoherenceEquation1 :=tinfty10+tinfty20; \n" }{MPLTEXT 1 0 21 "tinfty20:=-tinfty10:\n" }{MPLTEXT 1 0 32 "Pinfty 01 := -tinfty11-tinfty21;\n" }{MPLTEXT 1 0 32 "Pinfty11 := -tinfty12-t infty22;\n" }{MPLTEXT 1 0 32 "Pinfty21 := -tinfty13-tinfty23;\n" } {MPLTEXT 1 0 32 "Pinfty31 := -tinfty14-tinfty24;\n" }{MPLTEXT 1 0 31 " Pinfty62 := tinfty14*tinfty24;\n" }{MPLTEXT 1 0 49 "Pinfty52 := tinfty 13*tinfty24+tinfty14*tinfty23;\n" }{MPLTEXT 1 0 67 "Pinfty42 := tinfty 12*tinfty24+tinfty13*tinfty23+tinfty14*tinfty22;\n" }{MPLTEXT 1 0 85 " Pinfty32 := tinfty11*tinfty24+tinfty12*tinfty23+tinfty13*tinfty22+tinf ty14*tinfty21;\n" }{MPLTEXT 1 0 103 "Pinfty22 := tinfty20*tinfty14+tin fty10*tinfty24+tinfty11*tinfty23+tinfty12*tinfty22+tinfty13*tinfty21; \n" }{MPLTEXT 1 0 55 "P1:=x-> Pinfty01+Pinfty11*x+Pinfty21*x^2+Pinfty3 1*x^3:\n" }{MPLTEXT 1 0 94 "P2:=x-> Pinfty02+Pinfty12*x+Pinfty22*x^2+P infty32*x^3+Pinfty42*x^4+Pinfty52*x^5+Pinfty62*x^6:\n" }{MPLTEXT 1 0 43 "tdP2:=unapply(P2(x)-Pinfty02-Pinfty12*x,x):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 29 "mu1:= -(nu1*q2-nu2)/(q1-q2);\n" }{MPLTEXT 1 0 28 "mu2:= (nu1*q1-nu2)/(q1-q2);\n" }{MPLTEXT 1 0 117 "n ubis:=-(1/4)*(tinfty13-tinfty23)/(tinfty14-tinfty24)^2*(alpha14-alpha2 4)+1/3*(alpha13-alpha23)/(tinfty14-tinfty24):\n" }{MPLTEXT 1 0 67 "c4b is:=1/4*(alpha14*tinfty24-alpha24*tinfty14)/(tinfty14-tinfty24):" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 92 "c3bis:=-(1/4)*(tinfty13*tinfty24 -tinfty14*tinfty23)/(tinfty14-tinfty24)^2*(alpha14-alpha24)\n" } {MPLTEXT 1 0 62 "+1/3*(alpha13*tinfty24-alpha23*tinfty14)/(tinfty14-ti nfty24):\n" }{MPLTEXT 1 0 173 "c2bis:=-(1/4)*((tinfty14-tinfty24)*(tin fty24*tinfty12-tinfty14*tinfty22)-(-tinfty23+tinfty13)*(tinfty13*tinft y24-tinfty14*tinfty23))/(tinfty14-tinfty24)^3*(alpha14-alpha24)\n" } {MPLTEXT 1 0 85 "-(1/3)*(tinfty13*tinfty24-tinfty14*tinfty23)/(tinfty1 4-tinfty24)^2*(alpha13-alpha23)\n" }{MPLTEXT 1 0 62 "+(alpha12*tinfty2 4-alpha22*tinfty14)/(2*(tinfty14-tinfty24)):\n" }{MPLTEXT 1 0 63 "c1bi s:=(tinfty24*alpha11-tinfty14*alpha21)/(tinfty14-tinfty24)\n" } {MPLTEXT 1 0 85 "-(1/2)*(tinfty13*tinfty24-tinfty14*tinfty23)/(tinfty1 4-tinfty24)^2*(alpha12-alpha22)\n" }{MPLTEXT 1 0 166 "-(1/3)*((tinfty1 4-tinfty24)*(tinfty24*tinfty12-tinfty14*tinfty22)-(-tinfty23+tinfty13) *(tinfty13*tinfty24-tinfty14*tinfty23))/(tinfty14-tinfty24)^3*(alpha13 -alpha23)\n" }{MPLTEXT 1 0 131 "-1/4*(tinfty24*tinfty13^3-tinfty14*tin fty23^3+ (2*tinfty14+tinfty24)*tinfty13*tinfty23^2-(2*tinfty24+tinfty1 4)*tinfty13^2*tinfty23\n" }{MPLTEXT 1 0 88 "-(tinfty14-tinfty24)*(2*ti nfty12*tinfty24-tinfty14*tinfty22-tinfty22*tinfty24)*tinfty13\n" } {MPLTEXT 1 0 88 "-(tinfty14-tinfty24)*(2*tinfty22*tinfty14-tinfty14*ti nfty12-tinfty12*tinfty24)*tinfty23\n" }{MPLTEXT 1 0 102 "+(tinfty14-ti nfty24)^2*(tinfty11*tinfty24-tinfty14*tinfty21))/(tinfty14-tinfty24)^4 *(alpha14-alpha24):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 8 "nu1bis:=" }{MPLTEXT 1 0 113 "-(1/4)*((tinfty12-tinfty2 2)*(tinfty14-tinfty24) -(tinfty13-tinfty23)^2 )/(tinfty14-tinfty24)^3* (alpha14-alpha24)\n" }{MPLTEXT 1 0 67 "-(1/3)*(tinfty13-tinfty23)/(tin fty14-tinfty24)^2*(alpha13-alpha23)\n" }{MPLTEXT 1 0 43 "+(alpha12-alp ha22)/(2*(tinfty14-tinfty24)):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 8 " nu2bis:=" }{MPLTEXT 1 0 176 "-(1/4)*((tinfty11-tinfty21)*(tinfty14-tin fty24)^2-2*(tinfty12-tinfty22)*(tinfty14-tinfty24)*(tinfty13-tinfty23) +(tinfty13-tinfty23)^3)/(tinfty14-tinfty24)^4*(alpha14-alpha24)\n" } {MPLTEXT 1 0 112 "-(1/3)*((tinfty12-tinfty22)*(tinfty14-tinfty24) -(ti nfty13-tinfty23)^2)/(tinfty14-tinfty24)^3*(alpha13-alpha23)\n" } {MPLTEXT 1 0 68 "-(1/2)*(-tinfty23+tinfty13)/(tinfty14-tinfty24)^2*(al pha12-alpha22)\n" }{MPLTEXT 1 0 38 "+(alpha11-alpha21)/(tinfty14-tinft y24)" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 10 "c4:=c4bis:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "d P1dlambda:=unapply(diff(P1(lambda),lambda),lambda):\n" }{MPLTEXT 1 0 53 "dP2dlambda:=unapply(diff(P2(lambda),lambda),lambda):\n" }{MPLTEXT 1 0 56 "dtdP2dlambda:=unapply(diff(tdP2(lambda),lambda),lambda):" } {MPLTEXT 1 0 1 "\n" }{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 8 "L[2,1]:=" }{MPLTEXT 1 0 105 "-P2(lambda)+Pinfty02+Pinfty12*lambda +C1 *lambda+C0 -h*lambda^2*tinfty14-p1*h/(lambda-q1)-p2*h/(lambda-q2)" } {MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "L[2,2]:= " } {MPLTEXT 1 0 42 "P1(lambda) +h/(lambda-q1)+h/(lambda-q2):\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "A:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 8 "A[1,1]:=" }{MPLTEXT 1 0 68 "1/4*(alpha14*tinfty24-alpha24*tinft y14)/(tinfty14-tinfty24)*lambda^4" }{MPLTEXT 1 0 74 "+c3*lambda^3+c2*l ambda^2+c1*lambda +c0+ rho1/(lambda-q1)+rho2/(lambda-q2):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 8 "A[1,2]:=" }{MPLTEXT 1 0 67 "(alpha14-alpha 24)/4/(-tinfty24+tinfty14)*lambda+nu+ mu1/(lambda-q1)" }{MPLTEXT 1 0 19 "+ mu2/(lambda-q2):\n" }{MPLTEXT 1 0 8 "A[1,2]:=" }{MPLTEXT 1 0 103 "(alpha14-alpha24)/4/(-tinfty24+tinfty14)*lambda+nu+ (nu1*(lambda- (q1+q2))+nu2)/(lambda-q1)/(lambda-q2):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "A[2,1]:=AA21(lambda):\n" }{MPLTEXT 1 0 22 "A[2,2]:=AA22(lambd a):\n" }{MPLTEXT 1 0 26 "dAdlambda:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 88 "for i from 1 to 2 do for j from 1 to 2 do dAdlambda[i,j]:=diff(A[i ,j],lambda): od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 58 "nuinft yMinus1:=- residue(A[1,2]/lambda^2,lambda=infinity);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 10 "nuinfty0:=" }{MPLTEXT 1 0 40 "- residue(A[1,2]/ lambda,lambda=infinity)" }{MPLTEXT 1 0 1 ";" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 10 "nuinfty1:=" }{MPLTEXT 1 0 44 "- residue(A[1,2]/lambda ^0,lambda=infinity);\n" }{MPLTEXT 1 0 10 "nuinfty2:=" }{MPLTEXT 1 0 46 "- residue(A[1,2]/lambda^(-1),lambda=infinity);" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 5 "mu1:=" }{MPLTEXT 1 0 26 "residue(A[1,2],lambda=q1 );" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 5 "mu2:=" }{MPLTEXT 1 0 26 "res idue(A[1,2],lambda=q2);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 3 "L;\n" }{MPLTEXT 1 0 3 "A;\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 36 "Q2:=unapply( -p1*(lambda-q2)/(q1-q2)" }{MPLTEXT 1 0 33 "-p2*(lambda-q1)/(q2-q1), lambda):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "simplify(Q2(q1)); \n" }{MPLTEXT 1 0 18 "simplify(Q2(q2));\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 44 "J[2,1]:=Q2(lambda)/( lambda-q1)/(lambda-q2):\n" }{MPLTEXT 1 0 35 "J[2,2]:=1/(lambda-q1)/(la mbda-q2):\n" }{MPLTEXT 1 0 26 "dJdlambda:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 86 "for i from 1 to 2 do for j from 1 to 2 do dJdlambda[i,j]:=diff (J[i,j],lambda): od: od:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 19 "LJ:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 12 "LJ[1,1]:=0:\n " }{MPLTEXT 1 0 12 "LJ[1,2]:=0:\n" }{MPLTEXT 1 0 49 "LJ[2,2]:=diff(J[2 ,2],q1)*Lq1+diff(J[2,2],p1)*Lp1+" }{MPLTEXT 1 0 39 "diff(J[2,2],q2)*Lq 2+diff(J[2,2],p2)*Lp2" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 9 "LJ[2,1]: =" }{MPLTEXT 1 0 40 "diff(J[2,1],q1)*Lq1+diff(J[2,1],p1)*Lp1+" } {MPLTEXT 1 0 39 "diff(J[2,1],q2)*Lq2+diff(J[2,1],p2)*Lp2" }{MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 3 "LJ:" }{MPLTEXT 1 0 1 "\n " }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 79 "checkL:=simplify(Multiply(Mul tiply(J,L),J^(-1))+h*Multiply(dJdlambda,J^(-1))):\n" }{MPLTEXT 1 0 70 "checkA:=simplify(Multiply(Multiply(J,A),J^(-1))+Multiply(LJ,J^(-1))): \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3CoherenceEquation1G6\",&I)tinfty10GF$\"\"\"I)tinfty20G F$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty01G6\",&I)tinfty11GF$! \"\"I)tinfty21GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty11G6\",& I)tinfty12GF$!\"\"I)tinfty22GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I) Pinfty21G6\",&I)tinfty13GF$!\"\"I)tinfty23GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty31G6\",&I)tinfty14GF$!\"\"I)tinfty24GF$F'" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty62G6\"*&I)tinfty14GF$\"\"\"I)ti nfty24GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty52G6\",&*&I)tinf ty13GF$\"\"\"I)tinfty24GF$F(F(*&I)tinfty14GF$F(I)tinfty23GF$F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty42G6\",(*&I)tinfty12GF$\"\"\"I) tinfty24GF$F(F(*&I)tinfty13GF$F(I)tinfty23GF$F(F(*&I)tinfty14GF$F(I)ti nfty22GF$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty32G6\",**&I)ti nfty11GF$\"\"\"I)tinfty24GF$F(F(*&I)tinfty12GF$F(I)tinfty23GF$F(F(*&I) tinfty13GF$F(I)tinfty22GF$F(F(*&I)tinfty14GF$F(I)tinfty21GF$F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty22G6\",,*&I)tinfty10GF$\"\"\"I) tinfty14GF$F(!\"\"*&F'F(I)tinfty24GF$F(F(*&I)tinfty11GF$F(I)tinfty23GF $F(F(*&I)tinfty12GF$F(I)tinfty22GF$F(F(*&I)tinfty13GF$F(I)tinfty21GF$F (F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$mu1G6\",$*&,&*&I$nu1GF$\"\"\"I #q2GF$F*F*I$nu2GF$!\"\"F*,&I#q1GF$F*F+F-F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$mu2G6\"*&,&*&I$nu1GF$\"\"\"I#q1GF$F)F)I$nu2GF$!\"\"F), &F*F)I#q2GF$F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.nuinftyMinus1G6\" ,$*&,&I(alpha14GF$\"\"\"I(alpha24GF$!\"\"F),&I)tinfty14GF$F)I)tinfty24 GF$F+F+#F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuinfty0G6\"I#nuGF $" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuinfty1G6\"I$nu1GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuinfty2G6\",**&I$nu1GF$\"\"\",&I#q1GF$!\"\"I #q2GF$F+F(F(I$nu2GF$F(*&F'F(F*F(F(*&F'F(F,F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$mu1G6\"*&,**(I$nu1GF$\"\"\"I#q2GF$F)I)tinfty14GF$F)!\" %*(F(F)F*F)I)tinfty24GF$F)\"\"%*&I$nu2GF$F)F+F)F/*&F1F)F.F)F,F),**&I#q 1GF$F)F+F)F/*&F5F)F.F)F,*&F*F)F+F)F,*&F*F)F.F)F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$mu2G6\"*&,**(I$nu1GF$\"\"\"I#q1GF$F)I)tinfty14GF$F )!\"%*(F(F)F*F)I)tinfty24GF$F)\"\"%*&I$nu2GF$F)F+F)F/*&F1F)F.F)F,F),** &F*F)F+F)F,*&F*F)F.F)F/*&I#q2GF$F)F+F)F/*&F7F)F.F)F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_sy slibGF'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'/%+columnspanGFJ-F56( -F86%FJF;F>FAFDFFFHFKFAFDFF-F26'-F56(-F,66-I#moGF$6.Q*&uminus0;F'F;F>/ %&fenceGQ&falseF'/%*separatorGFgn/%)stretchyGFgn/%*symmetricGFgn/%(lar geopGFgn/%.movablelimitsGFgn/%'accentGFgn/%'lspaceGQ,0.2222222emF'/%'r spaceGFfo-F,6%-F#6%-F,6,FW-F,6%-I#miGF$6&Q)tinfty10F'/%'italicGQ%trueF 'F;/F?Q'italicF'-FX6-Q1⁢F'F>FenFhnFjnF\\oF^oF`oFbo/FeoQ &0.0emF'/FhoF^q-Fbp6&Q)tinfty14F'FepF;Fhp-FX6.Q\"+F'F;F>FenFhnFjnF\\oF ^oF`oFboFdoFgo-F,6%FapFjp-Fbp6&Q)tinfty24F'FepF;FhpFcq-F,6%-Fbp6&Q)tin fty11F'FepF;FhpFjp-Fbp6&Q)tinfty23F'FepF;FhpFcq-F,6%-Fbp6&Q)tinfty12F' FepF;FhpFjp-Fbp6&Q)tinfty22F'FepF;FhpFcq-F,6%-Fbp6&Q)tinfty13F'FepF;Fh pFjp-Fbp6&Q)tinfty21F'FepF;FhpF;F>Fjp-I%msupGF$6%-Fbp6&Q)λF'/Ff pFgnF;F>-F86%Q\"2F'F;F>/%1superscriptshiftGF:-FX6.Q(−F'F;F>FenFh nFjnF\\oF^oF`oFboFdoFgo-F,6%-F#6%-F,6)-F,6%F]rFjpFhqFcq-F,6%FerFjpF`rF cq-F,6%F]sFjpFhrFcq-F,6%F`qFjpF`sF;F>Fjp-Fds6%Ffs-F86%Q\"3F'F;F>F]tF_t -F,6%-F#6%-F,6'-F,6%FerFjpFhqFcq-F,6%F]sFjpF`rFcq-F,6%F`qFjpFhrF;F>Fjp -Fds6%Ffs-F86%Q\"4F'F;F>F]tF_t-F,6%-F#6%-F,6%-F,6%F]sFjpFhqFcq-F,6%F`q FjpF`rF;F>Fjp-Fds6%Ffs-F86%Q\"5F'F;F>F]tF_t-F,6'F`qFjpFhqFjp-Fds6%Ffs- F86%Q\"6F'F;F>F]tFcq-F,6%-Fbp6&Q#C1F'FepF;FhpFjpFfsFcq-Fbp6&Q#C0F'FepF ;FhpF_t-F,6'-Fbp6&Q\"hF'FepF;FhpFjpFcsFjpF`qF_t-I&mfracGF$6)-F,6%-Fbp6 &Q#p1F'FepF;FhpFjpFfx-F,6#-F,6%FfsF_t-Fbp6&Q#q1F'FepF;Fhp/%.linethickn essGFJ/%+denomalignGQ'centerF'/%)numalignGF\\z/%)bevelledGFgnF;F_t-Fjx 6)-F,6%-Fbp6&Q#p2F'FepF;FhpFjpFfx-F,6#-F,6%FfsF_t-Fbp6&Q#q2F'FepF;FhpF hyFjyF]zF_zF;FAFDFFFHFK-F56(-F,60FWF]rF_tF`sFcq-F,6%-F#6%-F,6&FWFerF_t FhrF;F>FjpFfsFcq-F,6%-F#6%-F,6&FWF]sF_tF`rF;F>FjpFcsFcq-F,6%-F#6%-F,6& FWF`qF_tFhqF;F>FjpF`uFcq-Fjx6)-F,6#FfxFayFhyFjyF]zF_zF;Fcq-Fjx6)Fg\\lF hzFhyFjyF]zF_zF;FAFDFFFHFKFAFDFF/%&alignGQ%axisF'/FBQ)baselineF'/FEF \\z/FGQ'|frleft|hrF'/%/alignmentscopeGFgp/%,columnwidthGQ%autoF'/%&wid thGFg]l/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ% noneF'/%,columnlinesGFb^l/%&frameGFb^l/%-framespacingGQ,0.4em~0.5exF'/ %*equalrowsGFgn/%-equalcolumnsGFgn/%-displaystyleGFgn/%%sideGQ&rightF' /%0minlabelspacingGF_^lF;F>/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_sys libGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-F,6/-F,6%-I&m fracGF$6)-I#mnGF$6%Q\"1F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'nor malF'-F?6%Q\"4F'FBFE/%.linethicknessGFA/%+denomalignGQ'centerF'/%)numa lignGFO/%)bevelledGQ&falseF'FB-I#moGF$6-Q1⁢F'FE/%&fence GFT/%*separatorGFT/%)stretchyGFT/%*symmetricGFT/%(largeopGFT/%.movable limitsGFT/%'accentGFT/%'lspaceGQ&0.0emF'/%'rspaceGFco-F<6)-F,6%-F#6%-F ,6%-F,6%-I#miGF$6&Q*α14F'/%'italicGQ%trueF'FB/FFQ'italicF'FU-Fap 6&Q)tinfty24F'FdpFBFgp-FV6.Q(−F'FBFEFYFenFgnFinF[oF]oF_o/FboQ,0. 2222222emF'/FeoF`q-F,6%-Fap6&Q*α24F'FdpFBFgpFU-Fap6&Q)tinfty14F' FdpFBFgpFBFEFU-I%msupGF$6%-Fap6&Q)λF'/FepFTFBFEFH/%1superscript shiftGQ\"0F'-F,6#-F,6%FgqF\\qFipFKFMFPFRFB-FV6.Q\"+F'FBFEFYFenFgnFinF[ oF]oF_oF_qFaq-F,6%-Fap6&Q#c3F'FdpFBFgpFU-F[r6%F]r-F?6%Q\"3F'FBFEFarFhr -F,6%-Fap6&Q#c2F'FdpFBFgpFU-F[r6%F]r-F?6%Q\"2F'FBFEFarFhr-F,6%-Fap6&Q# c1F'FdpFBFgpFUF]rFhr-Fap6&Q#c0F'FdpFBFgpFhr-F<6)-F,6#-Fap6&Q'ρ1F'F dpFBFgp-F,6#-F,6%F]rF\\q-Fap6&Q#q1F'FdpFBFgpFKFMFPFRFBFhr-F<6)-F,6#-Fa p6&Q'ρ2F'FdpFBFgp-F,6#-F,6%F]rF\\q-Fap6&Q#q2F'FdpFBFgpFKFMFPFRFB/% )rowalignGQ!F'/%,columnalignGFev/%+groupalignGFev/%(rowspanGFA/%+colum nspanGFA-F56(-F,6'-F,6%F;FU-F<6)-F,6%-F#6%-F,6%F`pF\\qFdqFBFEFUF]rFdrF KFMFPFRFBFhr-Fap6&Q%νF'F`rFBFEFhr-F<6)-F,6#-F,6%-F,6%-Fap6&Q&ν1F 'FdpFBFgpFU-F#6%-F,6'F]rF\\qFbuF\\qF`vFBFEFhr-Fap6&Q&ν2F'FdpFBFgp-F ,6%-F#6%F`uFBFEFU-F#6%F^vFBFEFKFMFPFRFBFcvFfvFhvFjvF\\wFcvFfvFhv-F26'- F56(-F,6%-Fap6&Q%AA21F'FdpFBFgp-FV6-Q0⁡F'FEFYFenFgnFinF[ oF]oF_oFaoFdo-F#6%-F,6#F]rFBFEFcvFfvFhvFjvF\\w-F56(-F,6%-Fap6&Q%AA22F' FdpFBFgpF`zFczFcvFfvFhvFjvF\\wFcvFfvFhv/%&alignGQ%axisF'/FdvQ)baseline F'/FgvFO/FivQ'|frleft|hrF'/%/alignmentscopeGFfp/%,columnwidthGQ%autoF' /%&widthGFj[l/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowli nesGQ%noneF'/%,columnlinesGFe\\l/%&frameGFe\\l/%-framespacingGQ,0.4em~ 0.5exF'/%*equalrowsGFT/%-equalcolumnsGFT/%-displaystyleGFT/%%sideGQ&ri ghtF'/%0minlabelspacingGFb\\lFBFE/%%openGQ\"[F'/%&closeGQ\"]F'" }} {PARA 11 "" 1 "" {XPPMATH 20 ",$I#p1G6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$I#p2G6\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 238 69 " The compatibility equation is \\mathcal\{L\}L=h\\partial_\\lambda A+[A ,L]\n" }{TEXT 238 111 "Since the first line of L is trivial, we may ea sily obtain A[2,1] et A[2,2] to obtain the full expression for A" } {TEXT 238 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "LL:=h*dAdl ambda+(Multiply(A,L)-Multiply(L,A)):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "Entry11:=LL[1,1]:\n" }{MPLTEXT 1 0 17 "Entry12:=LL[1,2]:" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "AA21:=unapp ly(solve(Entry11=0,AA21(lambda)),lambda):\n" }{MPLTEXT 1 0 40 "AA21bis :=h*dAdlambda[1,1]+A[1,2]*L[2,1]:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "simplify(AA21(lambda)-AA21bis);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "AA22:=unapply(solve(Entry12=0,AA22(lambda)),lambda):\n" } {MPLTEXT 1 0 47 "AA22bis:=h*dAdlambda[1,2]+A[1,1]+A[1,2]*L[2,2]:" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "simplify(AA22(lambda)-AA22bis); \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "simplify(Entry11);\n" } {MPLTEXT 1 0 19 "simplify(Entry12);\n" }{MPLTEXT 1 0 47 "LL:=h*dAdlamb da+(Multiply(A,L)-Multiply(L,A)):\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 238 95 "We now compute the acti on of \\mathcal\{L\} on L[2,2] et L[2,1] to obtain the evolution equa tions" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 26 "Evolution of entry L_\{ 2,2\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Entry22:=simplify( LL[2,2]):\n" }{MPLTEXT 1 0 129 "simplify(Entry22-(h^2*diff(A[1,2],lamb da$2)+2*h*diff(A[1,1],lambda)+h*A[1,2]*diff(L[2,2],lambda)+h*L[2,2]*di ff(A[1,2],lambda)));\n" }{MPLTEXT 1 0 80 "Entry22TermLambdaMinusq1Cube :=factor(residue(Entry22*(lambda-q1)^2,lambda=q1));\n" }{MPLTEXT 1 0 80 "Entry22TermLambdaMinusq1Square:=factor(residue(Entry22*(lambda-q1) ,lambda=q1));\n" }{MPLTEXT 1 0 62 "Entry22TermLambdaMinusq1:=factor(re sidue(Entry22,lambda=q1));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 80 "E ntry22TermLambdaMinusq2Cube:=factor(residue(Entry22*(lambda-q2)^2,lamb da=q2));\n" }{MPLTEXT 1 0 80 "Entry22TermLambdaMinusq2Square:=factor(r esidue(Entry22*(lambda-q2),lambda=q2));\n" }{MPLTEXT 1 0 61 "Entry22Te rmLambdaMinusq2:=factor(residue(Entry22,lambda=q2));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 76 "Entry22TermLambdaInfty5:=f actor(-residue(Entry22/lambda^6,lambda=infinity));" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 77 "Entry22TermLambdaInfty4:=factor(-residue(Entry22 /lambda^5,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry22TermLambdaInf ty3:=factor(-residue(Entry22/lambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry22TermLambdaInfty2:=factor(-residue(Entry22/lambda^3,lamb da=infinity));\n" }{MPLTEXT 1 0 77 "Entry22TermLambdaInfty1:=factor(-r esidue(Entry22/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 75 "Entry2 2TermLambdaInfty0:=factor(-residue(Entry22/lambda,lambda=infinity));\n " }{MPLTEXT 1 0 72 "Entry22TermLambdaInftyMinus1:=factor(-residue(Entr y22,lambda=infinity));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 101 "simplify( Entry22-(Entry22TermLambdaMinusq1Square/(l ambda-q1)^2+Entry22TermLambdaMinusq1/(lambda-q1)\n" }{MPLTEXT 1 0 1 "+ " }{MPLTEXT 1 0 81 "Entry22TermLambdaMinusq2Square/(lambda-q2)^2+Entry 22TermLambdaMinusq2/(lambda-q2)" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 159 "+Entry22TermLambdaInfty0+Entry22TermLambdaInfty1*lambda+Entry22Te rmLambdaInfty2*lambda^2+Entry22TermLambdaInfty3*lambda^3+Entry22TermLa mbdaInfty4*lambda^4) );\n" }{MPLTEXT 1 0 8 "L[2,2];\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I=Entry22TermLambdaMinusq1CubeG6\"\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I?Entry22TermLambdaMinusq1SquareG6 \",$**,`p**I$nu1GF$\"\"\"I#q1GF$\"\"$I#q2GF$F*I)tinfty14GF$\"\"#\"\"%* *F)F*F+F,F-F*I)tinfty24GF$F/!\"%*,F)F*F+F/F-F*I)tinfty13GF$F*F.F*F0*,F )F*F+F/F-F*F5F*F2F*F3*,F)F*F+F/F-F*F.F*I)tinfty23GF$F*F0*,F)F*F+F/F-F* F8F*F2F*F3*(I$nu2GF$F*F+F,F.F/F3*(F;F*F+F,F2F/F0*,F)F*F+F*F-F*I)tinfty 12GF$F*F.F*F0*,F)F*F+F*F-F*F>F*F2F*F3*,F)F*F+F*F-F*F.F*I)tinfty22GF$F* F0*,F)F*F+F*F-F*FAF*F2F*F3**F;F*F+F/F5F*F.F*F3**F;F*F+F/F5F*F2F*F0**F; F*F+F/F.F*F8F*F3**F;F*F+F/F8F*F2F*F0*(I(alpha14GF$F*I\"hGF$F*F+F/F***F HF*FIF*F+F*F-F*!\"\"*(I(alpha24GF$F*FIF*F+F/FK**FMF*FIF*F+F*F-F*F***FI F*I#nuGF$F*F+F*F.F*F0**FIF*FPF*F+F*F2F*F3**FIF*FPF*F-F*F.F*F3**FIF*FPF *F-F*F2F*F0**F)F*F-F*I)tinfty11GF$F*F.F*F0**F)F*F-F*FUF*F2F*F3**F)F*F- F*F.F*I)tinfty21GF$F*F0**F)F*F-F*FXF*F2F*F3**F;F*F+F*F>F*F.F*F3**F;F*F +F*F>F*F2F*F0**F;F*F+F*F.F*FAF*F3**F;F*F+F*FAF*F2F*F0*(FIF*F)F*F.F*F0* (FIF*F)F*F2F*F3*(F;F*FUF*F.F*F3*(F;F*FUF*F2F*F0*(F;F*F.F*FXF*F3*(F;F*F XF*F2F*F0*(F+F*I%rho1GF$F*F.F*\"\")*(F+F*F_oF*F2F*!\")*(F-F*F_oF*F.F*F bo*(F-F*F_oF*F2F*F`oF*FIF*,&F.F*F2FKFK,&F+F*F-FKFK#FKF0" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9Entry22TermLambdaMinusq1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I=Entry22TermLambdaMinusq2CubeG6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I?Entry22TermLambdaMinusq2SquareG6\",$**,`p** I$nu1GF$\"\"\"I#q1GF$F*I#q2GF$\"\"$I)tinfty14GF$\"\"#!\"%**F)F*F+F*F,F -I)tinfty24GF$F/\"\"%*,F)F*F+F*F,F/I)tinfty13GF$F*F.F*F0*,F)F*F+F*F,F/ F5F*F2F*F3*,F)F*F+F*F,F/F.F*I)tinfty23GF$F*F0*,F)F*F+F*F,F/F8F*F2F*F3* (I$nu2GF$F*F,F-F.F/F3*(F;F*F,F-F2F/F0*,F)F*F+F*F,F*I)tinfty12GF$F*F.F* F0*,F)F*F+F*F,F*F>F*F2F*F3*,F)F*F+F*F,F*F.F*I)tinfty22GF$F*F0*,F)F*F+F *F,F*FAF*F2F*F3**F;F*F,F/F5F*F.F*F3**F;F*F,F/F5F*F2F*F0**F;F*F,F/F.F*F 8F*F3**F;F*F,F/F8F*F2F*F0**I(alpha14GF$F*I\"hGF$F*F+F*F,F*F**(FHF*FIF* F,F/!\"\"**I(alpha24GF$F*FIF*F+F*F,F*FK*(FMF*FIF*F,F/F***FIF*I#nuGF$F* F+F*F.F*F3**FIF*FPF*F+F*F2F*F0**FIF*FPF*F,F*F.F*F0**FIF*FPF*F,F*F2F*F3 **F)F*F+F*I)tinfty11GF$F*F.F*F0**F)F*F+F*FUF*F2F*F3**F)F*F+F*F.F*I)tin fty21GF$F*F0**F)F*F+F*FXF*F2F*F3**F;F*F,F*F>F*F.F*F3**F;F*F,F*F>F*F2F* F0**F;F*F,F*F.F*FAF*F3**F;F*F,F*FAF*F2F*F0*(FIF*F)F*F.F*F0*(FIF*F)F*F2 F*F3*(F;F*FUF*F.F*F3*(F;F*FUF*F2F*F0*(F;F*F.F*FXF*F3*(F;F*FXF*F2F*F0*( F+F*I%rho2GF$F*F.F*\"\")*(F+F*F_oF*F2F*!\")*(F,F*F_oF*F.F*Fbo*(F,F*F_o F*F2F*F`oF*FIF*,&F.F*F2FKFK,&F+F*F,FKFK#FKF3" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9Entry22TermLambdaMinusq2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22TermLambdaInfty5G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22TermLambdaInfty4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22TermLambdaInfty3G6\",$*&,&I(alpha14GF$\"\"\"I(a lpha24GF$F)F)I\"hGF$F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry2 2TermLambdaInfty2G6\",$*(I\"hGF$\"\"\",2*&I#nuGF$F(I)tinfty14GF$\"\"# \"\"%*&F+F(I)tinfty24GF$F-!\"%*&I(alpha14GF$F(I)tinfty13GF$F(F(*&F3F(I )tinfty23GF$F(F(*&I(alpha24GF$F(F4F(!\"\"*&F8F(F6F(F9*&I#c3GF$F(F,F(! \")*&FI8Entry22TermLambdaInfty1G6\",$*(I\"hGF$\"\"\",:*(I#nuGF$F(I)tinf ty13GF$F(I)tinfty14GF$F(\"\"%*(F+F(F,F(I)tinfty24GF$F(!\"%*(F+F(F-F(I) tinfty23GF$F(F.*(F+F(F3F(F0F(F1*&I$nu1GF$F(F-\"\"#F.*&F6F(F0F7F1*&I(al pha14GF$F(I)tinfty12GF$F(F(*&F:F(I)tinfty22GF$F(F(*&I(alpha24GF$F(F;F( !\"\"*&F?F(F=F(F@*&I#c2GF$F(F-F(!\")*&FCF(F0F(\"\")F(,&F-F(F0F@F@#F@F7 " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22TermLambdaInfty0G6\",$*(I \"hGF$\"\"\",B*(I#nuGF$F(I)tinfty12GF$F(I)tinfty14GF$F(\"\"%*(F+F(F,F( I)tinfty24GF$F(!\"%*(F+F(F-F(I)tinfty22GF$F(F.*(F+F(F3F(F0F(F1*(I$nu1G F$F(I)tinfty13GF$F(F-F(F.*(F6F(F7F(F0F(F1*(F6F(F-F(I)tinfty23GF$F(F.*( F6F(F:F(F0F(F1*&I$nu2GF$F(F-\"\"#F.*&F=F(F0F>F1*&I(alpha14GF$F(I)tinft y11GF$F(F(*&FAF(I)tinfty21GF$F(F(*&I(alpha24GF$F(FBF(!\"\"*&FFF(FDF(FG *&I#c1GF$F(F-F(!\")*&FJF(F0F(\"\")F(,&F-F(F0FGFG#FGF." }}{PARA 11 "" 1 "" {XPPMATH 20 ">I=Entry22TermLambdaInftyMinus1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0I)tinfty 11G6\"!\"\"I)tinfty21GF$F%*&,&I)tinfty12GF$F%I)tinfty22GF$F%\"\"\"I'la mbdaGF$F+F+*&,&I)tinfty13GF$F%I)tinfty23GF$F%F+F,\"\"#F+*&,&I)tinfty14 GF$F%I)tinfty24GF$F%F+F,\"\"$F+*&I\"hGF$F+,&F,F+I#q1GF$F%F%F+*&F8F+,&F ,F+I#q2GF$F%F%F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 238 263 "Since the d eformation operator is \\hbar (alpha14\\partial_\{t_\{\\infty^\{(1)\} ,4\} +alpha24\\partial_\{t_\{\\infty^\{(2)\},4\}+alpha13*\\partial_\{t _\{\\infty^\{(1)\},3\} +alpha23*\\partial_\{t_\{\\infty^\{(2)\},3\}+al pha12\\partial_\{t_\{\\infty^\{(1)\},2\} +alpha22*\\partial_\{t_\{\\in fty^\{(2)\},2\}" }{TEXT 238 75 "+alpha11*\\partial_\{t_\{\\infty^\{(1) \},1\} +alpha21*\\partial_\{t_\{\\infty^\{(2)\},1\}" }{TEXT 238 40 ") \+ we can compute its action on L_\{2,2\}. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "L22OrderLambda4:=factor(-residue(L[2,2]/lambda^5,lamb da=infinity));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 68 "L22OrderLambda3 :=factor(-residue(L[2,2]/lambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 68 "L22OrderLambda2:=factor(-residue(L[2,2]/lambda^3,lambda=infinity)) :\n" }{MPLTEXT 1 0 68 "L22OrderLambda1:=factor(-residue(L[2,2]/lambda^ 2,lambda=infinity)):\n" }{MPLTEXT 1 0 68 "L22OrderLambda0:=factor(-res idue(L[2,2]/lambda^1,lambda=infinity));\n" }{MPLTEXT 1 0 244 "simplify (h*(alpha14*diff(L22OrderLambda3,tinfty14)+alpha24*diff(L22OrderLambda 3,tinfty24)+alpha13*diff(L22OrderLambda3,tinfty13)+alpha23*diff(L22Ord erLambda3,tinfty23)+alpha12diff(L22OrderLambda3,tinfty12)+alpha22*diff (L22OrderLambda3,tinfty22)" }{MPLTEXT 1 0 78 "+alpha11*diff(L22OrderLa mbda3,tinfty11)+alpha21*diff(L22OrderLambda3,tinfty21)" }{MPLTEXT 1 0 29 ")- Entry22TermLambdaInfty3);\n" }{MPLTEXT 1 0 18 "Equation1:=facto r(" }{MPLTEXT 1 0 323 "simplify(h*(alpha14*diff(L22OrderLambda2,tinfty 14)+alpha24*diff(L22OrderLambda2,tinfty24)+alpha13*diff(L22OrderLambda 2,tinfty13)+alpha23*diff(L22OrderLambda2,tinfty23)+alpha12*diff(L22Ord erLambda2,tinfty12)+alpha22*diff(L22OrderLambda2,tinfty22)+alpha11*dif f(L22OrderLambda2,tinfty11)+alpha21*diff(L22OrderLambda2,tinfty21)" } {MPLTEXT 1 0 27 ")- Entry22TermLambdaInfty2)" }{MPLTEXT 1 0 3 ");\n" } {MPLTEXT 1 0 11 "Equation2:=" }{MPLTEXT 1 0 7 "factor(" }{MPLTEXT 1 0 350 "simplify(h*(alpha14*diff(L22OrderLambda1,tinfty14)+alpha24*diff(L 22OrderLambda1,tinfty24)+alpha13*diff(L22OrderLambda1,tinfty13)+alpha2 3*diff(L22OrderLambda1,tinfty23)+alpha12*diff(L22OrderLambda1,tinfty12 )+alpha22*diff(L22OrderLambda1,tinfty22)+alpha11*diff(L22OrderLambda1, tinfty11)+alpha21*diff(L22OrderLambda1,tinfty21))- Entry22TermLambdaIn fty1)" }{MPLTEXT 1 0 2 ");" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 370 "Eq uation3:=factor(simplify(h*(alpha14*diff(L22OrderLambda0,tinfty14)+alp ha24*diff(L22OrderLambda0,tinfty24)+alpha13*diff(L22OrderLambda0,tinft y13)+alpha23*diff(L22OrderLambda0,tinfty23)+alpha12*diff(L22OrderLambd a0,tinfty12)+alpha22*diff(L22OrderLambda0,tinfty22)+alpha11*diff(L22Or derLambda0,tinfty11)+alpha21*diff(L22OrderLambda0,tinfty21))- Entry22T ermLambdaInfty0));" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L22OrderLambda4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L2 2OrderLambda3G6\",&I)tinfty14GF$!\"\"I)tinfty24GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L22OrderLambda0G6\",&I)tinfty11GF$!\"\"I)tinfty21GF $F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&I\"hG6\"\"\"\"-I,alpha12diffGF$ 6$,&I)tinfty14GF$!\"\"I)tinfty24GF$F+I)tinfty12GF$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation1G6\",$*(I\"hGF$\"\"\",:*&I#nuGF$F(I)tinfty 14GF$\"\"#!#7*&F+F(I)tinfty24GF$F-\"#7*&I(alpha13GF$F(F,F(\"\"%*&F3F(F 0F(!\"%*&I(alpha14GF$F(I)tinfty13GF$F(!\"$*&F8F(I)tinfty23GF$F(F:*&I(a lpha23GF$F(F,F(F4*&F>F(F0F(F6*&I(alpha24GF$F(F9F(\"\"$*&FAF(FI*Equation2G6\",$*(I\"hGF$\"\"\",B*(I#nuGF$F(I)tinfty13 GF$F(I)tinfty14GF$F(!\"%*(F+F(F,F(I)tinfty24GF$F(\"\"%*(F+F(F-F(I)tinf ty23GF$F(F.*(F+F(F3F(F0F(F1*&I$nu1GF$F(F-\"\"#F.*&F6F(F0F7F1*&I(alpha1 2GF$F(F-F(F7*&F:F(F0F(!\"#*&I(alpha14GF$F(I)tinfty12GF$F(!\"\"*&F>F(I) tinfty22GF$F(F@*&I(alpha22GF$F(F-F(F7*&FDF(F0F(F<*&I(alpha24GF$F(F?F(F (*&FGF(FBF(F(*&I#c2GF$F(F-F(\"\")*&FJF(F0F(!\")F(,&F-F(F0F@F@#F@F7" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation3G6\",$*(I\"hGF$\"\"\",J*(I#n uGF$F(I)tinfty12GF$F(I)tinfty14GF$F(!\"%*(F+F(F,F(I)tinfty24GF$F(\"\"% *(F+F(F-F(I)tinfty22GF$F(F.*(F+F(F3F(F0F(F1*(I$nu1GF$F(I)tinfty13GF$F( F-F(F.*(F6F(F7F(F0F(F1*(F6F(F-F(I)tinfty23GF$F(F.*(F6F(F:F(F0F(F1*&I$n u2GF$F(F-\"\"#F.*&F=F(F0F>F1*&I(alpha11GF$F(F-F(F1*&FAF(F0F(F.*&I(alph a14GF$F(I)tinfty11GF$F(!\"\"*&FDF(I)tinfty21GF$F(FF*&I(alpha21GF$F(F-F (F1*&FJF(F0F(F.*&I(alpha24GF$F(FEF(F(*&FMF(FHF(F(*&I#c1GF$F(F-F(\"\")* &FPF(F0F(!\")F(,&F-F(F0FFFF#FFF1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Lq1:=factor(Entry22TermLambdaMinusq1Square/h):\n" }{MPLTEXT 1 0 46 "Lq2:=factor(Entry22TermLambdaMinusq2Square/h):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 8 "Lq1bis:=" }{MPLTEXT 1 0 14 "-2*rho1-P1(q1)" } {MPLTEXT 1 0 4 "*mu1" }{MPLTEXT 1 0 19 "-nu*h-nu1*h/(q1-q2)" }{MPLTEXT 1 0 47 "-h*(alpha14-alpha24)/4/(tinfty14-tinfty24)*q1;\n" }{MPLTEXT 1 0 8 "Lq2bis:=" }{MPLTEXT 1 0 37 "-2*rho2-P1(q2)*mu2-nu*h+nu1*h/(q1-q 2)" }{MPLTEXT 1 0 46 "-h*(alpha14-alpha24)/4/(tinfty14-tinfty24)*q2;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "factor(simplify((Lq1-Lq1bis))); \n" }{MPLTEXT 1 0 31 "factor(simplify((Lq2-Lq2bis)));" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'Lq1bisG6\",,I%rho1GF$!\"#*(, ,I)tinfty11GF$!\"\"I)tinfty21GF$F+*&,&I)tinfty12GF$F+I)tinfty22GF$F+\" \"\"I#q1GF$F1F1*&,&I)tinfty13GF$F+I)tinfty23GF$F+F1F2\"\"#F1*&,&I)tinf ty14GF$F+I)tinfty24GF$F+F1F2\"\"$F1F1,**(I$nu1GF$F1I#q2GF$F1F:F1!\"%*( F?F1F@F1F;F1\"\"%*&I$nu2GF$F1F:F1FC*&FEF1F;F1FAF1,**&F2F1F:F1FC*&F2F1F ;F1FA*&F@F1F:F1FA*&F@F1F;F1FCF+F+*&I\"hGF$F1I#nuGF$F1F+*(F?F1FMF1,&F2F 1F@F+F+F+**FMF1,&I(alpha14GF$F1I(alpha24GF$F+F1,&F:F1F;F+F+F2F1#F+FC" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'Lq2bisG6\",,I%rho2GF$!\"#*(,,I)tinf ty11GF$!\"\"I)tinfty21GF$F+*&,&I)tinfty12GF$F+I)tinfty22GF$F+\"\"\"I#q 2GF$F1F1*&,&I)tinfty13GF$F+I)tinfty23GF$F+F1F2\"\"#F1*&,&I)tinfty14GF$ F+I)tinfty24GF$F+F1F2\"\"$F1F1,**(I$nu1GF$F1I#q1GF$F1F:F1!\"%*(F?F1F@F 1F;F1\"\"%*&I$nu2GF$F1F:F1FC*&FEF1F;F1FAF1,**&F@F1F:F1FA*&F@F1F;F1FC*& F2F1F:F1FC*&F2F1F;F1FAF+F+*&I\"hGF$F1I#nuGF$F1F+*(F?F1FMF1,&F@F1F2F+F+ F1**FMF1,&I(alpha14GF$F1I(alpha24GF$F+F1,&F:F1F;F+F+F2F1#F+FC" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 238 38 "Let us now look at \\ma thcal\{L\}[L[2,1]]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Entry 21:=simplify(LL[2,1]):\n" }{MPLTEXT 1 0 135 "simplify(Entry21-(h^2*dif f(A[1,1],lambda$2)+2*h*L[2,1]*diff(A[1,2],lambda)+h*A[1,2]*diff(L[2,1] ,lambda)-h*L[2,2]*diff(A[1,1],lambda)));" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 80 "Entry21TermLambdaMinusq1Cube:=factor(residue(Entry21* (lambda-q1)^2,lambda=q1));\n" }{MPLTEXT 1 0 80 "Entry21TermLambdaMinus q1Square:=factor(residue(Entry21*(lambda-q1),lambda=q1)):\n" }{MPLTEXT 1 0 62 "Entry21TermLambdaMinusq1:=factor(residue(Entry21,lambda=q1)): \n" }{MPLTEXT 1 0 80 "Entry21TermLambdaMinusq2Cube:=factor(residue(Ent ry21*(lambda-q2)^2,lambda=q2));\n" }{MPLTEXT 1 0 80 "Entry21TermLambda Minusq2Square:=factor(residue(Entry21*(lambda-q2),lambda=q2)):\n" } {MPLTEXT 1 0 61 "Entry21TermLambdaMinusq2:=factor(residue(Entry21,lamb da=q2)):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 76 " Entry21TermLambdaInfty7:=factor(-residue(Entry21/lambda^8,lambda=infin ity));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty6 :=factor(-residue(Entry21/lambda^7,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty5:=factor(-residue(Entry21/lambda^6,lambda =infinity));\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty4:=factor(-res idue(Entry21/lambda^5,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry21T ermLambdaInfty3:=factor(-residue(Entry21/lambda^4,lambda=infinity)):\n " }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty2:=factor(-residue(Entry21/l ambda^3,lambda=infinity)):\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty 1:=factor(-residue(Entry21/lambda^2,lambda=infinity)):\n" }{MPLTEXT 1 0 75 "Entry21TermLambdaInfty0:=factor(-residue(Entry21/lambda,lambda=i nfinity)):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 144 "simplify( Entry2 1-(Entry21TermLambdaMinusq1Cube/(lambda-q1)^3+Entry21TermLambdaMinusq1 Square/(lambda-q1)^2+Entry21TermLambdaMinusq1/(lambda-q1)\n" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 124 "Entry21TermLambdaMinusq2Cube/(lambda-q2) ^3+Entry21TermLambdaMinusq2Square/(lambda-q2)^2+Entry21TermLambdaMinus q2/(lambda-q2)" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 122 "+Entry21TermLa mbdaInfty0+Entry21TermLambdaInfty1*lambda+Entry21TermLambdaInfty2*lamb da^2+Entry21TermLambdaInfty3*lambda^3\n" }{MPLTEXT 1 0 100 "+Entry21Te rmLambdaInfty4*lambda^4+Entry21TermLambdaInfty5*lambda^5+Entry21TermLa mbdaInfty6*lambda^6\n" }{MPLTEXT 1 0 33 "+Entry21TermLambdaInfty7*lamb da^7" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 5 ") );\n" }{MPLTEXT 1 0 8 "L [2,1];\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I=Entry21TermLambdaMinusq1CubeG6\",$* (,**(I$nu1GF$\"\"\"I#p1GF$F*I#q2GF$F*F**&I$nu2GF$F*F+F*!\"\"*&I#q1GF$F *I%rho1GF$F*F/*&F,F*F2F*F*F*I\"hGF$\"\"#,&F1F*F,F/F/!\"$" }}{PARA 11 " " 1 "" {XPPMATH 20 ">I=Entry21TermLambdaMinusq2CubeG6\",$*(,**(I$nu1GF $\"\"\"I#p2GF$F*I#q1GF$F*F**&I$nu2GF$F*F+F*!\"\"*&F,F*I%rho2GF$F*F**&I #q2GF$F*F1F*F/F*I\"hGF$\"\"#,&F,F*F3F/F/\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty7G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty6G6\",$*&I\"hGF$\"\"\",&*&I(alph a14GF$F(I)tinfty24GF$F(F(*&I(alpha24GF$F(I)tinfty14GF$F(F(F(!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty5G6\",$*(I\"hGF $\"\"\",6*(I#nuGF$F(I)tinfty14GF$\"\"#I)tinfty24GF$F(\"#C*(F+F(F,F(F.F -!#C*(I(alpha14GF$F(I)tinfty13GF$F(F.F(\"\"$*(F3F(F,F(I)tinfty23GF$F( \"\"(*(F3F(F7F(F.F(!\"%*(I(alpha24GF$F(F4F(F,F(\"\"%*(FI8Entry21TermLambdaInfty4G6\",$*( I\"hGF$\"\"\",J**I#nuGF$F(I)tinfty13GF$F(I)tinfty14GF$F(I)tinfty24GF$F (\"#5*(F+F(F,F(F.\"\"#!#5*(F+F(F-F1I)tinfty23GF$F(F/**F+F(F-F(F4F(F.F( F2*(I$nu1GF$F(F-F1F.F(\"\")*(F7F(F-F(F.F1!\")*(I(alpha14GF$F(I)tinfty1 2GF$F(F.F(F(*(F " 0 "" {MPLTEXT 1 0 55 "rho1:=factor(solve(E ntry21TermLambdaMinusq1Cube,rho1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 40 "simplify(Entry21TermLambdaMinusq1Cube);\n" }{MPLTEXT 1 0 18 "rho 1bis:=-p1*mu1:\n" }{MPLTEXT 1 0 24 "simplify(rho1-rho1bis);\n" } {MPLTEXT 1 0 56 "rho2:=factor(solve(Entry21TermLambdaMinusq2Cube,rho2) );\n" }{MPLTEXT 1 0 39 "simplify(Entry21TermLambdaMinusq2Cube);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "rho2bis:=-p2*mu2:\n" }{MPLTEXT 1 0 23 "simplify(rho2-rho2bis);" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%rho1G6\"*(I#p1GF$\"\"\",&*&I$nu1GF$F'I#q2GF$F'F'I$n u2GF$!\"\"F',&I#q1GF$F'F+F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%rho2G6\",$*(I#p2GF$\"\"\",&*&I$nu1GF$F(I#q1GF$F(F(I$nu2GF$!\"\"F(, &F,F(I#q2GF$F.F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "L21OrderLambda7:=-residue(L[2,1]/lambda^8,lambda=infinity);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 59 "L21OrderLambda6:=-residue(L[2,1] /lambda^7,lambda=infinity);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 59 "L2 1OrderLambda5:=-residue(L[2,1]/lambda^6,lambda=infinity);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 60 "L21OrderLambda4:=-residue(L[2,1]/lambda^5 ,lambda=infinity):\n" }{MPLTEXT 1 0 60 "L21OrderLambda3:=-residue(L[2, 1]/lambda^4,lambda=infinity):\n" }{MPLTEXT 1 0 60 "L21OrderLambda2:=-r esidue(L[2,1]/lambda^3,lambda=infinity):\n" }{MPLTEXT 1 0 60 "L21Order Lambda1:=-residue(L[2,1]/lambda^2,lambda=infinity):\n" }{MPLTEXT 1 0 60 "L21OrderLambda0:=-residue(L[2,1]/lambda^1,lambda=infinity):\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 322 "simplify(h *(alpha14*diff(L21OrderLambda6,tinfty14)+alpha24*diff(L21OrderLambda6, tinfty24)+alpha13*diff(L21OrderLambda6,tinfty13)+alpha23*diff(L21Order Lambda6,tinfty23)+alpha12diff(L21OrderLambda6,tinfty12)+alpha22*diff(L 21OrderLambda6,tinfty22)+alpha11*diff(L21OrderLambda6,tinfty11)+alpha2 1*diff(L21OrderLambda6,tinfty21)" }{MPLTEXT 1 0 28 ")- Entry21TermLamb daInfty6);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "Equation4:=factor(" }{MPLTEXT 1 0 323 "simplify(h*(alpha14*diff(L2 1OrderLambda5,tinfty14)+alpha24*diff(L21OrderLambda5,tinfty24)+alpha13 *diff(L21OrderLambda5,tinfty13)+alpha23*diff(L21OrderLambda5,tinfty23) +alpha12*diff(L21OrderLambda5,tinfty12)+alpha22*diff(L21OrderLambda5,t infty22)+alpha11*diff(L21OrderLambda5,tinfty11)+alpha21*diff(L21OrderL ambda5,tinfty21)" }{MPLTEXT 1 0 30 ")- Entry21TermLambdaInfty5));\n" } {MPLTEXT 1 0 18 "Equation5:=factor(" }{MPLTEXT 1 0 323 "simplify(h*(al pha14*diff(L21OrderLambda4,tinfty14)+alpha24*diff(L21OrderLambda4,tinf ty24)+alpha13*diff(L21OrderLambda4,tinfty13)+alpha23*diff(L21OrderLamb da4,tinfty23)+alpha12*diff(L21OrderLambda4,tinfty12)+alpha22*diff(L21O rderLambda4,tinfty22)+alpha11*diff(L21OrderLambda4,tinfty11)+alpha21*d iff(L21OrderLambda4,tinfty21)" }{MPLTEXT 1 0 29 ")- Entry21TermLambdaI nfty4));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "Equation6:=factor(" } {MPLTEXT 1 0 323 "simplify(h*(alpha14*diff(L21OrderLambda3,tinfty14)+a lpha24*diff(L21OrderLambda3,tinfty24)+alpha13*diff(L21OrderLambda3,tin fty13)+alpha23*diff(L21OrderLambda3,tinfty23)+alpha12*diff(L21OrderLam bda3,tinfty12)+alpha22*diff(L21OrderLambda3,tinfty22)+alpha11*diff(L21 OrderLambda3,tinfty11)+alpha21*diff(L21OrderLambda3,tinfty21)" } {MPLTEXT 1 0 27 ")- Entry21TermLambdaInfty3)" }{MPLTEXT 1 0 3 ");\n" } {MPLTEXT 1 0 368 "Equation7:=factor(simplify(h*(alpha14*diff(L21OrderL ambda2,tinfty14)+alpha24*diff(L21OrderLambda2,tinfty24)+alpha13*diff(L 21OrderLambda2,tinfty13)+alpha23*diff(L21OrderLambda2,tinfty23)+alpha1 2*diff(L21OrderLambda2,tinfty12)+alpha22*diff(L21OrderLambda2,tinfty22 )+alpha11*diff(L21OrderLambda2,tinfty11)+alpha21*diff(L21OrderLambda2, tinfty21))- Entry21TermLambdaInfty2)" }{MPLTEXT 1 0 2 ");" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 " " {XPPMATH 20 ">I0L21OrderLambda7G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L21OrderLambda6G6\",$*&I)tinfty14GF$\"\"\"I)tinfty24GF $F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L21OrderLambda5G6\",&*&I) tinfty13GF$\"\"\"I)tinfty24GF$F(!\"\"*&I)tinfty14GF$F(I)tinfty23GF$F(F *" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&-I,alpha12diffG6\"6$,$*&I)tinfty1 4GF%\"\"\"I)tinfty24GF%F*!\"\"I)tinfty12GF%F*I\"hGF%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation4G6\",$*(I\"hGF$\"\"\",:*(I#nuGF$F(I)tinf ty14GF$\"\"#I)tinfty24GF$F(!#C*(F+F(F,F(F.F-\"#C*(I(alpha13GF$F(F,F(F. F(\"\"%*&F3F(F.F-!\"%*(I(alpha14GF$F(I)tinfty13GF$F(F.F(!\"$*(F8F(F,F( I)tinfty23GF$F(F:*&I(alpha23GF$F(F,F-F4*(F>F(F,F(F.F(F6*(I(alpha24GF$F (F9F(F.F(\"\"$*(FAF(F,F(FI*Equation5G6\",$*(I \"hGF$\"\"\",V**I#nuGF$F(I)tinfty13GF$F(I)tinfty14GF$F(I)tinfty24GF$F( !#5*(F+F(F,F(F.\"\"#\"#5*(F+F(F-F1I)tinfty23GF$F(F/**F+F(F-F(F4F(F.F(F 2*(I$nu1GF$F(F-F1F.F(!\")*(F7F(F-F(F.F1\"\")*(I(alpha12GF$F(F-F(F.F(F1 *&F*(I(alpha14GF$ F(I)tinfty12GF$F(F.F(!\"\"*(FCF(F,F(F4F(!\"$*(FCF(F-F(I)tinfty22GF$F(F E*&I(alpha22GF$F(F-F1F1*(FKF(F-F(F.F(F>*(I(alpha23GF$F(F,F(F-F(F1*(FNF (F,F(F.F(F>*(I(alpha24GF$F(FDF(F.F(F(*(FQF(F,F(F4F(\"\"$*(FQF(F-F(FIF( F(*&I#c2GF$F(F-F1\"\"%*&FVF(F.F1!\"%*(I#c3GF$F(F,F(F-F(\"\"'*(FenF(F,F (F.F(!\"'*(FenF(F-F(F4F(Ffn*(FenF(F4F(F.F(FhnF(,&F-F(F.FEFE#FEF1" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation6G6\",$*(I\"hGF$\"\"\",`p**I# nuGF$F(I)tinfty12GF$F(I)tinfty14GF$F(I)tinfty24GF$F(!#;*(F+F(F,F(F.\" \"#\"#;**F+F(I)tinfty13GF$F(F-F(I)tinfty23GF$F(F/**F+F(F4F(F5F(F.F(F2* (F+F(F-F1I)tinfty22GF$F(F/**F+F(F-F(F8F(F.F(F2**I$nu1GF$F(F4F(F-F(F.F( !#7*(F;F(F4F(F.F1\"#7*(F;F(F-F1F5F(F<**F;F(F-F(F5F(F.F(F>*(I$nu2GF$F(F -F1F.F(!\")*(FBF(F-F(F.F1\"\")*(I(alpha11GF$F(F-F(F.F(\"\"%*&FGF(F.F1! \"%*(I(alpha12GF$F(F-F(F5F(FH*(FLF(F5F(F.F(FJ*(I(alpha13GF$F(F-F(F8F(F H*(FOF(F8F(F.F(FJ*(I(alpha14GF$F(I)tinfty11GF$F(F.F(!\"\"*(FRF(F,F(F5F (!\"&*(FRF(F4F(F8F(FV*(FRF(F-F(I)tinfty21GF$F(FT*&I(alpha21GF$F(F-F1FH *(FenF(F-F(F.F(FJ*(I(alpha22GF$F(F4F(F-F(FH*(FhnF(F4F(F.F(FJ*(I(alpha2 3GF$F(F,F(F-F(FH*(F[oF(F,F(F.F(FJ*(I(alpha24GF$F(FSF(F.F(F(*(F^oF(F,F( F5F(\"\"&*(F^oF(F4F(F8F(F`o*(F^oF(F-F(FYF(F(*&I#c1GF$F(F-F1FH*&FdoF(F. F1FJ*(I#c2GF$F(F4F(F-F(FE*(FgoF(F4F(F.F(FC*(FgoF(F-F(F5F(FE*(FgoF(F5F( F.F(FC*(I#c3GF$F(F,F(F-F(F>*(F\\pF(F,F(F.F(F<*(F\\pF(F-F(F8F(F>*(F\\pF (F8F(F.F(FI*Equ ation7G6\",$*(I\"hGF$\"\"\",\\q**I#nuGF$F(I)tinfty11GF$F(I)tinfty14GF$ F(I)tinfty24GF$F(!\"$*(F+F(F,F(F.\"\"#\"\"$**F+F(I)tinfty12GF$F(F-F(I) tinfty23GF$F(F/**F+F(F4F(F5F(F.F(F2**F+F(I)tinfty13GF$F(F-F(I)tinfty22 GF$F(F/**F+F(F8F(F9F(F.F(F2*(F+F(F-F1I)tinfty21GF$F(F/**F+F(F-F(F " 0 "" {MPLTEXT 1 0 57 "Lp1Function:=unapply(-Entry21TermLambdaMinusq1/h,C0,C 1):\n" }{MPLTEXT 1 0 56 "Lp2Function:=unapply(-Entry21TermLambdaMinusq 2/h,C0,C1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Equation8:=s implify(Entry21TermLambdaMinusq1Square-(-p1*h*Lq1)):\n" }{MPLTEXT 1 0 64 "Equation9:=simplify(Entry21TermLambdaMinusq2Square-(-p2*h*Lq2)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1153 "C0:= -(q1^6*q2*tinfty14 *tinfty24-q1*q2^6*tinfty14*tinfty24+q1^5*q2*tinfty13*tinfty24+q1^5*q2* tinfty14*tinfty23-q1*q2^5*tinfty13*tinfty24-q1*q2^5*tinfty14*tinfty23+ q1^4*q2*tinfty12*tinfty24+q1^4*q2*tinfty13*tinfty23+q1^4*q2*tinfty14*t infty22-q1*q2^4*tinfty12*tinfty24-q1*q2^4*tinfty13*tinfty23-q1*q2^4*ti nfty14*tinfty22+p1*q1^3*q2*tinfty14+p1*q1^3*q2*tinfty24-p2*q1*q2^3*tin fty14-p2*q1*q2^3*tinfty24+q1^3*q2*tinfty11*tinfty24+q1^3*q2*tinfty12*t infty23+q1^3*q2*tinfty13*tinfty22+q1^3*q2*tinfty14*tinfty21-q1*q2^3*ti nfty11*tinfty24-q1*q2^3*tinfty12*tinfty23-q1*q2^3*tinfty13*tinfty22-q1 *q2^3*tinfty14*tinfty21+h*q1^2*q2*tinfty14-h*q1*q2^2*tinfty14+p1*q1^2* q2*tinfty13+p1*q1^2*q2*tinfty23-p2*q1*q2^2*tinfty13-p2*q1*q2^2*tinfty2 3-q1^2*q2*tinfty10*tinfty14+q1^2*q2*tinfty10*tinfty24+q1^2*q2*tinfty11 *tinfty23+q1^2*q2*tinfty12*tinfty22+q1^2*q2*tinfty13*tinfty21+q1*q2^2* tinfty10*tinfty14-q1*q2^2*tinfty10*tinfty24-q1*q2^2*tinfty11*tinfty23- q1*q2^2*tinfty12*tinfty22-q1*q2^2*tinfty13*tinfty21+p1*q1*q2*tinfty12+ p1*q1*q2*tinfty22-p2*q1*q2*tinfty12-p2*q1*q2*tinfty22+p1^2*q2+p1*q2*ti nfty11+p1*q2*tinfty21-p2^2*q1-p2*q1*tinfty11-p2*q1*tinfty21+h*p1-h*p2) /(q1-q2):\n" }{MPLTEXT 1 0 991 "C1:=(q1^6*tinfty14*tinfty24-q2^6*tinft y14*tinfty24+q1^5*tinfty13*tinfty24+q1^5*tinfty14*tinfty23-q2^5*tinfty 13*tinfty24-q2^5*tinfty14*tinfty23+q1^4*tinfty12*tinfty24+q1^4*tinfty1 3*tinfty23+q1^4*tinfty14*tinfty22-q2^4*tinfty12*tinfty24-q2^4*tinfty13 *tinfty23-q2^4*tinfty14*tinfty22+p1*q1^3*tinfty14+p1*q1^3*tinfty24-p2* q2^3*tinfty14-p2*q2^3*tinfty24+q1^3*tinfty11*tinfty24+q1^3*tinfty12*ti nfty23+q1^3*tinfty13*tinfty22+q1^3*tinfty14*tinfty21-q2^3*tinfty11*tin fty24-q2^3*tinfty12*tinfty23-q2^3*tinfty13*tinfty22-q2^3*tinfty14*tinf ty21+h*q1^2*tinfty14-h*q2^2*tinfty14+p1*q1^2*tinfty13+p1*q1^2*tinfty23 -p2*q2^2*tinfty13-p2*q2^2*tinfty23-q1^2*tinfty10*tinfty14+q1^2*tinfty1 0*tinfty24+q1^2*tinfty11*tinfty23+q1^2*tinfty12*tinfty22+q1^2*tinfty13 *tinfty21+q2^2*tinfty10*tinfty14-q2^2*tinfty10*tinfty24-q2^2*tinfty11* tinfty23-q2^2*tinfty12*tinfty22-q2^2*tinfty13*tinfty21+p1*q1*tinfty12+ p1*q1*tinfty22-p2*q2*tinfty12-p2*q2*tinfty22+p1^2+p1*tinfty11+p1*tinft y21-p2^2-p2*tinfty11-p2*tinfty21)/(q1-q2):\n" }{MPLTEXT 1 0 21 "simpli fy(Equation8);\n" }{MPLTEXT 1 0 21 "simplify(Equation9);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 58 "C0bis:=( q1*(p2^2-P1(q2)*p2)-q2*(p1^2-p1* P1(q1)))/(q1-q2)\n" }{MPLTEXT 1 0 53 "-(q2*(P2(q1)-Pinfty02)-q1*(P2(q2 )-Pinfty02))/(q1-q2)\n" }{MPLTEXT 1 0 37 "-h*(p1-p2)/(q1-q2)-h*tinfty1 4*q1*q2:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 58 "C0ter:=( q1*(p2^2-P 1(q2)*p2)-q2*(p1^2-p1*P1(q1)))/(q1-q2)\n" }{MPLTEXT 1 0 15 "+P2(q1)+P2 (q2)\n" }{MPLTEXT 1 0 40 "-(q1*P2(q1)-q2*P2(q2))/(q1-q2)-Pinfty02\n" } {MPLTEXT 1 0 37 "-h*(p1-p2)/(q1-q2)-h*tinfty14*q1*q2:\n" }{MPLTEXT 1 0 18 "factor(C0-C0bis);\n" }{MPLTEXT 1 0 18 "factor(C0-C0ter);\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 97 "C1bis:=(p1^2-p1*P1(q1)+P2(q1) - \+ (p2^2 -p2*P1(q2) +P2(q2)))/(q1-q2) -Pinfty12+(q1+q2)*tinfty14*h:\n" } {MPLTEXT 1 0 30 "factor(series(C1-C1bis,p1=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "mu1:=simplify(mu1);\n" }{MPLTEXT 1 0 5 "mu2:=" } {MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 5 "mu2);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$mu1G6\"*&,&*&I$nu1GF$\"\"\"I#q2GF$F)!\"\"I$nu2GF$F)F), &I#q1GF$F)F*F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$mu2G6\"*&,&*&I$nu 1GF$\"\"\"I#q1GF$F)F)I$nu2GF$!\"\"F),&F*F)I#q2GF$F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Lp1:=factor(simplify(Lp1Function(C0 bis,C1bis))):\n" }{MPLTEXT 1 0 48 "Lp2:=factor(simplify(Lp2Function(C0 bis,C1bis))):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 77 "Lp1bis:=mu1*(p1^ 2-p1*P1(q1)-p2^2 +p2*P1(q2))/(q1-q2) +mu1*diff(P1(q1),q1)*p1\n" } {MPLTEXT 1 0 113 "+q1^3*(tinfty24*alpha14-tinfty14*alpha24)*h/(tinfty1 4-tinfty24) +1/4*p1*h/(tinfty14-tinfty24)*(alpha14-alpha24)\n" } {MPLTEXT 1 0 73 "+h*(3*c3*q1^2+2*c2*q1+c1 - nu1*(p1-p2)/(q1-q2)^2 +tin fty14*(nu1*q2-nu2))\n" }{MPLTEXT 1 0 1 "-" }{MPLTEXT 1 0 46 "mu1*(q1-q 2)*diff((P2(q1)-P2(q2))/(q1-q2),q1):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 8 "Lp2bis:=" }{MPLTEXT 1 0 44 "mu2*(p1^2-p1*P1(q1)-p2^2 +p2*P1(q2 ))/(q1-q2)" }{MPLTEXT 1 0 26 " + mu2*diff(P1(q2),q2)*p2\n" }{MPLTEXT 1 0 113 "+q2^3*(tinfty24*alpha14-tinfty14*alpha24)*h/(tinfty14-tinfty2 4) +1/4*p2*h/(tinfty14-tinfty24)*(alpha14-alpha24)\n" }{MPLTEXT 1 0 73 "+h*(3*c3*q2^2+2*c2*q2+c1 + nu1*(p1-p2)/(q1-q2)^2 +tinfty14*(nu1*q1 -nu2))\n" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 44 "mu2*(q1-q2)*diff((P2(q 1)-P2(q2))/(q1-q2),q2)" }{MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "factor(Lp1-Lp1bis);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "factor(Lp2-Lp2bis);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 10 "Lp1ter:=h *" }{MPLTEXT 1 0 94 "(mu2+mu1)*(p2-p1)/(q2-q1)^2+ mu1*(diff(P1(q1),q1) *p1-diff(tdP2(q1),q1)+C1bis-2*h*tinfty14*q1)\n" }{MPLTEXT 1 0 3 "+h*" }{MPLTEXT 1 0 13 "nuinftyMinus1" }{MPLTEXT 1 0 3 "*p1" }{MPLTEXT 1 0 37 "+h*(4*c4*q1^3+3*c3*q1^2+2*c2*q1+c1):\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 104 "Lp2ter:=h*(mu1+mu2)*(p1-p2)/(q1-q2)^2+ mu2*(diff(P1( q2),q2)*p2-diff(tdP2(q2),q2)+C1bis-2*h*tinfty14*q2)\n" }{MPLTEXT 1 0 3 "+h*" }{MPLTEXT 1 0 13 "nuinftyMinus1" }{MPLTEXT 1 0 3 "*p2" } {MPLTEXT 1 0 4 "+h*(" }{MPLTEXT 1 0 10 "4*c4*q2^3+" }{MPLTEXT 1 0 22 " 3*c3*q2^2+2*c2*q2+c1):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 19 "factor(Lp1-Lp1ter);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "factor(Lp2-Lp2ter);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 93 "Lq1ter:=2*mu1*(p1-P1(q1)/2)-nu*h-nu1*h/(q1-q2)-h*(a lpha14-alpha24)/4/(tinfty14-tinfty24)*q1:\n" }{MPLTEXT 1 0 93 "Lq2ter: =2*mu2*(p2-P1(q2)/2)-nu*h+nu1*h/(q1-q2)-h*(alpha14-alpha24)/4/(tinfty1 4-tinfty24)*q2:\n" }{MPLTEXT 1 0 22 "simplify(Lq1-Lq1ter);\n" } {MPLTEXT 1 0 21 "simplify(Lq2-Lq2ter);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 86 "Hamiltonian:=-h*(mu1+mu2)*(p1-p2)/(q1-q2 ) -h*nu*(p1+p2)-h*nuinftyMinus1*(q1*p1+q2*p2)\n" }{MPLTEXT 1 0 47 "+mu 1*(p1^2-P1(q1)*p1+tdP2(q1)+h*tinfty14*q1^2)\n" }{MPLTEXT 1 0 47 "+mu2* (p2^2-P1(q2)*p2+tdP2(q2)+h*tinfty14*q2^2)\n" }{MPLTEXT 1 0 70 "-h*(c4* q1^4+c3*q1^3+c2*q1^2+c1*q1)-h*(c4*q2^4+c3*q2^3+c2*q2^2+c1*q2):\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 47 "factor(simplify(Lp1-(-diff(Hamil tonian,q1))));\n" }{MPLTEXT 1 0 38 "simplify(Lq1-(diff(Hamiltonian,p1) ));\n" }{MPLTEXT 1 0 47 "factor(simplify(Lp2-(-diff(Hamiltonian,q2)))) ;\n" }{MPLTEXT 1 0 37 "simplify(Lq2-(diff(Hamiltonian,p2)));" } {MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 238 70 "We have obtained the evolution eq uations for the Darboux coordinates:\n" }{TEXT 232 6 "L[q1]=" }{TEXT 233 83 "2*mu1*(p1-P1(q1)/2)-nu*h-nu1*h/(q1-q2)-h*(alpha14-alpha24)/4/( tinfty14-tinfty24)*q1" }{TEXT 234 1 "\n" }{TEXT 235 6 "L[q2]=" }{TEXT 236 83 "2*mu2*(p2-P1(q2)/2)-nu*h+nu1*h/(q1-q2)-h*(alpha14-alpha24)/4/( tinfty14-tinfty24)*q2" }{TEXT 237 1 "\n" }}{PARA 0 "" 0 "" {TEXT 212 8 "L[p1] = " }{TEXT 212 69 "mu1*(p1^2-p1*P1(q1)-p2^2 +p2*P1(q2))/(q1-q 2) +mu1*diff(P1(q1),q1)*p1\n" }{TEXT 212 45 "-mu1*(q1-q2)*diff((P2(q1) -P2(q2))/(q1-q2),q1)" }{TEXT 212 1 "\n" }{TEXT 212 113 "+q1^3*(tinfty2 4*alpha14-tinfty14*alpha24)*h/(tinfty14-tinfty24) +1/4*p1*h/(tinfty14 -tinfty24)*(alpha14-alpha24)\n" }{TEXT 212 73 "+h*(3*c3*q1^2+2*c2*q1+c 1 - nu1*(p1-p2)/(q1-q2)^2 +tinfty14*(nu1*q2-nu2))\n" }{TEXT 212 1 "\n" }{TEXT 212 6 "L[p2]=" }{TEXT 212 70 "mu2*(p1^2-p1*P1(q1)-p2^2 +p2*P1( q2))/(q1-q2) + mu2*diff(P1(q2),q2)*p2\n" }{TEXT 212 45 "+mu2*(q1-q2)*d iff((P2(q1)-P2(q2))/(q1-q2),q2)" }{TEXT 212 1 "\n" }{TEXT 212 113 "+q2 ^3*(tinfty24*alpha14-tinfty14*alpha24)*h/(tinfty14-tinfty24) +1/4*p2* h/(tinfty14-tinfty24)*(alpha14-alpha24)\n" }{TEXT 212 72 "+h*(3*c3*q2^ 2+2*c2*q2+c1 + nu1*(p1-p2)/(q1-q2)^2 +tinfty14*(nu1*q1-nu2))" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Equation1Function:=unapply(E quation1,nu,c1,c2,c3,nu1,nu2):\n" }{MPLTEXT 1 0 59 "Equation2Function: =unapply(Equation2,nu,c1,c2,c3,nu1,nu2):\n" }{MPLTEXT 1 0 59 "Equation 3Function:=unapply(Equation3,nu,c1,c2,c3,nu1,nu2):\n" }{MPLTEXT 1 0 59 "Equation4Function:=unapply(Equation4,nu,c1,c2,c3,nu1,nu2):\n" } {MPLTEXT 1 0 59 "Equation5Function:=unapply(Equation5,nu,c1,c2,c3,nu1, nu2):\n" }{MPLTEXT 1 0 59 "Equation6Function:=unapply(Equation6,nu,c1, c2,c3,nu1,nu2):\n" }{MPLTEXT 1 0 59 "Equation7Function:=unapply(Equati on7,nu,c1,c2,c3,nu1,nu2):\n" }{MPLTEXT 1 0 68 "simplify(Equation1Funct ion(nubis,c1bis,c2bis,c3bis,nu1bis,nu2bis));\n" }{MPLTEXT 1 0 68 "simp lify(Equation2Function(nubis,c1bis,c2bis,c3bis,nu1bis,nu2bis));\n" } {MPLTEXT 1 0 68 "simplify(Equation3Function(nubis,c1bis,c2bis,c3bis,nu 1bis,nu2bis));\n" }{MPLTEXT 1 0 68 "simplify(Equation4Function(nubis,c 1bis,c2bis,c3bis,nu1bis,nu2bis));\n" }{MPLTEXT 1 0 68 "simplify(Equati on5Function(nubis,c1bis,c2bis,c3bis,nu1bis,nu2bis));\n" }{MPLTEXT 1 0 68 "simplify(Equation6Function(nubis,c1bis,c2bis,c3bis,nu1bis,nu2bis)) ;\n" }{MPLTEXT 1 0 67 "simplify(Equation7Function(nubis,c1bis,c2bis,c3 bis,nu1bis,nu2bis));" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 239 90 "Computation of the Lax matrix \\td\{L\} in the geome tric gauge without apparent singularities" }}{EXCHG {PARA 0 "" 0 "" {TEXT 212 20 "Symplectic reduction" }}}{EXCHG {PARA 0 "> " 0 "" {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 12 "alpha14:=0:\n" }{MPLTEXT 1 0 12 "alpha24:=0:\n" } {MPLTEXT 1 0 12 "alpha13:=0:\n" }{MPLTEXT 1 0 12 "alpha23:=0:\n" } {MPLTEXT 1 0 19 "alpha22:=-alpha12:\n" }{MPLTEXT 1 0 19 "alpha21:=-alp ha11:\n" }{MPLTEXT 1 0 20 "tinfty12:=1/2*tau2;\n" }{MPLTEXT 1 0 21 "ti nfty22:=-tinfty12;\n" }{MPLTEXT 1 0 20 "tinfty11:=1/2*tau1;\n" } {MPLTEXT 1 0 21 "tinfty21:=-tinfty11;\n" }{MPLTEXT 1 0 20 "tinfty20:=- tinfty10;" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)tinft y14G6\"\"\"\"" }}{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\",$I)tinfty10GF$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "C0ter:=simplify(C0ter);\n" }{MPLTEXT 1 0 23 " C1bis:=simplify(C1bis);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 138 "Cinft y0ter:=-(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)" } {MPLTEXT 1 0 28 "*tau1+ 2*tinfty10-h)*q1*q2;\n" }{MPLTEXT 1 0 81 "Cinf ty1bis:=(p1^2-p2^2)/(q1-q2)-( (q1^2+q2^2+1/2*tau2)^2-q1^2*q2^2 +(2*tin fty10-h" }{MPLTEXT 1 0 34 "))*(q1+q2)-(q1^2+q1*q2+q2^2)*tau1;" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 28 "simplify(C0ter-Cinfty0ter);\n" } {MPLTEXT 1 0 28 "simplify(C1bis-Cinfty1bis);\n" }{MPLTEXT 1 0 21 "nu:= simplify(nubis);\n" }{MPLTEXT 1 0 5 "nu1:=" }{MPLTEXT 1 0 9 "simplify( " }{MPLTEXT 1 0 9 "nu1bis);\n" }{MPLTEXT 1 0 5 "nu2:=" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 9 "nu2bis);\n" }{MPLTEXT 1 0 15 "nuinftyMi nus1:=" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 16 "nuinftyMinus1); \n" }{MPLTEXT 1 0 10 "nuinfty0:=" }{MPLTEXT 1 0 9 "simplify(" } {MPLTEXT 1 0 11 "nuinfty0);\n" }{MPLTEXT 1 0 10 "nuinfty1:=" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 11 "nuinfty1);\n" }{MPLTEXT 1 0 30 "n uinfty2:=simplify(nuinfty2);\n" }{MPLTEXT 1 0 20 "mu1:=simplify(mu1); \n" }{MPLTEXT 1 0 20 "mu2:=simplify(mu2);\n" }{MPLTEXT 1 0 21 "c4:=sim plify(c4bis);\n" }{MPLTEXT 1 0 4 "c3:=" }{MPLTEXT 1 0 9 "simplify(" } {MPLTEXT 1 0 8 "c3bis);\n" }{MPLTEXT 1 0 4 "c2:=" }{MPLTEXT 1 0 9 "sim plify(" }{MPLTEXT 1 0 8 "c2bis);\n" }{MPLTEXT 1 0 4 "c1:=" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 7 "c1bis);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&C0terG6\"*&,0*&I#q1GF$\"\"'I#q2GF$\"\"\"\"\"%*(I%tau2G F$F+F(F,F*F+F,*(I%tau1GF$F+F(\"\"$F*F+F,*(,(*$F.\"\"##!\"\"F,I\"hGF$F+ I)tinfty10GF$!\"#F+F*F+F(F5!\"%*&,,*$F*F)F;*&F.F+F*F,F;*&F0F+F*F1F;*&, (F4F7F8F,F9!\")F+F*F5F+*$I#p2GF$F5F,F+F(F+F+*&I#p1GF$F5F*F+F;*&F8F+,&F GF+FEF7F+F;F+,&F(F,F*F;F7" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&C1bisG6 \"*&,6*$I#q1GF$\"\"'!\"%*&I%tau2GF$\"\"\"F(\"\"%F**&I%tau1GF$F-F(\"\"$ F**&,(*$F,\"\"#!\"\"I\"hGF$F.I)tinfty10GF$!\")F-F(F5F-*$I#q2GF$F)F.*&F ,F-F;F.F.*&F0F-F;F1F.*&,(F4F-F7F*F8\"\")F-F;F5F-*$I#p1GF$F5F.*$I#p2GF$ F5F*F-,&F(F.F;F*F6" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+Cinfty0terG6\", **(,&*$I#p1GF$\"\"#\"\"\"*$I#p2GF$F*!\"\"F+I#q1GF$F+,&F/F+I#q2GF$F.F.F .*(I\"hGF$F+,&F)F+F-F.F+F0F.F.F(F+*(,.*&,&*$F/\"\"&F+*$F1F:F.F+F0F.F+* $I%tau2GF$F*#F+\"\"%*(,&*$F/\"\"$F+*$F1FCF.F+F0F.F=F+F+*(,&*$F/F*F+*$F 1F*F.F+F0F.I%tau1GF$F+F+I)tinfty10GF$F*F3F.F+F/F+F1F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+Cinfty1bisG6\",(*&,&*$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$F.F+,&F0F+F1F+F+F.*&,(F6F+*&F0 F+F1F+F+F7F+F+I%tau1GF$F+F." }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "> I#nuG6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$nu1G6\",$I(alpha12GF $#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$nu2G6\"I(alpha11GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.nuinftyMinus1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuinfty0G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuinfty1G6\",$I(alpha12GF$#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuinfty2G6\"I(alpha11GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$mu1G6\"*&,&*&I(alpha12GF$\"\"\"I#q2GF$F)!\"\"I(alpha11 GF$\"\"#F),&I#q1GF$F-F*!\"#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$mu2G 6\"*&,&*&I(alpha12GF$\"\"\"I#q1GF$F)F)I(alpha11GF$!\"#F),&F*\"\"#I#q2G F$F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "> I#c2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c1G6\"\"\"!" }}} {SECT 0 {PARA 4 "" 0 "" {TEXT 240 13 "Matrix \\td\{L\}" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "G1:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 12 "G1[1,1]:=1:\n" }{MPLTEXT 1 0 12 "G1[2,2]:=1:\n" }{MPLTEXT 1 0 12 "G1[1,2]:=0:\n" }{MPLTEXT 1 0 9 "G1[2,1]:=" }{MPLTEXT 1 0 8 "tinfty 14" }{MPLTEXT 1 0 14 "*lambda+eta0:\n" }{MPLTEXT 1 0 33 "eta0:=tinfty1 4*(q1+q2)+tinfty13;\n" }{MPLTEXT 1 0 3 "G1;" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 27 "dG1dlambda:=Matrix(2,2,0):\n" } {MPLTEXT 1 0 89 "for i from 1 to 2 do for j from 1 to 2 do dG1dlambda[ i,j]:=diff(G1[i,j],lambda): od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 84 "tdL:=simplify(Multiply(Multiply(G1,checkL),G1^(-1))+h*Multiply (dG1dlambda,G1^(-1))):" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%eta0G6\",&I#q1GF$\"\"\"I#q2GF$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(-I#mnGF$6%Q\"1F'/%+fo regroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%)rowalignGQ!F'/%,column alignGFC/%+groupalignGFC/%(rowspanGF:/%+columnspanGF:-F56(-F86%Q\"0F'F ;F>FAFDFFFHFJFAFDFF-F26'-F56(-F,6%-I#miGF$6&Q'η0F'/%'italicGQ%true F'F;/F?Q'italicF'-I#moGF$6.Q\"+F'F;F>/%&fenceGQ&falseF'/%*separatorGF` o/%)stretchyGF`o/%*symmetricGF`o/%(largeopGF`o/%.movablelimitsGF`o/%'a ccentGF`o/%'lspaceGQ,0.2222222emF'/%'rspaceGF_p-FX6&Q)λF'/FfnF` oF;F>FAFDFFFHFJF4FAFDFF/%&alignGQ%axisF'/FBQ)baselineF'/FEQ'centerF'/F GQ'|frleft|hrF'/%/alignmentscopeGFgn/%,columnwidthGQ%autoF'/%&widthGFc q/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF' /%,columnlinesGF^r/%&frameGF^r/%-framespacingGQ,0.4em~0.5exF'/%*equalr owsGF`o/%-equalcolumnsGF`o/%-displaystyleGF`o/%%sideGQ&rightF'/%0minla belspacingGF[rF;F>/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "series(tdL[1,1],lambda=infinity):\n" }{MPLTEXT 1 0 7 "series(" }{MPLTEXT 1 0 27 "tdL[1,2],lambda=infinity):\n" } {MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 20 "tdL[2,2]+tdL[1,1]);\n" } {MPLTEXT 1 0 33 "series(tdL[2,1],lambda=infinity):" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "tdL 11bis:=-lambda^3+( (p1-p2)/(q1-q2)+ (q1^2+q1*q2+q2^2))*lambda +(p2*q1- p1*q2)/(q1-q2)- q1*q2*(q1+q2);\n" }{MPLTEXT 1 0 28 "simplify(tdL[1,1]- tdL11bis);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "factor(tdL[1,2]);\n " }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 66 "tdL21bis:= (tau2 +2*(p1-p2)/( q1-q2)+2*(q1^2+q1*q2+q2^2))*lambda^2\n" }{MPLTEXT 1 0 75 "+((q1+q2)*ta u2 +tau1+2*(p1*q1-p2*q2)/(q1-q2)+2*(q1+q2)*(q1^2+q2^2))*lambda\n" } {MPLTEXT 1 0 70 "+(q1+q2)*tau1 +1/4*tau2^2 +(q1^2+q1*q2+q2^2)*tau2-(p1 -p2)^2/(q1-q2)^2\n" }{MPLTEXT 1 0 66 "-2*(q2*p1-q1*p2)*(q1+q2)/(q1-q2) +q1^4+q2^4-q1^2*q2^2+2*(tinfty10)\n" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 28 "simplify(tdL[2,1]-tdL21bis);" }}{PARA 11 "" 1 "" {XPPMATH 20 " >I)tdL11bisG6\",**$I'lambdaGF$\"\"$!\"\"*&,**&,&I#p1GF$\"\"\"I#p2GF$F) F/,&I#q1GF$F/I#q2GF$F)F)F/*$F2\"\"#F/*&F2F/F3F/F/*$F3F5F/F/F'F/F/*&,&* &F.F/F3F/F)*&F0F/F2F/F/F/F1F)F/*(F2F/F3F/,&F2F/F3F/F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&I'lambda G6\"\"\"\"I#q1GF%!\"\"F&,&F$F&I#q2GF%F(F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)tdL21bisG6\",8*&,,I%tau2GF$\"\"\"*&,&I#p1GF$F)I#p2GF$! \"\"F),&I#q1GF$F)I#q2GF$F.F.\"\"#*$F0F2F2*&F0F)F1F)F2*$F1F2F2F)I'lambd aGF$F2F)*&,**&,&F0F)F1F)F)F(F)F)I%tau1GF$F)*&,&*&F,F)F0F)F)*&F-F)F1F)F .F)F/F.F2*&F:F),&F3F)F5F)F)F2F)F6F)F)*&F:F)F;F)F)*$F(F2#F)\"\"%*&,(F3F )F4F)F5F)F)F(F)F)*&F+F2F/!\"#F.*(,&*&F,F)F1F)F)*&F-F)F0F)F.F)F:F)F/F.F I*$F0FEF)*$F1FEF)*&F0F2F1F2F.I)tinfty10GF$F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "tdL21 bisOrderLambda2:=-residue(tdL[2,1]/lambda^3,lambda=infinity):\n" } {MPLTEXT 1 0 25 "tdL21bisOrderLambda2aux:=" }{MPLTEXT 1 0 45 "(tau2 -2 *(p1-p2)/(q1-q2)+2*(q1^2+q1*q2+q2^2))" }{MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 66 "tdL21bisOrderLambda1:=-residue(tdL[2,1]/l ambda^2,lambda=infinity):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 92 "tdL2 1bisOrderLambda1aux:=(q1+q2)*tau2 +tau1- 2*(p1*q1-p2*q2)/(q1-q2)+2*(q1 +q2)*(q1^2+q2^2):\n" }{MPLTEXT 1 0 65 "tdL21bisOrderLambda0:=-residue( tdL[2,1]/lambda,lambda=infinity):\n" }{MPLTEXT 1 0 25 "tdL21bisOrderLa mbda0aux:=" }{MPLTEXT 1 0 68 "(q1+q2)*tau1 +1/4*tau2^2 +(q1^2+q1*q2+q2 ^2)*tau2-(p1-p2)^2/(q1-q2)^2" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 4 "+2 *(" }{MPLTEXT 1 0 6 "q2*p1-" }{MPLTEXT 1 0 5 "q1*p2" }{MPLTEXT 1 0 53 ")*(q1+q2)/(q1-q2)+q1^4+q2^4-q1^2*q2^2+2*(tinfty10-h):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 14 "factor(series(" }{MPLTEXT 1 0 21 "tdL21bisOrd erLambda0-" }{MPLTEXT 1 0 23 "tdL21bisOrderLambda0aux" }{MPLTEXT 1 0 7 ",p1)):\n" }{MPLTEXT 1 0 18 "simplify(tdL[2,1]-" }{MPLTEXT 1 0 10 "t dL21bis);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 239 59 "Computation of the auxiliary matrices \\td\{A\} _1 and \\td\{A\}_2" }}{EXCHG {PARA 0 "" 0 "" {TEXT 212 205 "After symp lectic reduction, there are two independent non-trivial direction corr esponding to either \\partial_\{\\tau_1\} or \\partial_\{\\tau_2\}. Th is gives rise to two auxiliary matrices \\td\{A\}_1 and \\td\{A\}_2.\n " }{TEXT 212 139 "Warning: Compiling both cases create interferences. \+ Skip the subsection relatively to tau_1 if you want the deformation re latively to tau_2" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 240 45 "Study of t he deformation relatively to \\tau_1" }{TEXT 240 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "alpha12:=0:\n" }{MPLTEXT 1 0 12 "alpha22:= 0:\n" }{MPLTEXT 1 0 7 "alpha11" }{MPLTEXT 1 0 7 ":=1/2:\n" }{MPLTEXT 1 0 7 "alpha21" }{MPLTEXT 1 0 8 ":=-1/2:\n" }{MPLTEXT 1 0 11 "nu:=nubi s;\n" }{MPLTEXT 1 0 13 "nu1:=nu1bis;\n" }{MPLTEXT 1 0 13 "nu2:=nu2bis; \n" }{MPLTEXT 1 0 30 "nuinftyMinus1:=nuinftyMinus1;\n" }{MPLTEXT 1 0 20 "nuinfty0:=nuinfty0;\n" }{MPLTEXT 1 0 20 "nuinfty1:=nuinfty1;\n" } {MPLTEXT 1 0 30 "nuinfty2:=simplify(nuinfty2);\n" }{MPLTEXT 1 0 5 "mu1 :=" }{MPLTEXT 1 0 5 "mu1;\n" }{MPLTEXT 1 0 5 "mu2:=" }{MPLTEXT 1 0 5 " mu2;\n" }{MPLTEXT 1 0 7 "c0:=0:\n" }{MPLTEXT 1 0 8 "c1:=c1;\n" } {MPLTEXT 1 0 8 "c2:=c2;\n" }{MPLTEXT 1 0 8 "c3:=c3;\n" }{MPLTEXT 1 0 8 "c4:=c4;\n" }{MPLTEXT 1 0 19 "tdL:=simplify(tdL);" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "Lq1final:=Lq1ter;\n" } {MPLTEXT 1 0 17 "Lq2final:=Lq2ter;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 28 "Lp1final:=simplify(Lp1bis);\n" }{MPLTEXT 1 0 28 "Lp2final:=simpl ify(Lp2bis);\n" }{MPLTEXT 1 0 36 "Hamiltonian:=simplify(Hamiltonian); \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 10 "dq1dtau1:=" }{MPLTEXT 1 0 8 "Lq1final" }{MPLTEXT 1 0 4 "/h:\n" }{MPLTEXT 1 0 4 "dq2d" }{MPLTEXT 1 0 4 "tau1" }{MPLTEXT 1 0 10 ":=Lq2final" }{MPLTEXT 1 0 3 "/h:" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 4 "dp1d" }{MPLTEXT 1 0 4 "tau1" } {MPLTEXT 1 0 2 ":=" }{MPLTEXT 1 0 8 "Lp1final" }{MPLTEXT 1 0 3 "/h:" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 4 "dp2d" }{MPLTEXT 1 0 4 "tau1" } {MPLTEXT 1 0 13 ":=Lp2final/h:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 25 "dG1dtau1:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 103 "f or i from 1 to 2 do for j from 1 to 2 do dG1dtau1[i,j]:=simplify( diff (G1[i,j],tau1)+diff(G1[i,j],q1)*" }{MPLTEXT 1 0 8 "dq1dtau1" }{MPLTEXT 1 0 89 "+diff(G1[i,j],p1)*dp1dtau1+diff(G1[i,j],q2)*dq2dtau1+diff(G1[ i,j],p2)*dp2dtau1): od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 83 "tdA1:=simplify(Multiply(Multiply(G1,checkA),G1^( -1))+h*Multiply(dG1dtau1,G1^(-1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "> I#nuG6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$nu1G6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I$nu2G6\"#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.nuinftyMinus1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuinfty0G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I) nuinfty1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuinfty2G6\"#\" \"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$mu1G6\"*$,&I#q1GF$\"\"#I #q2GF$!\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$mu2G6\",$*$,&I#q1G F$\"\"#I#q2GF$!\"#!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c1G6\"\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c2G6\"\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 ">I#c3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c4G6 \"\"\"!" }}{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(-I&mfracGF$6)-F,6#-F,6+-F,6%-F#6%-F,6&-I#moGF$6.Q*&uminus0;F'/ %+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%* separatorGFP/%)stretchyGFP/%*symmetricGFP/%(largeopGFP/%.movablelimits GFP/%'accentGFP/%'lspaceGQ,0.2222222emF'/%'rspaceGFin-I#miGF$6&Q#q1F'/ %'italicGQ%trueF'FH/FLQ'italicF'-FE6.Q\"+F'FHFKFNFQFSFUFWFYFenFgnFjn-F ]o6&Q#q2F'F`oFHFcoFHFK-FE6-Q1⁢F'FKFNFQFSFUFWFYFen/FhnQ& 0.0emF'/F[oF_p-I%msupGF$6%-F]o6&Q)λF'/FaoFPFHFK-I#mnGF$6%Q\"3F' FHFK/%1superscriptshiftGQ\"0F'Feo-F,6%-F#6%-F,6)-F,6#-Fbp6%F\\oFhpF\\q -FE6.Q(−F'FHFKFNFQFSFUFWFYFenFgnFjn-F,6#-Fbp6%FhoFhpF\\qFeo-F]o6 &Q#p1F'F`oFHFcoFiq-F]o6&Q#p2F'F`oFHFcoFHFKF[pFdpFiq-F,6%FgqF[pFhoFeo-F ,6%-F#6%-F,6%F\\rFeoFcrFHFKF[pF\\oFiq-F,6%F`rF[pFho-F,6#-F,6%F\\oFiqFh o/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFis/%)beve lledGFPFH/%)rowalignGQ!F'/%,columnalignGF`t/%+groupalignGF`t/%(rowspan GFfs/%+columnspanGFfs-F56(-F,6%-F#6%-F,6&FDFdpFeoF\\oFHFKF[p-F#6%-F,6& FDFdpFeoFhoFHFKF^tFatFctFetFgtF^tFatFct-F26'-F56(-F,6%-F86)-Fip6%FfsFH FK-Fip6%Q\"4F'FHFKFdsFgsFjsF\\tFHF[p-F86)-F,6#-F,6;-F,6%F_vF[p-Fbp6%F \\o-Fip6%Q\"6F'FHFKF\\qFeo-F,6%-F#6%-F,6&FD-F,6%-Fip6%Q\"8F'FHFKF[pFho Feo-F,6%FgwF[pFdpFHFKF[p-Fbp6%F\\o-Fip6%Q\"5F'FHFKF\\qFeo-F,6%-F#6%-F, 6'-F,6%FgwF[p-Fbp6%Fdp-Fip6%Q\"2F'FHFKF\\qFiq-F,6'FgwF[pFdpF[pFhoFeo-F ,6%F_vF[p-F]o6&Q'τ2F'F`oFHFcoFHFKF[p-Fbp6%F\\oF_vF\\qFeo-F,6%-F#6% -F,6+-F,6%FgwF[pF^rFeo-F,6%-F#6%-F,6&FDFgxFiqF`yFHFKF[pFhoFeo-F,6'F_vF [pFbyF[pFdpFeo-F,6%FgwF[pFcrFeo-F,6%F_vF[p-F]o6&Q'τ1F'F`oFHFcoFHFK F[pFgqFeo-F,6%-F#6%-F,6+-F,6%-F#6%-F,6(FD-F,6'F_vF[pFdpF[pFbyFiq-F,6%F gwF[pF`rFiqFizFHFKF[pFhoFeo-F,6'F_vF[pFbyF[pFixFeo-F,6%-F#6%-F,6%F \\\\lFeoFizFHFKF[pFdpFeo-F,6#-Fbp6%FbyF[yF\\qFeo-F,6%FgwF[p-F]o6&Q)tin fty10F'F`oFHFcoFHFKF[p-Fbp6%F\\oF[yF\\qFeo-F,6%-F#6%-F,6.FD-F,6%FgwF[p -Fbp6%FhoF^xF\\qFiq-F,6'FgwF[pFdpF[p-Fbp6%FhoF_vF\\qFeo-F,6%FazF[pF^rF eo-F,6%-F#6%-F,6(FDFj[lFiqFgzFiqFizFHFKF[p-Fbp6%FhoF[yF\\qFeo-F,6%-F#6 %-F,6*FD-F,6'FgwF[pFbyF[pFixFeo-F,6%-F#6%-F,6(FDF\\\\lFiqFgzFiq-F,6%Fg wF[pF[[lFHFKF[pFdpFiq-F,6%F[yF[pFh\\lFiq-F,6%-Fip6%Q#16F'FHFKF[pF\\]lF HFKF[pFhoFeo-F,6'FgwF[p-F#6%-F,6%F`rFiqFcrFHFKF[pFixFHFKF[pF\\oFeo-F,6 %F_vF[p-Fbp6%FhoF\\wF\\qFeo-F,6'FgwF[pFdpF[pFi]lFeo-F,6%-F#6%-F,6%FgxF eoF`yFHFKF[pF]^lFeo-F,6%-F#6%-F,6'Fj[lFeoF\\\\lFeoFizFHFKF[pF^rFeo-F,6 %-F#6%-F,6)F^\\lFeo-F,6%-F#6%-F,6%FgzFeoFizFHFKF[pFdpFeoFf\\lFeoFj\\lF HFKF[pFg^lFiq-F,6)FgwF[pFixF[pFb`lF[pFhoFiq-F,6%F_vF[p-Fbp6%Fb`lF[yF \\q-F,6#-Fbp6%-F#6%FbsFHFKF[yF\\qFdsFgsFjsF\\tFHF^tFatFctFetFgt-F56(-F 86)-F,6#-F,6+-F,6%F^clF[pFapFeo-F,6%-F#6%-F,6*FDFeqFeoF\\rFiqF`rFeoFcr FHFKF[pFdpFeoFfrFeo-F,6%-F#6%-F,6&FDF\\rFiqFcrFHFKF[pF\\oFeoF^sF`sFdsF gsFjsF\\tFHF^tFatFctFetFgtF^tFatFct/%&alignGQ%axisF'/F_tQ)baselineF'/F btFis/FdtQ'|frleft|hrF'/%/alignmentscopeGFbo/%,columnwidthGQ%autoF'/%& widthGFbel/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlines GQ%noneF'/%,columnlinesGF]fl/%&frameGF]fl/%-framespacingGQ,0.4em~0.5ex F'/%*equalrowsGFP/%-equalcolumnsGFP/%-displaystyleGFP/%%sideGQ&rightF' /%0minlabelspacingGFjelFHFK/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 " " 1 "" {XPPMATH 20 ">I)Lq1finalG6\"*&I#p1GF$\"\"\",&I#q1GF$F'I#q2GF$! \"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Lq2finalG6\",$*&,&I#q1GF$\" \"\"I#q2GF$!\"\"F+I#p2GF$F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Lp1f inalG6\",$*&,:*$I#q1GF$\"\"'\"#?*&F)\"\"&I#q2GF$\"\"\"!#C*&I%tau2GF$F/ F)\"\"%\"#7*&,&*&F.F/F2F/!#;I%tau1GF$\"\")F/F)\"\"$F/*&,**&F.F/F9F/!#7 *$F2\"\"#F/I\"hGF$!\"%I)tinfty10GF$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$FAFCF/,&F)F/F.FHFI#F/F:" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Lp2finalG6\",$*&,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+!\"$*&F;F5F-F.!\"#*&F;F.F0F.#F>F5F4#!\" \"F+F6F.F8F@F.F)F.F9*$F;F*\"#?*&F-F.F;F+\"#7*&F0F.F;F1F9*&F3F.F;F5F.*$ I#p1GF$F5F7*$I#p2GF$F5F+F.,&F)F.F;FDF@#F.F9" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,HamiltonianG6\"*&,6*$I#q1GF$\"\"'!\"%*&I%tau2GF$\"\"\" F(\"\"%F**&I%tau1GF$F-F(\"\"$F**&,(*$F,\"\"#!\"\"I\"hGF$F.I)tinfty10GF $!\")F-F(F5F-*$I#q2GF$F)F.*&F,F-F;F.F.*&F0F-F;F1F.*&,(F4F-F7F*F8\"\")F -F;F5F-*$I#p1GF$F5F.*$I#p2GF$F5F*F-,&F(F@F;F9F6" }}{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#moGF$6.Q*&umi nus0;F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%&fenceGQ&fa lseF'/%*separatorGFE/%)stretchyGFE/%*symmetricGFE/%(largeopGFE/%.movab lelimitsGFE/%'accentGFE/%'lspaceGQ,0.2222222emF'/%'rspaceGFT-F,6%-I&mf racGF$6)-I#mnGF$6%Q\"1F'F=F@-Fgn6%Q\"2F'F=F@/%.linethicknessGFin/%+den omalignGQ'centerF'/%)numalignGFao/%)bevelledGFEF=-F:6-Q1&InvisibleTime s;F'F@FCFFFHFJFLFNFP/FSQ&0.0emF'/FVFjo-F,6#-I#miGF$6&Q#q1F'/%'italicGQ %trueF'F=/FAQ'italicF'-F:6.Q(−F'F=F@FCFFFHFJFLFNFPFRFU-F,6%FYFfo -F,6#-F_p6&Q#q2F'FbpF=FepFgp-F,6%FYFfo-F,6#-F_p6&Q)λF'/FcpFEF=F @/%)rowalignGQ!F'/%,columnalignGF[r/%+groupalignGF[r/%(rowspanGFin/%+c olumnspanGFin-F56(FYFiqF\\rF^rF`rFbrFiqF\\rF^r-F26'-F56(-FZ6)-F,6#-F,6 +-F,6%-F_p6&Q'τ2F'FbpF=FepFfo-F#6%-F,6%F^pFgpF^qF=F@-F:6.Q\"+F'F=F @FCFFFHFJFLFNFPFRFU-F,6%FjnFfo-I%msupGF$6%F^p-Fgn6%Q\"3F'F=F@/%1supers criptshiftGQ\"0F'Fgp-F,6%FjnFfo-F_t6%F^qFatFdtFis-F,6%FjnFfo-F_p6&Q#p1 F'FbpF=FepFgp-F,6%FjnFfo-F_p6&Q#p2F'FbpF=Fep-F,6#-F,6%-F,6%FjnFfoF^pFg p-F,6%FjnFfoF^qF]oF_oFboFdoF=FiqF\\rF^rF`rFbr-F56(-F,6'FWFisFjpFisFaqF iqF\\rF^rF`rFbrFiqF\\rF^r/%&alignGQ%axisF'/FjqQ)baselineF'/F]rFao/F_rQ '|frleft|hrF'/%/alignmentscopeGFdp/%,columnwidthGQ%autoF'/%&widthGF]w/ %+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/% ,columnlinesGFhw/%&frameGFhw/%-framespacingGQ,0.4em~0.5exF'/%*equalrow sGFE/%-equalcolumnsGFE/%-displaystyleGFE/%%sideGQ&rightF'/%0minlabelsp acingGFewF=F@/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 "simplify(Hamiltonian-1/2*C1);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 47 "factor(simplify(Lp1-(-diff(Hamiltonian,q1))));\n" } {MPLTEXT 1 0 38 "simplify(Lq1-(diff(Hamiltonian,p1)));\n" }{MPLTEXT 1 0 47 "factor(simplify(Lp2-(-diff(Hamiltonian,q2))));\n" }{MPLTEXT 1 0 38 "simplify(Lq2-(diff(Hamiltonian,p2)));\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 28 "hdq1dtau1theo:= p1/(q1-q2);\n" }{MPLTEXT 1 0 29 "hdq2 dtau1theo:= -p2/(q1-q2);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 244 "hd p1dtau1theo:=1/(8*(q1-q2)^2)*(20*q1^6-24*q1^5*q2+12*tau2*q1^4+(-16*q2* tau2+8*tau1)*q1^3+(-12*q2*tau1+tau2^2-4*h+8*tinfty10)*q1^2+8*q2*(-(1/4 )*tau2^2+h-2*tinfty10)*q1+4*q2^6+4*tau2*q2^4+4*tau1*q2^3+(tau2^2-4*h+8 *tinfty10)*q2^2+4*p1^2-4*p2^2);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 240 "hdp2dtau1theo:=1/(8*(q1-q2)^2)*(4*q1^6+4*tau2*q1^4+4*tau1*q1^3+(t au2^2-4*h+8*tinfty10)*q1^2+8*q2*(-3*q2^4-2*q2^2*tau2-(3/2*q2)*tau1-(1/ 4)*tau2^2+h-2*tinfty10)*q1+20*q2^6+12*tau2*q2^4+8*tau1*q2^3+(tau2^2-4* h+8*tinfty10)*q2^2-4*p1^2+4*p2^2);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 176 "Ham1theo:=(-4*q1^6+4*q2^6-4*q1^4*tau2+4*q2^4*tau2-4*q1^3*tau1 -q1^2*tau2^2+4*q2^3*tau1+q2^2*tau2^2+4*h*q1^2-4*h*q2^2-8*q1^2*tinfty10 +8*q2^2*tinfty10+4*p1^2-4*p2^2)/(8*(q1-q2));\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 40 "factor(simplify(Lq1ter-hdq1dtau1theo));\n" }{MPLTEXT 1 0 40 "factor(simplify(Lq2ter-hdq2dtau1theo));\n" }{MPLTEXT 1 0 42 "f actor(simplify(Lp1final-hdp1dtau1theo));\n" }{MPLTEXT 1 0 42 "factor(s implify(Lp2final-hdp2dtau1theo));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 27 "factor(simplify(Hamiltonian" }{MPLTEXT 1 0 13 "-Ham1theo));\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 43 "simplify(diff(Ham1theo,q1)+hdp1d tau1theo);\n" }{MPLTEXT 1 0 43 "simplify(diff(Ham1theo,q2)+hdp2dtau1th eo);\n" }{MPLTEXT 1 0 43 "simplify(diff(Ham1theo,p1)-hdq1dtau1theo);\n " }{MPLTEXT 1 0 43 "simplify(diff(Ham1theo,p2)-hdq2dtau1theo);\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 46 "simplify(diff(Hamiltonian,q1)+hd p1dtau1theo);\n" }{MPLTEXT 1 0 46 "simplify(diff(Hamiltonian,q2)+hdp2d tau1theo);\n" }{MPLTEXT 1 0 46 "simplify(diff(Hamiltonian,p1)-hdq1dtau 1theo);\n" }{MPLTEXT 1 0 45 "simplify(diff(Hamiltonian,p2)-hdq2dtau1th eo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I.hdq1dtau1theoG6\"*&I#p1GF$\"\"\",&I#q 1GF$F'I#q2GF$!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.hdq2dtau1theo G6\",$*&,&I#q1GF$\"\"\"I#q2GF$!\"\"F+I#p2GF$F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.hdp1dtau1theoG6\",$*&,:*$I#q1GF$\"\"'\"#?*&F)\"\"&I#q2 GF$\"\"\"!#C*&I%tau2GF$F/F)\"\"%\"#7*&,&*&F.F/F2F/!#;I%tau1GF$\"\")F/F )\"\"$F/*&,**&F.F/F9F/!#7*$F2\"\"#F/I\"hGF$!\"%I)tinfty10GF$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$FAFCF/,&F)F/F.FHFI#F/F:" } }{PARA 11 "" 1 "" {XPPMATH 20 ">I.hdp2dtau1theoG6\",$*&,8*$I#q1GF$\"\" '\"\"%*&I%tau2GF$\"\"\"F)F+F+*&I%tau1GF$F.F)\"\"$F+*&,(*$F-\"\"#F.I\"h GF$!\"%I)tinfty10GF$\"\")F.F)F5F.*(I#q2GF$F.,.*$F;F+!\"$*&F;F5F-F.!\"# *&F;F.F0F.#F>F5F4#!\"\"F+F6F.F8F@F.F)F.F9*$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)Ham1theoG6\"*&,>*$I#q1GF$\"\"'!\"%*$I #q2GF$F)\"\"%*&F(F-I%tau2GF$\"\"\"F**&F,F-F/F0F-*&F(\"\"$I%tau1GF$F0F* *&F(\"\"#F/F6!\"\"*&F,F3F4F0F-*&F,F6F/F6F0*&I\"hGF$F0F(F6F-*&F;F0F,F6F **&F(F6I)tinfty10GF$F0!\")*&F,F6F>F0\"\")*$I#p1GF$F6F-*$I#p2GF$F6F*F0, &F(FAF,F?F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 240 45 "Study of the deformation relatively to \\tau_2" }{TEXT 240 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "alpha12" }{MPLTEXT 1 0 7 ":=1/2:\n" }{MPLTEXT 1 0 15 "alpha22:=-1/2:\n" }{MPLTEXT 1 0 12 "alp ha11:=0:\n" }{MPLTEXT 1 0 7 "alpha21" }{MPLTEXT 1 0 5 ":=0:\n" } {MPLTEXT 1 0 11 "nu:=nubis;\n" }{MPLTEXT 1 0 13 "nu1:=nu1bis;\n" } {MPLTEXT 1 0 13 "nu2:=nu2bis;\n" }{MPLTEXT 1 0 30 "nuinftyMinus1:=nuin ftyMinus1;\n" }{MPLTEXT 1 0 20 "nuinfty0:=nuinfty0;\n" }{MPLTEXT 1 0 20 "nuinfty1:=nuinfty1;\n" }{MPLTEXT 1 0 30 "nuinfty2:=simplify(nuinft y2);\n" }{MPLTEXT 1 0 10 "mu1:=mu1;\n" }{MPLTEXT 1 0 10 "mu2:=mu2;\n" }{MPLTEXT 1 0 6 "c0:=0:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 8 "c1:=c1; \n" }{MPLTEXT 1 0 8 "c2:=c2;\n" }{MPLTEXT 1 0 7 "c3:=c3;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 8 "c4:=c4;\n" }{MPLTEXT 1 0 18 "Lq1final:=Lq1te r;\n" }{MPLTEXT 1 0 18 "Lq2final:=Lq2ter;\n" }{MPLTEXT 1 0 28 "Lp1fina l:=simplify(Lp1bis);\n" }{MPLTEXT 1 0 28 "Lp2final:=simplify(Lp2bis); \n" }{MPLTEXT 1 0 36 "Hamiltonian:=simplify(Hamiltonian);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "dq1dtau2:=Lq1final/h:\n" }{MPLTEXT 1 0 22 "dq2dtau2:=Lq2final/h:\n" }{MPLTEXT 1 0 22 "dp1dtau2:=Lp1final/h:\n " }{MPLTEXT 1 0 22 "dp2dtau2:=Lp2final/h:\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 25 "dG1dtau2:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 200 "for i \+ from 1 to 2 do for j from 1 to 2 do dG1dtau2[i,j]:=simplify( diff(G1[i ,j],tau2)+diff(G1[i,j],q1)*dq1dtau2+diff(G1[i,j],p1)*dp1dtau2+diff(G1[ i,j],q2)*dq2dtau2+diff(G1[i,j],p2)*dp2dtau2): od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "tdL:=simplify(tdL);" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 83 "tdA2:=simplify(Multiply(Multiply(G1,checkA),G1^(-1))+ h*Multiply(dG1dtau2,G1^(-1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#nuG 6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$nu1G6\"#\"\"\"\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I$nu2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.nuinftyMinus1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuinfty0G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuinfty1G6 \"#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuinfty2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$mu1G6\",$*&I#q2GF$\"\"\",&I#q1GF$\" \"#F'!\"#!\"\"#F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$mu2G6\",$*&I#q 1GF$\"\"\",&F'\"\"#I#q2GF$!\"#!\"\"#F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c2G6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I#c3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Lq1fin alG6\",&*(,&I#q1GF$\"\"\"I#q2GF$!\"\"F+I#p1GF$F)F*F)#F+\"\"#*&I\"hGF$F )F'F+#F+\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Lq2finalG6\",&*(,&I# q1GF$\"\"\"I#q2GF$!\"\"F+I#p2GF$F)F(F)#F)\"\"#*&I\"hGF$F)F'F+#F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Lp1finalG6\",$*&,0*$I#q2GF$\"\"(! \"%*&I%tau2GF$\"\"\"F)\"\"&F+*&I%tau1GF$F.F)\"\"%F+*&,(*$F-\"\"#!\"\"I \"hGF$F2I)tinfty10GF$!\")F.F)\"\"$F.*(,.*$I#q1GF$F2!\"$*&F?F6F-F.!\"#* &F1F.F?F.#F@F6F5#F7F2F8F.F9FBF.F?F.F)F6F:*&,.*$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)F7FB#F.\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Lp2f inalG6\",$*&,0*$I#q1GF$\"\"(!\"%*&I%tau2GF$\"\"\"F)\"\"&F+*&I%tau1GF$F .F)\"\"%F+*&,(*$F-\"\"#!\"\"I\"hGF$F2I)tinfty10GF$!\")F.F)\"\"$F.*(,.* $I#q2GF$F2!\"$*&F?F6F-F.!\"#*&F?F.F1F.#F@F6F5#F7F2F8F.F9FBF.F?F.F)F6F: *&,.*$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,HamiltonianG6\"*&,0*&I#q1GF$\"\"'I#q2GF$\"\" \"\"\"%*(I%tau2GF$F+F(F,F*F+F,*(I%tau1GF$F+F(\"\"$F*F+F,*(,(*$F.\"\"## !\"\"F,I\"hGF$F+I)tinfty10GF$!\"#F+F*F+F(F5!\"%*&,,*$F*F)F;*&F.F+F*F,F ;*&F0F+F*F1F;*&,(F4F7F8F,F9!\")F+F*F5F+*$I#p2GF$F5F,F+F(F+F+*&I#p1GF$F 5F*F+F;*&F8F+,&FGF+FEF7F+F;F+,&F(\"#;F*!#;F7" }}{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(-I&mfracGF$6)-F,6#-F, 6+-F,6%-F#6%-F,6&-I#moGF$6.Q*&uminus0;F'/%+foregroundGQ([0,0,0]F'/%,ma thvariantGQ'normalF'/%&fenceGQ&falseF'/%*separatorGFP/%)stretchyGFP/%* symmetricGFP/%(largeopGFP/%.movablelimitsGFP/%'accentGFP/%'lspaceGQ,0. 2222222emF'/%'rspaceGFin-I#miGF$6&Q#q1F'/%'italicGQ%trueF'FH/FLQ'itali cF'-FE6.Q\"+F'FHFKFNFQFSFUFWFYFenFgnFjn-F]o6&Q#q2F'F`oFHFcoFHFK-FE6-Q1 ⁢F'FKFNFQFSFUFWFYFen/FhnQ&0.0emF'/F[oF_p-I%msupGF$6%-F] o6&Q)λF'/FaoFPFHFK-I#mnGF$6%Q\"3F'FHFK/%1superscriptshiftGQ\"0F 'Feo-F,6%-F#6%-F,6)-F,6#-Fbp6%F\\oFhpF\\q-FE6.Q(−F'FHFKFNFQFSFUF WFYFenFgnFjn-F,6#-Fbp6%FhoFhpF\\qFeo-F]o6&Q#p1F'F`oFHFcoFiq-F]o6&Q#p2F 'F`oFHFcoFHFKF[pFdpFiq-F,6%FgqF[pFhoFeo-F,6%-F#6%-F,6%F\\rFeoFcrFHFKF[ pF\\oFiq-F,6%F`rF[pFho-F,6#-F,6%F\\oFiqFho/%.linethicknessGQ\"1F'/%+de nomalignGQ'centerF'/%)numalignGFis/%)bevelledGFPFH/%)rowalignGQ!F'/%,c olumnalignGF`t/%+groupalignGF`t/%(rowspanGFfs/%+columnspanGFfs-F56(-F, 6%-F#6%-F,6&FDFdpFeoF\\oFHFKF[p-F#6%-F,6&FDFdpFeoFhoFHFKF^tFatFctFetFg tF^tFatFct-F26'-F56(-F,6%-F86)-Fip6%FfsFHFK-Fip6%Q\"4F'FHFKFdsFgsFjsF \\tFHF[p-F86)-F,6#-F,6;-F,6%F_vF[p-Fbp6%F\\o-Fip6%Q\"6F'FHFKF\\qFeo-F, 6%-F#6%-F,6&FD-F,6%-Fip6%Q\"8F'FHFKF[pFhoFeo-F,6%FgwF[pFdpFHFKF[p-Fbp6 %F\\o-Fip6%Q\"5F'FHFKF\\qFeo-F,6%-F#6%-F,6'-F,6%FgwF[p-Fbp6%Fdp-Fip6%Q \"2F'FHFKF\\qFiq-F,6'FgwF[pFdpF[pFhoFeo-F,6%F_vF[p-F]o6&Q'τ2F'F`oF HFcoFHFKF[p-Fbp6%F\\oF_vF\\qFeo-F,6%-F#6%-F,6+-F,6%FgwF[pF^rFeo-F,6%-F #6%-F,6&FDFgxFiqF`yFHFKF[pFhoFeo-F,6'F_vF[pFbyF[pFdpFeo-F,6%FgwF[pFcrF eo-F,6%F_vF[p-F]o6&Q'τ1F'F`oFHFcoFHFKF[pFgqFeo-F,6%-F#6%-F,6+-F,6% -F#6%-F,6(FD-F,6'F_vF[pFdpF[pFbyFiq-F,6%FgwF[pF`rFiqFizFHFKF[pFhoFeo-F ,6'F_vF[pFbyF[pFixFeo-F,6%-F#6%-F,6%F\\\\lFeoFizFHFKF[pFdpFeo-F,6#-Fbp 6%FbyF[yF\\qFeo-F,6%FgwF[p-F]o6&Q)tinfty10F'F`oFHFcoFHFKF[p-Fbp6%F\\oF [yF\\qFeo-F,6%-F#6%-F,6.FD-F,6%FgwF[p-Fbp6%FhoF^xF\\qFiq-F,6'FgwF[pFdp F[p-Fbp6%FhoF_vF\\qFeo-F,6%FazF[pF^rFeo-F,6%-F#6%-F,6(FDFj[lFiqFgzFiqF izFHFKF[p-Fbp6%FhoF[yF\\qFeo-F,6%-F#6%-F,6*FD-F,6'FgwF[pFbyF[pFixFeo-F ,6%-F#6%-F,6(FDF\\\\lFiqFgzFiq-F,6%FgwF[pF[[lFHFKF[pFdpFiq-F,6%F[yF[pF h\\lFiq-F,6%-Fip6%Q#16F'FHFKF[pF\\]lFHFKF[pFhoFeo-F,6'FgwF[p-F#6%-F,6% F`rFiqFcrFHFKF[pFixFHFKF[pF\\oFeo-F,6%F_vF[p-Fbp6%FhoF\\wF\\qFeo-F,6'F gwF[pFdpF[pFi]lFeo-F,6%-F#6%-F,6%FgxFeoF`yFHFKF[pF]^lFeo-F,6%-F#6%-F,6 'Fj[lFeoF\\\\lFeoFizFHFKF[pF^rFeo-F,6%-F#6%-F,6)F^\\lFeo-F,6%-F#6%-F,6 %FgzFeoFizFHFKF[pFdpFeoFf\\lFeoFj\\lFHFKF[pFg^lFiq-F,6)FgwF[pFixF[pFb` lF[pFhoFiq-F,6%F_vF[p-Fbp6%Fb`lF[yF\\q-F,6#-Fbp6%-F#6%FbsFHFKF[yF\\qFd sFgsFjsF\\tFHF^tFatFctFetFgt-F56(-F86)-F,6#-F,6+-F,6%F^clF[pFapFeo-F,6 %-F#6%-F,6*FDFeqFeoF\\rFiqF`rFeoFcrFHFKF[pFdpFeoFfrFeo-F,6%-F#6%-F,6&F DF\\rFiqFcrFHFKF[pF\\oFeoF^sF`sFdsFgsFjsF\\tFHF^tFatFctFetFgtF^tFatFct /%&alignGQ%axisF'/F_tQ)baselineF'/FbtFis/FdtQ'|frleft|hrF'/%/alignment scopeGFbo/%,columnwidthGQ%autoF'/%&widthGFbel/%+rowspacingGQ&1.0exF'/% .columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF]fl/%&fram eGF]fl/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFP/%-equalcolumnsGFP /%-displaystyleGFP/%%sideGQ&rightF'/%0minlabelspacingGFjelFHFK/%%openG Q\"[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(-I&mfracGF$6)-F,6#-F,60-I#moGF$6.Q*&uminus0;F'/%+ foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%*se paratorGFJ/%)stretchyGFJ/%*symmetricGFJ/%(largeopGFJ/%.movablelimitsGF J/%'accentGFJ/%'lspaceGQ,0.2222222emF'/%'rspaceGFY-F,6#-I%msupGF$6%-I# miGF$6&Q#q2F'/%'italicGQ%trueF'FB/FFQ'italicF'-I#mnGF$6%Q\"3F'FBFE/%1s uperscriptshiftGQ\"0F'-F?6.Q(−F'FBFEFHFKFMFOFQFSFUFWFZ-F,6%-F\\o 6&Q#q1F'F_oFBFbo-F?6-Q1⁢F'FEFHFKFMFOFQFSFU/FXQ&0.0emF'/ FenFgp-Fin6%F[o-Feo6%Q\"2F'FBFEFho-F?6.Q\"+F'FBFEFHFKFMFOFQFSFUFWFZ-F, 6%-F#6%-F,6%-F,6#-Fin6%-F\\o6&Q)λF'/F`oFJFBFEF[qFhoF^q-F,6#-Fin 6%F`pF[qFhoFBFEFcpF[oF^q-F,6#-Fin6%F`pFdoFhoF[p-F,6%FiqFcpF`pF^q-F\\o6 &Q#p1F'F_oFBFboF[p-F\\o6&Q#p2F'F_oFBFbo-F,6#-F,6%-F,6%-Feo6%Q\"4F'FBFE FcpF`pF[p-F,6%FesFcpF[o/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF' /%)numalignGF_t/%)bevelledGFJFB/%)rowalignGQ!F'/%,columnalignGFft/%+gr oupalignGFft/%(rowspanGF\\t/%+columnspanGF\\t-F56(-F,6'-F,6%-F86)-Feo6 %F\\tFBFEFesFjsF]tF`tFbtFBFcp-F,6#F[rF[p-F,6%FeuFcp-F,6#F`pF[p-F,6%Feu Fcp-F,6#F[oFdtFgtFitF[uF]uFdtFgtFit-F26'-F56(-F86)-F,6#-F,63-F,6%F[qFc p-Fin6%F`pFesFhoF^q-F,6'F[qFcpF[rFcpFerF^q-F,6%FarFcp-F\\o6&Q'τ2F' F_oFBFboF^q-F,6%-F#6%-F,6'-F,6%F[rFcpFewF^q-F,6%F[qFcpFirF^q-F\\o6&Q'& tau;1F'F_oFBFboFBFEFcpF`pF[p-F,6%F[qFcp-Fin6%F[oFesFhoF[p-F,6'F[qFcpF[ rFcpFhnF[p-F,6%FipFcpFewF^q-F,6%-F#6%-F,6(F>F^xF[p-F,6%F[qFcpF\\sF[pFb xFBFEFcpF[oF^q-F,6'F[qFcp-F#6%-F,6%FirF[pF\\sFBFEFcpF[rF_sFjsF]tF`tFbt FBFdtFgtFitF[uF]u-F56(-F86)-F,6#-F,60F>FcrF[p-F,6%F[oFcpFarF^q-F,6%-F# 6%-F,6%FgqF^q-F,6#FipFBFEFcpF`pF[p-F,6%FiqFcpF[oF^qFfnF[pFirF^qF\\sF_s FjsF]tF`tFbtFBFdtFgtFitF[uF]uFdtFgtFit/%&alignGQ%axisF'/FetQ)baselineF '/FhtF_t/FjtQ'|frleft|hrF'/%/alignmentscopeGFao/%,columnwidthGQ%autoF' /%&widthGF[\\l/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowl inesGQ%noneF'/%,columnlinesGFf\\l/%&frameGFf\\l/%-framespacingGQ,0.4em ~0.5exF'/%*equalrowsGFJ/%-equalcolumnsGFJ/%-displaystyleGFJ/%%sideGQ&r ightF'/%0minlabelspacingGFc\\lFBFE/%%openGQ\"[F'/%&closeGQ\"]F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "series(simplify(series(tdA2[ 1,1],lambda=infinity)),lambda=infinity);" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 68 "series(simplify(series(tdA2[1,2],lambda=infinity)),la mbda=infinity);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 31 "simplify(tdA2[ 2,2]+tdA2[1,1]);\n" }{MPLTEXT 1 0 69 "series(simplify(series(tdA2[2,1] ,lambda=infinity)),lambda=infinity):\n" }{MPLTEXT 1 0 97 "tdA221bis:=1 /2* ((p1-p2)/(q1-q2)+ q1^2+q1*q2+q2^2+1/2*tau2)*lambda +1/4*((q1+q2)*t au2+tau1)+1/2*(" }{MPLTEXT 1 0 6 "p1*q1-" }{MPLTEXT 1 0 40 "p2*q2)/(q1 -q2)+1/2*(q1+q2)*(q1^2+q2^2);\n" }{MPLTEXT 1 0 27 "simplify(series(tdA 2[2,1]- " }{MPLTEXT 1 0 19 "tdA221bis,lambda));" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*(,&I#q1G6\"!\"\"I#q2GF&\"\"\"F),&F% \"\"%F(!\"%F'I'lambdaGF&\"\"#F)*&,.*$F%\"\"$F)*&F%F.F(F)F)*&F%F)F(F.F' *$F(F2F'I#p1GF&F)I#p2GF&F'F)F*F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ",( I'lambdaG6\"#\"\"\"\"\"%I#q1GF$#!\"\"F'I#q2GF$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*tdA221bisG6\",, *&,,*&,&I#p1GF$\"\"\"I#p2GF$!\"\"F+,&I#q1GF$F+I#q2GF$F-F-F+*$F/\"\"#F+ *&F/F+F0F+F+*$F0F2F+I%tau2GF$#F+F2F+I'lambdaGF$F+F6*&,&F/F+F0F+F+F5F+# F+\"\"%I%tau1GF$F:*&,&*&F*F+F/F+F+*&F,F+F0F+F-F+F.F-F6*&F9F+,&F1F+F4F+ F+F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "factor(simplify(Lp1-(-diff(Hamiltonian,q1))));\n" }{MPLTEXT 1 0 38 "simplify(Lq1-(diff(Hamiltonian,p1)));\n" }{MPLTEXT 1 0 47 "factor(simplify(Lp2-(-diff(Hamiltonian,q2))));\n" }{MPLTEXT 1 0 38 "simplify(Lq2-(diff(Hamiltonian,p2)));\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 30 "simplify(Hamiltonian-1/4*C0);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 43 "hdq1dtau2theo := (-2*p1*q2-h)/(4*q1-4*q2):\n" } {MPLTEXT 1 0 42 "hdq2dtau2theo := (2*p2*q1+h)/(4*q1-4*q2):\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 266 "hdp1dtau2theo:=1/(16*(q1-q2)^2) *(-4*q2^7-4*tau2*q2^5-4*tau1*q2^4+(-tau2^2+4*h-8*tinfty10)*q2^3-(8*(-3 *q1^4-2*q1^2*tau2-(3/2*tau1)*q1-(1/4)*tau2^2+h-2*tinfty10))*q1*q2^2+(- 20*q1^6-12*tau2*q1^4-8*tau1*q1^3+(-tau2^2+4*h-8*tinfty10)*q1^2-4*p1^2+ 4*p2^2)*q2-4*h*(p1-p2));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 266 "hd p2dtau2theo:=1/(16*(q1-q2)^2)*(-4*q1^7-4*tau2*q1^5-4*tau1*q1^4+(-tau2^ 2+4*h-8*tinfty10)*q1^3-(8*(-3*q2^4-2*tau2*q2^2-(3/2*tau1)*q2-(1/4)*tau 2^2+h-2*tinfty10))*q2*q1^2+(-20*q2^6-12*tau2*q2^4-8*tau1*q2^3+(-tau2^2 +4*h-8*tinfty10)*q2^2+4*p1^2-4*p2^2)*q1+4*h*(p1-p2));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 234 "Ham2theo:=-(-4*q1^6*q2+4*q1*q2^6-4*q1^4*q2 *tau2+4*q1*q2^4*tau2-4*q1^3*q2*tau1-q1^2*q2*tau2^2+4*q1*q2^3*tau1+q1*q 2^2*tau2^2+4*h*q1^2*q2-4*h*q1*q2^2-8*q1^2*q2*tinfty10+8*q1*q2^2*tinfty 10+4*p1^2*q2-4*p2^2*q1+4*h*p1-4*h*p2)/(16*(q1-q2));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 40 "factor(simplify(Lq1ter-hdq1dtau2theo));\n" } {MPLTEXT 1 0 40 "factor(simplify(Lq2ter-hdq2dtau2theo));\n" }{MPLTEXT 1 0 42 "factor(simplify(Lp1final-hdp1dtau2theo));\n" }{MPLTEXT 1 0 42 "factor(simplify(Lp2final-hdp2dtau2theo));\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 40 "factor(simplify(Hamiltonian-Ham2theo));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 43 "simplify(diff(Ham2theo,q1)+hdp1dtau2theo) ;\n" }{MPLTEXT 1 0 43 "simplify(diff(Ham2theo,q2)+hdp2dtau2theo);\n" } {MPLTEXT 1 0 43 "simplify(diff(Ham2theo,p1)-hdq1dtau2theo);\n" } {MPLTEXT 1 0 43 "simplify(diff(Ham2theo,p2)-hdq2dtau2theo);\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 46 "simplify(diff(Hamiltonian,q1)+hd p1dtau2theo);\n" }{MPLTEXT 1 0 46 "simplify(diff(Hamiltonian,q2)+hdp2d tau2theo);\n" }{MPLTEXT 1 0 46 "simplify(diff(Hamiltonian,p1)-hdq1dtau 2theo);\n" }{MPLTEXT 1 0 45 "simplify(diff(Hamiltonian,p2)-hdq2dtau2th eo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I.hdp1dtau2theoG6\",$*&,0*$I#q2GF$\"\"( !\"%*&I%tau2GF$\"\"\"F)\"\"&F+*&I%tau1GF$F.F)\"\"%F+*&,(*$F-\"\"#!\"\" I\"hGF$F2I)tinfty10GF$!\")F.F)\"\"$F.*(,.*$I#q1GF$F2!\"$*&F?F6F-F.!\"# *&F1F.F?F.#F@F6F5#F7F2F8F.F9FBF.F?F.F)F6F:*&,.*$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)F7FB#F.\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.hdp 2dtau2theoG6\",$*&,0*$I#q1GF$\"\"(!\"%*&I%tau2GF$\"\"\"F)\"\"&F+*&I%ta u1GF$F.F)\"\"%F+*&,(*$F-\"\"#!\"\"I\"hGF$F2I)tinfty10GF$!\")F.F)\"\"$F .*(,.*$I#q2GF$F2!\"$*&F?F6F-F.!\"#*&F?F.F1F.#F@F6F5#F7F2F8F.F9FBF.F?F. F)F6F:*&,.*$F?\"\"'!#?*&F-F.F?F2!#7*&F1F.F?F;F:*&F4F.F?F6F.*$I#p1GF$F6 F2*$I#p2GF$F6F+F.F)F.F.*&F8F.,&FPF.FRF7F.F2F.,&F)F.F?F7FB#F.\"#;" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I)Ham2theoG6\",$*&,B*&I#q1GF$\"\"'I#q2G F$\"\"\"!\"%*&F)F,F+F*\"\"%*(F)F/F+F,I%tau2GF$F,F-*(F)F,F+F/F1F,F/*(F) \"\"$F+F,I%tau1GF$F,F-*(F)\"\"#F+F,F1F7!\"\"*(F)F,F+F4F5F,F/*(F)F,F+F7 F1F7F,*(I\"hGF$F,F)F7F+F,F/*(F " 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 }