{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart;with(linalg) :" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "N:=15;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "x:=(k)->sum(a*sin(psi[n]),n=1..k);\ny:=(k)->sum(a*cos(psi[n]),n=1 ..k);\ny1:=(k)->sum(-a*psi1[n]*sin(psi[n]),n=1..k);\nx1:=(k)->sum(a*ps i1[n]*cos(psi[n]),n=1..k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGR6 #%\"kG6\"6$%)operatorG%&arrowGF(-%$sumG6$*&%\"aG\"\"\"-%$sinG6#&%$psiG 6#%\"nGF1/F8;F19$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGR6#% \"kG6\"6$%)operatorG%&arrowGF(-%$sumG6$*&%\"aG\"\"\"-%$cosG6#&%$psiG6# %\"nGF1/F8;F19$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1GR6#%\"k G6\"6$%)operatorG%&arrowGF(-%$sumG6$,$*(%\"aG\"\"\"&%%psi1G6#%\"nGF2-% $sinG6#&%$psiGF5F2!\"\"/F6;F29$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1GR6#%\"kG6\"6$%)operatorG%&arrowGF(-%$sumG6$*(%\"aG\"\"\"&%%ps i1G6#%\"nGF1-%$cosG6#&%$psiGF4F1/F5;F19$F(F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 157 "V:=-m*g*sum(y(k),k=1..N):\nT1:=1/2*m*sum(x1(k )^2+y1(k)^2,k=1..N):\nT2:=1/2*m*sum(x1(k-1)*x1(k)+y1(k-1)*y1(k)+1/3*(( x1(k-1)-x1(k))^2+(y1(k-1)-y1(k))^2),k=1..N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "C OS:=(x)->1-x^2/2;\nSIN:=(x)->x;\nscs:=\{g=1,a=1,m=1,sin=SIN,cos=COS\}; \nss0:=\{g=1,a=1,m=1,seq(psi[i]=0,i=1..N)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$COSGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"\"\"F-*$)9 $\"\"#\"\"\"#!\"\"F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SINGR 6#%\"xG6\"6$%)operatorG%&arrowGF(9$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$scsG<'/%\"gG\"\"\"/%\"aGF(/%\"mGF(/%$sinG%$SING/%$co sG%$COSG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ss0G<4/%\"gG\"\"\"/%\"a GF(/%\"mGF(/&%$psiG6#F(\"\"!/&F/6#\"\"#F1/&F/6#\"\"$F1/&F/6#\"\"%F1/&F /6#\"\"&F1/&F/6#\"\"'F1/&F/6#\"\"(F1/&F/6#\"\")F1/&F/6#\"\"*F1/&F/6#\" #5F1/&F/6#\"#6F1/&F/6#\"#7F1/&F/6#\"#8F1/&F/6#\"#9F1/&F/6#\"#:F1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "v:=simplify(eval(subs(scs,V) )):\nt1:=simplify(eval(subs(ss0,T1))):\nt2:=simplify(eval(subs(ss0,T2) )):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "B := matrix(N,N,(i, j) -> diff(v,psi[i],psi[j]));\nB1 := matrix(N,N,(i,j) -> if i=j then 1 /B[i,j] else 0 fi);\nA1 := matrix(N,N,(i,j) -> diff(t1,psi1[i],psi1[j] ));\nA2 := matrix(N,N,(i,j) -> diff(t2,psi1[i],psi1[j]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7171\"#:\"\"!F+F+F+F+F+F+F+F+ F+F+F+F+F+71F+\"#9F+F+F+F+F+F+F+F+F+F+F+F+F+71F+F+\"#8F+F+F+F+F+F+F+F+ F+F+F+F+71F+F+F+\"#7F+F+F+F+F+F+F+F+F+F+F+71F+F+F+F+\"#6F+F+F+F+F+F+F+ F+F+F+71F+F+F+F+F+\"#5F+F+F+F+F+F+F+F+F+71F+F+F+F+F+F+\"\"*F+F+F+F+F+F +F+F+71F+F+F+F+F+F+F+\"\")F+F+F+F+F+F+F+71F+F+F+F+F+F+F+F+\"\"(F+F+F+F +F+F+71F+F+F+F+F+F+F+F+F+\"\"'F+F+F+F+F+71F+F+F+F+F+F+F+F+F+F+\"\"&F+F +F+F+71F+F+F+F+F+F+F+F+F+F+F+\"\"%F+F+F+71F+F+F+F+F+F+F+F+F+F+F+F+\"\" $F+F+71F+F+F+F+F+F+F+F+F+F+F+F+F+\"\"#F+71F+F+F+F+F+F+F+F+F+F+F+F+F+F+ \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B1G-%'matrixG6#7171#\"\" \"\"#:\"\"!F-F-F-F-F-F-F-F-F-F-F-F-F-71F-#F+\"#9F-F-F-F-F-F-F-F-F-F-F- F-F-71F-F-#F+\"#8F-F-F-F-F-F-F-F-F-F-F-F-71F-F-F-#F+\"#7F-F-F-F-F-F-F- F-F-F-F-71F-F-F-F-#F+\"#6F-F-F-F-F-F-F-F-F-F-71F-F-F-F-F-#F+\"#5F-F-F- F-F-F-F-F-F-71F-F-F-F-F-F-#F+\"\"*F-F-F-F-F-F-F-F-71F-F-F-F-F-F-F-#F+ \"\")F-F-F-F-F-F-F-71F-F-F-F-F-F-F-F-#F+\"\"(F-F-F-F-F-F-71F-F-F-F-F-F -F-F-F-#F+\"\"'F-F-F-F-F-71F-F-F-F-F-F-F-F-F-F-#F+\"\"&F-F-F-F-71F-F-F -F-F-F-F-F-F-F-F-#F+\"\"%F-F-F-71F-F-F-F-F-F-F-F-F-F-F-F-#F+\"\"$F-F-7 1F-F-F-F-F-F-F-F-F-F-F-F-F-#F+\"\"#F-71F-F-F-F-F-F-F-F-F-F-F-F-F-F-F+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G-%'matrixG6#7171\"#:\"#9\"#8 \"#7\"#6\"#5\"\"*\"\")\"\"(\"\"'\"\"&\"\"%\"\"$\"\"#\"\"\"71F+F+F,F-F. 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0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 49 "V:=s*INT(cos(psi(s)),s,N)-INT(s*cos(psi(s)),s,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG,&*&%\"sG\"\"\"-%$INTG6%-%$cos G6#-%$psiG6#F'F'%\"NGF(F(-F*6%*&F'\"\"\"F,F(F'F2!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "v2:=INT(psi_(s)*cos(psi(s)),s,w)^2+ INT(psi_(s)*sin(psi(s)),s,w)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# v2G,&*$)-%$INTG6%*&-%%psi_G6#%\"sG\"\"\"-%$cosG6#-%$psiGF.F0F/%\"wG\" \"#\"\"\"F0*$)-F)6%*&F,F8-%$sinGF3F0F/F6F7F8F0" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "T:=1/2*INT(v2,w,N);" }}{PARA 11 "" 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