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0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Vess_ Title3" 256 261 1 {CSTYLE "" -1 -1 "" 0 0 255 0 255 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 258 "" 0 "" {TEXT -1 76 " \+ Vessiot Tutorial: Getting Started II" }}{PARA 260 "" 0 "" {TEXT 257 7 "Purpose" }}{PARA 257 "" 0 "" {TEXT -1 56 "an introduction to calculus on jet spaces with Vessiot." }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(Vessiot):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 256 "" 0 "" {TEXT 258 24 "Forms, Vectors, Biforms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 92 "All the standard arithmetic oper ations on forms, vectors, ... can be used on jet spaces." }}{PARA 0 "" 0 "" {TEXT 256 64 "Moveover on jet spaces we have a new Vessiot \+ object -- biforms." }}{PARA 0 "" 0 "" {TEXT 256 0 "" }{TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "coord_init([x,y],[u],euc):" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 7 "Vec tors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 26 "X:= u[0,1] &mult D_u[0,0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG*&&%\"uG6$\"\"!\"\"\"F*&&%$D_uG6#7$F)F)6#%!GF*" } }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 6 "op(X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%%vectG%$eucG7\"7#7$7#\"\"$&%\"uG6$\"\"!\"\"\" " }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 56 "Y:= v_zip([u[1,0], u[0,0]], [u[1,0], u[ 0,1]],plus,vect);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG,&*&&%\"uG6 $\"\"\"\"\"!F*&&%$D_uG6#7$F*F+6#%!GF*F**&&F(6$F+F+F*&&F.6#7$F+F*6#F2F* F*" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 20 "Z:=Lie_bracket(X,Y) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"ZG,&*&&%\"uG6$\"\"!F*\"\"\"&& %$D_uG6#7$F*F*6#%!GF+!\"\"*&&F(6$F*F+F+&&F.6#7$F*F+6#F2F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 5 "Forms" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 52 "alpha:= evalV(u[0,2]*form(u[2,1]) + u[0,1]*du[0,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG,&*&&%\"uG6$\"\"!\"\"\"F+&&%#duG6#7 $F*F+6#%!GF+F+*&&F(6$F*\"\"#F+&&F.6#7$F6F+6#F2F+F+" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 20 "beta1:=ext_d(alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&beta1G*(&&%#duG6#7$\"\"!\"\"#6#%!G\"\"\"%#^~GF/&&F(6 #7$F,F/6#F.F/" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 21 "beta2:=h ook(Y,alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&beta2G*&&%\"uG6$\" \"!\"\"\"F*&F'6$F)F)F*" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 31 "beta3:=Lie_derivative(Y,alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% &beta3G,&*&&%\"uG6$\"\"!\"\"\"F+&&%#duG6#7$F*F*6#%!GF+F+*&&F(6$F*F*F+& &F.6#7$F*F+6#F2F+F+" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 " " 0 "" {TEXT -1 7 "Biforms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 73 "The type (0,1) biforms are just the standard co ntact forms on jet space." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 23 " alpha:=biform(u[0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG&&%#CuG6#7$\"\"!F*6#%!G" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 30 " omega:=biform_to_form(alpha); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG,(*&&%\"uG6$\"\"\"\"\"!F* &%#dxG6#%!GF*!\"\"*&&F(6$F+F*F*&%#dyG6#F/F*F0&&%#duG6#7$F+F+6#F/F*" }} }{PARA 0 "" 0 "" {TEXT -1 78 "Note the difference between dx and Dx, o ne is a form the other a (1,0) biform." }}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 7 "op(dx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%%formG %$eucG\"\"\"7#7$7#F'F'" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 7 "op(Dx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%'biformG%$eucG7$\" \"\"\"\"!7#7$7#F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 22 "A type (2,3) biform: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 83 "alpha:= evalV( vol_biform() &w biform(u[1,0]) &w biform(u[2,2]) &w biform(u[1 ,4]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG*4&%#DxG6#%!G\"\"\" %#^~GF*&%#DyG6#F)F*F+F*&&%#CuG6#7$F*\"\"!6#F)F*F+F*&&F16#7$\"\"#F:6#F) F*F+F*&&F16#7$F*\"\"%6#F)F*" }}}{PARA 256 "" 0 "" {TEXT -1 0 "" }} {PARA 261 "" 0 "" {TEXT -1 30 "Biform_to_form, Form_to_biform" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 52 "These co mmands change one form type into the other." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 23 " alpha:=biform(u [1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG&&%#CuG6#7$\"\"\"F *6#%!G" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 28 "beta:=biform_to _form(alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG,(*&&%\"uG6$ \"\"#\"\"\"F+&%#dxG6#%!GF+!\"\"*&&F(6$F+F*F+&%#dyG6#F/F+F0&&%#duG6#7$F +F+6#F/F+" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 16 "omega:= du[1 ,0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG&&%#duG6#7$\"\"\"\"\" !6#%!G" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 37 "omega10:=form_t o_biform(omega,[1,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(omega10G, &*&&%\"uG6$\"\"#\"\"!\"\"\"&%#DxG6#%!GF,F,*&&F(6$F,F,F,&%#DyG6#F0F,F, " }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 37 "omega01:=form_to_bifo rm(omega,[0,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(omega01G&&%#CuG 6#7$\"\"\"\"\"!6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{SECT 1 {PARA 256 "" 0 "" {TEXT 259 50 "Total Derivatives and the Euler-Lagran ge Operator." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > \+ " 0 "" {MPLTEXT 1 0 26 "coord_init([x,y],[u],euc);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%0frame~name:~eucG" }}}{PARA 261 "" 0 "" {TEXT -1 0 " " }}{PARA 261 "" 0 "" {TEXT -1 17 "Total Derivatives" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 65 "u[1,1] denotes the m ixed partial of u with respect to x and y." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 11 "f:= u[1,1]:" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 8 "td(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"uG6$\"\"#\"\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 8 "td(f,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"uG6 $\"\"\"\"\"#" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 16 "g:= cos (u[0,0]):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 8 "td(g,x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$sinG6#&%\"uG6$\"\"!F+\"\"\"&F) 6$F,F+F,!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 20 "multi_td( f, [2,3]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"uG6$\"\"$\"\"%" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 26 "Euler Lagrange Expre ssions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 " " {MPLTEXT 1 0 24 "L:= u[1,0]^2 + u[0,1]^2:" }}}{EXCHG {PARA 0 "euc > \+ " 0 "" {MPLTEXT 1 0 7 "EL0(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#,& &%\"uG6$\"\"!\"\"#!\"#&F&6$F)F(F*" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 63 "The Euler-Lagrange operator annihil ates total derivatives." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 29 "f:= x^2*u[1,2] - y^3*u[0,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*&)%\"xG\"\"#\"\"\"&%\"uG6$ \"\"\"F)F.F.*&)%\"yG\"\"$F*&F,6$\"\"!F.F.!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "L:=td(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,(*&%\"xG\"\"\"&%\"uG6$F(\"\"#F(F,*&)%\"yG\"\"$\"\"\"&F*6$ F(F(F(!\"\"*&)F'F,F1&F*6$F,F,F(F(" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 7 "EL0(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#\"\"!" }} }}{SECT 1 {PARA 256 "" 0 "" {TEXT 260 77 "Calculus on Jet Spaces 1: Pr olongations of Transformations and Vector Fields." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "coord_in it([x,y],[u],euc):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 30 "Prolongation of Vector Fields." }}{PARA 0 "" 0 "" {TEXT 256 9 "A Scaling" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 39 "S:= evalV( x*D_x + 2*D_y + 3*D_u[ 0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG,(*&%\"xG\"\"\"&%$D_xG 6#%!GF(F(&%$D_yG6#F,\"\"#&&%$D_uG6#7$\"\"!F66#F,\"\"$" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 17 "S2:=pr_vect(S,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S2G,.*&%\"xG\"\"\"&%$D_xG6#%!GF(F(&%$D_yG6#F,\" \"#&&%$D_uG6#7$\"\"!F66#F,\"\"$*&&%\"uG6$F(F6F(&&F36#7$F(F66#F,F(!\"\" *&&F;6$F0F6F(&&F36#7$F0F66#F,F(!\"#*&&F;6$F(F(F(&&F36#7$F(F(6#F,F(FB" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 10 "A Rot ation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 23 "R:=evalV(-y*D_x+x*D_y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG,&*&%\"yG\"\"\"&%$D_xG6#%!GF(!\"\"*&%\"xGF(&%$D_yG6#F,F(F( " }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 17 "R2:=pr_vect(R,2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#R2G,0*&%\"yG\"\"\"&%$D_xG6#%!GF(!\" \"*&%\"xGF(&%$D_yG6#F,F(F(*&&%\"uG6$\"\"!F(F(&&%$D_uG6#7$F(F76#F,F(F-* &&F56$F(F7F(&&F:6#7$F7F(6#F,F(F(*&&F56$F(F(F(&&F:6#7$\"\"#F76#F,F(!\"# *&,&&F56$FMF7F(&F56$F7FMF-F(&&F:6#7$F(F(6#F,F(F(*&FG\"\"\"&&F:6#7$F7FM 6#F,F(FM" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 32 "Prolongation of Transformations" }}{PARA 0 "" 0 "" {TEXT 256 108 "Let's find the flow of S, prolong it to order 2 and sho w that the vector field for the prolonged flow is S2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 28 "psi:= vect_to_transform(S,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG7%/% \"xG*&-%$expG6#%\"tG\"\"\"F'F-/%\"yG,&F/F-F,\"\"#/&%\"uG6$\"\"!F6,&F3F -F,\"\"$" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 36 "psi2:=simplif y(pr_transform(psi,2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%psi2G7*/ %\"xG*&-%$expG6#%\"tG\"\"\"F'F-/%\"yG,&F/F-F,\"\"#/&%\"uG6$\"\"!F6,&F3 F-F,\"\"$/&F46$F-F6*&-F*6#,$F,!\"\"F-F:F-/&F46$F6F-FB/&F46$F1F6*&-F*6# ,$F,!\"#F-FEF-/&F46$F-F-*&F=\"\"\"FMF-/&F46$F6F1FR" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 45 "newS2:=transform_to_vect(psi2,[t], [t=0]) [1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&newS2G,.*&%\"xG\"\"\"&%$D_x G6#%!GF(F(&%$D_yG6#F,\"\"#&&%$D_uG6#7$\"\"!F66#F,\"\"$*&&%\"uG6$F(F6F( &&F36#7$F(F66#F,F(!\"\"*&&F;6$F0F6F(&&F36#7$F0F66#F,F(!\"#*&&F;6$F(F(F (&&F36#7$F(F(6#F,F(FB" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 16 "newS2 &minus S2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#0~G\"\"\"& %$D_xG6#%!GF%" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 51 "We check some differential invariants for S and R." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 22 "f:= x*u[0,1]*u[0,2]^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" fG*(%\"xG\"\"\"&%\"uG6$\"\"!F'F')&F)6$F+\"\"#F/\"\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 21 "Lie_derivative(S2,f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%\"xG\"\"\"&%\"uG6$\"\"!F%F%)&F'6$F)\"\"#F -\"\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 28 "g:=u[1,1]^2 \+ - u[2,0]*u[0,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,&*$)&%\"uG6 $\"\"\"F+\"\"#\"\"\"F+*&&F)6$F,\"\"!F+&F)6$F1F,F+!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 21 "Lie_derivative(R2,g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&&%\"uG6$\"\"\"F(F(&F&6$\"\"!\"\"#F(F,*&, &&F&6$F,F+F(F)!\"\"F(F%\"\"\"F,*&F%F2F/F(!\"#" }}}{EXCHG {PARA 0 "euc \+ > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 10 "ex pand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 261 "" 0 " " {TEXT -1 40 "The total derivative as a vector field." }}{PARA 261 " " 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 7 "X :=D_x:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 18 "total_X:=tota l(X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(total_XG,&&%$D_xG6#%!G\"\" \"*&&%\"uG6$F*\"\"!F*&&%$D_uG6#7$F/F/6#F)F*F*" }}}{EXCHG {PARA 0 "euc \+ > " 0 "" {MPLTEXT 1 0 29 "total_X2:=pr_vect(total_X,2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%)total_X2G,0&%$D_xG6#%!G\"\"\"*&&%\"uG6$F*\"\" !F*&&%$D_uG6#7$F/F/6#F)F*F**&&F-6$\"\"#F/F*&&F26#7$F*F/6#F)F*F**&&F-6$ F*F*F*&&F26#7$F/F*6#F)F*F**&&F-6$\"\"$F/F*&&F26#7$F9F/6#F)F*F**&&F-6$F 9F*F*&&F26#7$F*F*6#F)F*F**&&F-6$F*F9F*&&F26#7$F/F96#F)F*F*" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 11 "f:= u[0,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG&%\"uG6$\"\"!\"\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 8 "td(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"u G6$\"\"\"F&" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 27 "Lie_deri vative(total_X2,f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"uG6$\"\"\"F &" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT 261 51 "Calculus on Jet Spaces \+ 2: dH, dV, EL_form, I_parts." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 55 "dH is the horizontal exterior derivative on \+ biforms. " }}{PARA 0 "" 0 "" {TEXT 256 51 "dV is the vertical exterio r derivative on biforms." }}{PARA 0 "" 0 "" {TEXT 256 0 "" }}{PARA 0 " " 0 "" {TEXT 256 11 "d= dH + dV." }}{PARA 0 "" 0 "" {TEXT 256 0 "" }} {PARA 0 "" 0 "" {TEXT 256 61 "EL_form is the form version of the Eul er_Lagrange operator." }}{PARA 0 "" 0 "" {TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT 256 46 "I_parts is the integration by parts operator." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "coord_init([x,y],[u],euc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %0frame~name:~eucG" }}}{PARA 0 "" 0 "" {TEXT 256 26 "Define a few form s to use." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc>" 0 " " {MPLTEXT 1 0 24 "alpha:= u[1,0] &mult Dx:" }}}{EXCHG {PARA 0 "euc > \+ " 0 "" {MPLTEXT 1 0 22 "beta:= biform(u[1,3]):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 62 "lambda:= (1/2*u[1,0]* u[0,1] +exp(u[0,0])) \+ &mult vol_biform():" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 " " 0 "" {TEXT -1 9 "dH and dV" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 6 "alpha:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 10 "dH(alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**&%\"uG6$\"\"\"F(F(&%#DxG6#%!GF(%#^~GF(&%#DyG6#F,F(! \"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 10 "dV(alpha);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(&%#DxG6#%!G\"\"\"%#^~GF)&&%#CuG6#7 $F)\"\"!6#F(F)!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 5 "b eta;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&&%#CuG6#7$\"\"\"\"\"$6#%!G" } }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 9 "dH(beta);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&*(&%#DxG6#%!G\"\"\"%#^~GF)&&%#CuG6#7$\"\"#\"\" $6#F(F)F)*(&%#DyG6#F(F)F*F)&&F-6#7$F)\"\"%6#F(F)F)" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 9 "dV(beta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**%#0~G\"\"\"&&%#CuG6#7$\"\"!F+6#%!GF%%#^~GF%&&F(6#7$F%F+6#F-F% " }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 7 "E L_form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 88 "EL_form acts on biforms of top horizontal degree, 0 vertical degr ee -- ie Lagrangians. " }}{PARA 0 "" 0 "" {TEXT 256 25 "It returns a s ource form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > \+ " 0 "" {MPLTEXT 1 0 7 "lambda;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,& *&&%\"uG6$\"\"\"\"\"!F)&F'6$F*F)F)#F)\"\"#-%$expG6#&F'6$F*F*F)F)&%#DxG 6#%!GF)%#^~GF)&%#DyG6#F7F)" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 23 "Delta:=EL_form(lambda):" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }} {PARA 261 "" 0 "" {TEXT -1 7 "I_parts" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 256 71 "I-parts acts on biforms of top hori zontal degree, vertical degee >=1. " }}{PARA 0 "" 0 "" {TEXT 256 31 "I t annihilates dH exact forms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 42 "delta:=(x*u[0,1]) &mult Cu [1,1] &wedge Dy;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deltaG,$*,%\"xG \"\"\"&%\"uG6$\"\"!F(F(&%#DyG6#%!GF(%#^~GF(&&%#CuG6#7$F(F(6#F0F(!\"\" " }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 15 "eta:=dH(delta);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$etaG,&*.,&&%\"uG6$\"\"!\"\"\"!\"\"* &%\"xGF,&F)6$F,F,F,F-F,&%#DxG6#%!GF,%#^~GF,&%#DyG6#F5F,F6F,&&%#CuG6#7$ F,F,6#F5F,F,*0F/\"\"\"F(F,&F36#F5F,F6F,&F86#F5F,F6F,&&F<6#7$\"\"#F,6#F 5F,F-" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 13 "I_parts(eta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*.%#0~G\"\"\"&%#DxG6#%!GF%%#^~GF%&%#DyG6#F)F%F*F%&&%#CuG6#7$\"\" !F36#F)F%" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 50 "The Inverse Problem of the Calculus of Variations." }} {PARA 0 "" 0 "" {TEXT 256 79 "The Helmholtz conditions for a source fo rm are computed using I_Parts o dV" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 6 "Delta;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*.,&&%\"uG6$\"\"\"F(!\"\"-%$expG6#&F&6$\"\"!F/F( F(&%#DxG6#%!GF(%#^~GF(&%#DyG6#F3F(F4F(&&%#CuG6#7$F/F/6#F3F(" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 25 "Helm:=I_parts(dV(Delta)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%HelmG*2%#0~G\"\"\"&%#DxG6#%!GF '%#^~GF'&%#DyG6#F+F'F,F'&&%#CuG6#7$\"\"!F56#F+F'F,F'&&F26#7$F'F56#F+F' " }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 262 "" 0 "" {TEXT -1 64 "Calculus on Jet Spaces 3: Interior Products a nd Lie derivatives." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 256 145 "When working with biforms, it is important to remembe r that in general interior products and Lie derivatives will not pres erve the bigrading. " }}{PARA 0 "" 0 "" {TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT 256 55 "Vessiot has special commands to take this into acco unt." }}{PARA 257 "" 0 "" {TEXT 256 70 "vert_hook: interior product o f a vertical vector field and a biform." }}{PARA 257 "" 0 "" {TEXT 256 67 "total_hook: interior product of a total vector field and a bi form." }}{PARA 257 "" 0 "" {TEXT 256 120 "Lie_biform: Lie derivative \+ of a projectable vector field with respect to the prolongation of pro jectable vector field." }}{PARA 257 "" 0 "" {TEXT 256 111 "Lie_vert: \+ Lie derivative of biform with respect to an evolutionary vector gener alized vertical vector field. " }}{PARA 257 "" 0 "" {TEXT 256 76 "Lie_ total: Lie derivative of a biform with respect to a total vector fiel d." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 22 "coord_init([x,y],[u]):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 9 "Vert_hook" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 63 "A vertical vector field \+ is one with zero horizontal components." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 71 "Hooking with a vertical vector field lowers the ver tical degree by 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 64 "alpha:= evalV(a*Cu[0,0] &w Cu[1,0] + Cu[1,0] &w biform(u[1,2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alp haG,&**%\"aG\"\"\"&&%#CuG6#7$\"\"!F.6#%!GF(%#^~GF(&&F+6#7$F(F.6#F0F(F( *(&F36#F0F(F1F(&&F+6#7$F(\"\"#6#F0F(F(" }}}{EXCHG {PARA 0 "euc > " 0 " " {MPLTEXT 1 0 21 "hook(D_u[1,0],alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"aG\"\"\"&&%#CuG6#7$\"\"!F,6#%!GF&!\"\"&&F)6#7$F&\"\"#6#F. F&" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 32 "beta:=vert_hook(D _u[1,0],alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG,&*&%\"aG\" \"\"&&%#CuG6#7$\"\"!F.6#%!GF(!\"\"&&F+6#7$F(\"\"#6#F0F(" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 10 "Total_hook" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 68 "A total \+ vector field corresponds to the total derivative operator." }}{PARA 0 "" 0 "" {TEXT 256 65 "In Vessiot a total vector is constructed using the total command." }}{PARA 0 "" 0 "" {TEXT 256 69 "Hooking with a to tal vector field lowers the horizontal degree by 1. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 51 "alpha := evalV( a* Dx &w Cu[0,0] + b*Dy &w Cu[0,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG,&**%\"aG\"\"\"&%#DxG6#%!GF(%#^~GF(&&%#CuG6#7$ \"\"!F36#F,F(F(**%\"bGF(&%#DyG6#F,F(F-F(&&F06#7$F3\"\"#6#F,F(F(" }}} {EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 23 "total_hook(D_y, alpha);" } }{PARA 8 "" 1 "" {TEXT -1 64 "Error, (in total_hook) VESSIOT ERROR: ve ctor must be of type tot" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 21 "totalD_y:=total(D_y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)tota lD_yG,&&%$D_yG6#%!G\"\"\"*&&%\"uG6$\"\"!F*F*&&%$D_uG6#7$F/F/6#F)F*F*" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 33 "beta:=total_hook(tota lD_y,alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG*&%\"bG\"\"\"& &%#CuG6#7$\"\"!\"\"#6#%!GF'" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }} {PARA 261 "" 0 "" {TEXT -1 10 "Lie_biform" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 107 "Lie derivative of a projectable \+ vector field with respect to the prolongation of projectable vector f ield." }}{PARA 257 "" 0 "" {TEXT 256 149 "A vector field of the form \+ X= A(x)D_x + B(x,u)D_u is called projectable --- the coefficients of D_x are independent of the dependent variables. " }}{PARA 257 "" 0 "" {TEXT 256 14 "Lie_biform(X, " }{XPPEDIT 256 0 "alpha;" "6#%&alphaG " }{TEXT -1 5 ") = " }{TEXT 256 13 "ext_d hook(X," }{XPPEDIT 18 0 "al pha;" "6#%&alphaG" }{TEXT 256 21 ") + hook (X, ext_d(" }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 256 4 ")) ." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 26 "al pha:= Dx &wedge Cu[1,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG* (&%#DxG6#%!G\"\"\"%#^~GF*&&%#CuG6#7$F*\"\"#6#F)F*" }}}{EXCHG {PARA 0 " euc>" 0 "" {MPLTEXT 1 0 45 "S:= evalV(x*D_x + 2*y*D_y+3*u[0,0]*D_u[0,0 ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG,(*&%\"xG\"\"\"&%$D_xG6#% !GF(F(*&%\"yGF(&%$D_yG6#F,F(\"\"#*&&%\"uG6$\"\"!F7F(&&%$D_uG6#7$F7F76# F,F(\"\"$" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 20 "Lie_biform(S ,alpha);" }}{PARA 8 "" 1 "" {TEXT -1 97 "Error, (in Lie_biform) VESSIO T ERROR: vector must be prolonged to order at least that of biform" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 17 "S3:=pr_vect(S,3);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%#S3G,6*&%\"xG\"\"\"&%$D_xG6#%!GF(F(* &%\"yGF(&%$D_yG6#F,F(\"\"#*&&%\"uG6$\"\"!F7F(&&%$D_uG6#7$F7F76#F,F(\" \"$*&&F56$F(F7F(&&F:6#7$F(F76#F,F(F2*&&F56$F7F(F(&&F:6#7$F7F(6#F,F(F(* &&F56$F2F7F(&&F:6#7$F2F76#F,F(F(*&&F56$F7F2F(&&F:6#7$F7F26#F,F(!\"\"*& &F56$F2F(F(&&F:6#7$F2F(6#F,F(Fin*&&F56$F(F2F(&&F:6#7$F(F26#F,F(!\"#*&& F56$F7F>F(&&F:6#7$F7F>6#F,F(!\"$" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 27 "beta:=Lie_biform(S3,alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG,$*(&%#DxG6#%!G\"\"\"%#^~GF+&&%#CuG6#7$F+\"\"#6 #F*F+!\"\"" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 8 "Lie_vert" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 256 100 "Lie derivative of biform with respect to an evolutio nary vector generalized vertical vector field. " }}{PARA 257 "" 0 "" {TEXT 256 125 "An evolutionary vector field is one of the form A(u^k) D_u --- it is a vertical vector field with jet dependent coefficients " }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT 256 12 "Lie_vert(X, " } {XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 256 21 ") = dV (vert_hook (X" }{TEXT -1 1 "," }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 256 23 "))+ vert_ hook (X, dV(" }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" } {TEXT 256 2 "))" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 25 "Y:=u[1,2] &mult D_u[0,0]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG*&&%\"uG6$\"\"\"\"\"#F)&&%$D _uG6#7$\"\"!F06#%!GF)" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 6 "alpha;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(&%#DxG6#%!G\"\"\"%#^~GF(& &%#CuG6#7$F(\"\"#6#F'F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 18 "Lie_vert(Y,alpha);" }} {PARA 8 "" 1 "" {TEXT -1 95 "Error, (in Lie_vert) VESSIOT ERROR: vecto r must be prolonged to order at least that of biform" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 17 "Y3:=pr_vect(Y,3);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%#Y3G,6*&&%\"uG6$\"\"\"\"\"#F*&&%$D_uG6#7$\"\"! F16#%!GF*F**&&F(6$F+F+F*&&F.6#7$F*F16#F3F*F**&&F(6$F*\"\"$F*&&F.6#7$F1 F*6#F3F*F**&&F(6$F?F+F*&&F.6#7$F+F16#F3F*F**&&F(6$F+F?F*&&F.6#7$F*F*6# F3F*F**&&F(6$F*\"\"%F*&&F.6#7$F1F+6#F3F*F**&&F(6$FXF+F*&&F.6#7$F?F16#F 3F*F**&&F(6$F?F?F*&&F.6#7$F+F*6#F3F*F**&&F(6$F+FXF*&&F.6#7$F*F+6#F3F*F **&&F(6$F*\"\"&F*&&F.6#7$F1F?6#F3F*F*" }}}{EXCHG {PARA 0 "euc > " 0 " " {MPLTEXT 1 0 25 "beta:=Lie_vert(Y3,alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG*(&%#DxG6#%!G\"\"\"%#^~GF*&&%#CuG6#7$\"\"#\"\"% 6#F)F*" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 9 "Lie_total" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 66 " Lie derivative of a biform with respect to a total vec tor field." }}{PARA 257 "" 0 "" {TEXT 256 13 "Lie_total(X, " } {XPPEDIT 256 0 "alpha;" "6#%&alphaG" }{TEXT -1 2 ") " }{TEXT 256 20 " \+ = dH(total_hook(X," }{XPPEDIT 256 0 "alpha;" "6#%&alphaG" }{TEXT 256 23 "))+ total_hook (X, dH(" }{XPPEDIT 256 0 "alpha;" "6#%&alphaG" } {TEXT 256 4 ")) ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 5 "NOTE!" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 122 "One need not remember all these commands -- both hook and Lie_derivative automatically impl ement vert_hook or total_hook " }}{PARA 0 "" 0 "" {TEXT 256 42 "and \+ Lie_biform, Lie_vert, and Lie_total. " }}{PARA 0 "" 0 "" {TEXT 256 0 " " }}{PARA 0 "" 0 "" {TEXT 256 68 "It is important to remember exactly what these commands are doing!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc lid > " 0 "" {MPLTEXT 1 0 29 "hook(D_x, dx &wedge du[0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&&%#duG6#7$\"\"!F(6#%!G" }}}{EXCHG {PARA 0 " euc > " 0 "" {MPLTEXT 1 0 34 "hook(D_u[0,0], dx &wedge du[0,0]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$&%#dxG6#%!G!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 34 "hook(D_u[0,0], Dx &wedge Cu[0,0]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$&%#DxG6#%!G!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 36 "hook(total(D_x), Dx &wedge Cu[0,0]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#&&%#CuG6#7$\"\"!F(6#%!G" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT 262 19 "Homotopy Operators." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 65 "We briefly illustrate the ext_d, dH and dV homotopy operators." }}{PARA 0 "" 0 "" {TEXT 256 39 "See the helpfiles for more examples." }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "coord_init([x,y],[u],euc):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 261 "" 0 "" {TEXT -1 14 "homotopy_ext_d" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 32 "alpha:= (x ^2 * u[0,2]) &mult dy;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG*() %\"xG\"\"#\"\"\"&%\"uG6$\"\"!F(\"\"\"&%#dyG6#%!GF." }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 19 "beta:=ext_d(alpha);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%betaG,&*,%\"xG\"\"\"&%\"uG6$\"\"!\"\"#F(&%#dxG6#%! GF(%#^~GF(&%#dyG6#F1F(F-**)F'F-\"\"\"&F46#F1F(F2F(&&%#duG6#7$F,F-6#F1F (!\"\"" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 26 "eta:=homotopy_e xt_d(beta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$etaG,(**%\"xG\"\"\"& %\"uG6$\"\"!\"\"#F(%\"yGF(&%#dxG6#%!GF(#!\"\"F-*()F'F-\"\"\"F)F7&%#dyG 6#F2F(#\"\"$\"\"%*(F6F7F.F7&&%#duG6#7$F,F-6#F2F(#F4F=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 49 "Check that eta and \+ alpha differ by an exact form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 24 "delta:=alpha &minus eta;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deltaG,(**%\"xG\"\"\"&%\"uG6$\"\" !\"\"#F(%\"yGF(&%#dxG6#%!GF(#F(F-*()F'F-\"\"\"F)F6&%#dyG6#F2F(#F(\"\"% *(F5F6F.F6&&%#duG6#7$F,F-6#F2F(F:" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 13 "ext_d(delta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**%# 0~G\"\"\"&%#dxG6#%!GF%%#^~GF%&%#dyG6#F)F%" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 30 "sigma:= homotopy_ext_d(delta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaG,$*()%\"xG\"\"#\"\"\"&%\"uG6$\"\"!F)\"\"\"% \"yGF/#F/\"\"%" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 47 "check:= alpha &minus (eta &plus ext_d(sigma));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&checkG*&%#0~G\"\"\"&%#dxG6#%!GF'" }}}{EXCHG {PARA 261 "" 0 " " {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 11 "homotopy_dH" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 35 "alpha:= (1/u[0,2]^2) &mult Cu[0,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%&alphaG*&&&%#CuG6#7$\"\"!\"\"#6#%!G\"\"\"*$)&%\"uG6$F+F,\"\"#F/!\" \"" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 16 "beta:=dH(alpha);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG,**&**&%\"uG6$\"\"\"\"\"#F+&% #DxG6#%!GF+%#^~GF+&&%#CuG6#7$\"\"!F,6#F0F+\"\"\"*$)&F)6$F7F,\"\"$F9!\" \"!\"#*&*(&F.6#F0F+F1F+&&F46#7$F+F,6#F0F+F9*$)F<\"\"#F9F?F+*&**&F)6$F7 \"\"$F+&%#DyG6#F0F+F1F+&F36#F0F+F9*$)F<\"\"$F9F?F@*&*(&FS6#F0F+F1F+&&F 46#7$F7FQ6#F0F+F9*$)F<\"\"#F9F?F+" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 23 "eta:=homotopy_dH(beta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$etaG*&&&%#CuG6#7$\"\"!\"\"#6#%!G\"\"\"*$)&%\"uG6$F+F,\"\"#F/! \"\"" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 11 "homotopy_dV" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc >" 0 "" {MPLTEXT 1 0 5 "beta;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&* *&%\"uG6$\"\"\"\"\"#F)&%#DxG6#%!GF)%#^~GF)&&%#CuG6#7$\"\"!F*6#F.F)\"\" \"*$)&F'6$F5F*\"\"$F7!\"\"!\"#*&*(&F,6#F.F)F/F)&&F26#7$F)F*6#F.F)F7*$) F:\"\"#F7F=F)*&**&F'6$F5\"\"$F)&%#DyG6#F.F)F/F)&F16#F.F)F7*$)F:\"\"$F7 F=F>*&*(&FQ6#F.F)F/F)&&F26#7$F5FO6#F.F)F7*$)F:\"\"#F7F=F)" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 9 "dV(beta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*.%#0~G\"\"\"&%#DxG6#%!GF%%#^~GF%&&%#CuG6#7$\"\"!F06#F) F%F*F%&&F-6#7$F%F06#F)F%" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 50 "eta:=(-1) &mult homotopy_dV(beta,[1,infinity,[]]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$etaG,&*&*&&%\"uG6$\"\"\"\"\"#F+&%#DxG6#%!GF+ \"\"\"*$)&F)6$\"\"!F,\"\"#F1!\"\"!\"\"*&*&&F)6$F6\"\"$F+&%#DyG6#F0F+F1 *$)F4\"\"#F1F8F9" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 20 "dV( eta) &minus beta;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**%#0~G\"\"\"&%#D xG6#%!GF%%#^~GF%&&%#CuG6#7$\"\"!F06#F)F%" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 263 "" 0 "" {TEXT -1 56 "Diff erential Equations, Symmetries and Conservation Laws" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 65 "Differential Equation s define another Vessiot data structure. " }}{PARA 0 "" 0 "" {TEXT 256 78 "The 3 commands diffeq, pr_diffeq, and diffeq_to_transform are used together. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 72 "Use the command diffeq to define a differential equation data structure." }}{PARA 0 "" 0 "" {TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT 256 51 " Use pr_diffeq to prolong a differential equation." }} {PARA 0 "" 0 "" {TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT 256 79 "Use di ffeq_to_transform to change a differential equation to a transformatio n." }}{PARA 0 "" 0 "" {TEXT 256 2 " " }}{PARA 0 "" 0 "" {TEXT 256 103 "This transformation can then be used to pullback differential f orms on the free variational bicomplex" }}{PARA 0 "" 0 "" {TEXT 256 27 "to the equation manifold." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 14 "with(Vessiot):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "coord_init([x,t],[u],euc):" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }} {PARA 261 "" 0 "" {TEXT -1 6 "diffeq" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 31 "We consider the KdV equation. " }} {PARA 257 "" 0 "" {TEXT 256 70 "typical one solves for the highest de rivative -- in this case u_xxx ." }}{PARA 257 "" 0 "" {TEXT 256 73 "ho wever for evolution equations it is often desirable to solve for u_t ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 256 22 "W e try both methods." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 58 "KdVxxx:=diffeq([u[3,0] + u[0,0]*u[1,0] \+ -u[0,1]],[u[3,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'KdVxxxG7$<#& %\"uG6$\"\"$\"\"!7#,(F'\"\"\"*&&F(6$F+F+F.&F(6$F.F+F.F.&F(6$F+F.!\"\" " }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 56 "KdVt:=diffeq([u[3,0 ] + u[0,0]*u[1,0] -u[0,1]],[u[0,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%KdVtG7$<#&%\"uG6$\"\"!\"\"\"7#,(&F(6$\"\"$F*F+*&&F(6$F*F*F+&F(6$ F+F*F+F+F'!\"\"" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 " " {TEXT -1 9 "pr_diffeq" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 112 "This command differentiates the components of the original equation and the set of variables to be solved for." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 32 "pr_KdVxxx1:=pr_diffeq(KdVxxx,1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+pr_KdVxxx1G7$<%&%\"uG6$\"\"%\"\"!&F(6$\"\"$\"\"\"&F(6$F.F+7%, (F0F/*&&F(6$F+F+F/&F(6$F/F+F/F/&F(6$F+F/!\"\",**$)F7\"\"#\"\"\"F/*&F5F @&F(6$F?F+F/F/&F(6$F/F/F;F'F/,**&F7F@F9F/F/*&F5F@FDF/F/&F(6$F+F?F;F,F/ " }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 28 "pr_KdVt1:=pr_diffeq (KdVt,2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)pr_KdVt1G7$<(&%\"uG6$ \"\"#\"\"\"&F(6$F+F*&F(6$\"\"!\"\"$&F(6$F+F+&F(6$F0F*&F(6$F0F+7(,(&F(6 $F1F0F+*&&F(6$F0F0F+&F(6$F+F0F+F+F6!\"\",**$)F?F*\"\"\"F+*&F=FE&F(6$F* F0F+F+F2FA&F(6$\"\"%F0F+,**&F?FEF6F+F+*&F=FEF2F+F+F4FA&F(6$F1F+F+,*&F( 6$\"\"&F0F+F'FA*&F=FEF:F+F+*&FGFEF?FEF1,,&F(6$FKF+F+*&F=FEF'F+F+F,FA*& F2FEF?FEF**&F6FEFGFEF+,,&F(6$F1F*F+*&F=FEF,F+F+F.FA*&F2FEF6FEF**&F?FEF 4F+F+" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 19 "diffeq_to_trans form" }}{PARA 0 "" 0 "" {TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT 256 108 "This command constructs a transformation which defines an imbeddi ng of the equation manifold into jet space." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 41 "Phi_xxx:=diffe q_to_transform(pr_KdVxxx1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Phi_ xxxG73/%\"xGF'/%\"tGF)/&%\"uG6$\"\"!F.F+/&F,6$\"\"\"F.F0/&F,6$F.F2F4/& F,6$\"\"#F.F7/&F,6$F2F2F;/&F,6$F.F9F>/&F,6$\"\"$F.,&*&F+F2F0F2!\"\"F4F 2/&F,6$F9F2FH/&F,6$F2F9FK/&F,6$F.FCFN/&F,6$\"\"%F.,(*$)F0F9\"\"\"FF*&F +FWF7F2FFF;F2/&F,6$FCF2,(*&F0FWF4F2FF*&F+FWF;F2FFF>F2/&F,6$F9F9Fjn/&F, 6$F2FCF]o/&F,6$F.FSF`o" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 37 "Phi_t:=diffeq_to_transform(pr_KdVt1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&Phi_tG7./%\"xGF'/%\"tGF)/&%\"uG6$\"\"!F.F+/&F,6$\"\" \"F.F0/&F,6$F.F2,&&F,6$\"\"$F.F2*&F+F2F0F2F2/&F,6$\"\"#F.F\"\"\"F2*&F+FEF,,*&F0FEF7F2F2*&F+ FEFDFEF>*&)F+F>FEFF2,(&F,6$\" \"&F.F2*&F+FEF7FEF2*&F,0&F,6$FIF2F2*&F+FEFZF2F2*&FQ FEF7FEF2*(F*&F0FEFGFEF>*&FF2*&F+FEF]oF2F2*&FQFEFZFEF2*&)F+F9FEF7FEF2*(FQFEF0FEF*&F0FEFSF2F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 256 13 "We see that " }{XPPEDIT 256 0 "Phi;" "6#%$PhiG" } {TEXT 256 8 "_xxx and" }{TEXT -1 1 " " }{XPPEDIT 256 0 "Phi;" "6#%$Phi G" }{TEXT 256 90 "_t define different parametrizations of the first \+ prolongation of the equation manifold." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 26 "pullback(Phi_xxx, u[3,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&&%\"uG6$\"\"!F(\"\"\"&F&6$F)F(F)!\"\"&F&6$F(F)F)" } }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 23 "pullback(Phi_t,u[3,0]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"uG6$\"\"$\"\"!" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 25 "pullback(Phi_xxx,u[1,1]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"uG6$\"\"\"F&" }}}{EXCHG {PARA 0 "e uc > " 0 "" {MPLTEXT 1 0 24 "pullback(Phi_t, u[1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)&%\"uG6$\"\"\"\"\"!\"\"#\"\"\"F)*&&F'6$F*F*F) &F'6$F+F*F)F)&F'6$\"\"%F*F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 29 "Check a Point Symmetry of KdV" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 44 "S:= evalV( x*D_x +3*t*D_t-2*u[0,0]*D_u[0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG,(*&%\"xG\"\"\"&%$D_xG6#%!GF(F(*&%\"tGF(&%$D_tG6# F,F(\"\"$*&&%\"uG6$\"\"!F7F(&&%$D_uG6#7$F7F76#F,F(!\"#" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 17 "S3:=pr_vect(S,3);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%#S3G,:*&%\"xG\"\"\"&%$D_xG6#%!GF(F(*&%\"tGF(&% $D_tG6#F,F(\"\"$*&&%\"uG6$\"\"!F7F(&&%$D_uG6#7$F7F76#F,F(!\"#*&&F56$F( F7F(&&F:6#7$F(F76#F,F(!\"$*&&F56$F7F(F(&&F:6#7$F7F(6#F,F(!\"&*&&F56$\" \"#F7F(&&F:6#7$FTF76#F,F(!\"%*&&F56$F(F(F(&&F:6#7$F(F(6#F,F(!\"'*&&F56 $F7FTF(&&F:6#7$F7FT6#F,F(!\")*&&F56$F2F7F(&&F:6#7$F2F76#F,F(FP*&&F56$F TF(F(&&F:6#7$FTF(6#F,F(!\"(*&&F56$F(FTF(&&F:6#7$F(FT6#F,F(!\"**&&F56$F 7F2F(&&F:6#7$F7F26#F,F(!#6" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 40 "Delta:= u[3,0] + u[1,0]*u[0,0] - u[0,1]:" }}}{EXCHG {PARA 0 "e uc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 35 "LieS3_eq:=Lie_derivative(S3,Delta);" }}{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)LieS3_e qG,(*&&%\"uG6$\"\"!F*\"\"\"&F(6$F+F*F+!\"&&F(6$F*F+\"\"&&F(6$\"\"$F*F. " }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 34 "check:=pullback(Phi _xxx,LieS3_eq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&checkG\"\"!" }}} {PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 34 "Check a Conservation Law: Method 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 256 34 "Suppose just the density is given." }} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "H:=1/3*u[0,0]^3 -u[1,0]^ 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG,&*$)&%\"uG6$\"\"!F+\"\"$ \"\"\"#\"\"\"F,*$)&F)6$F/F+\"\"#F-!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 12 "Ht:=td(H,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #HtG,&*&)&%\"uG6$\"\"!F+\"\"#\"\"\"&F)6$F+\"\"\"F0F0*&&F)6$F0F0F0&F)6$ F0F+F0!\"#" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 22 "A:=pullba ck(Phi_t,Ht);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,,*&)&%\"uG6$\" \"!F+\"\"#\"\"\"&F)6$\"\"$F+\"\"\"F1*&)F(F0F-&F)6$F1F+F1F1*$)F4F0F-!\" #*(&F)6$F,F+F1F(F1F4F-F8*&F4F-&F)6$\"\"%F+F1F8" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 7 "EL0(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7 #,(*&&%\"uG6$\"\"#\"\"!\"\"\"&F'6$F+F*F+!\"#*&)&F'6$F*F*F)\"\"\"F,F3\" \"$*&,&*$F0F3F4F&F.F+F,F3!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#\"\"! " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 47 "Fin d the Dt component of the conservation law." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 29 "alpha:= A &mul t vol_biform();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG**,,*&)&% \"uG6$\"\"!F,\"\"#\"\"\"&F*6$\"\"$F,\"\"\"F2*&)F)F1F.&F*6$F2F,F2F2*$)F 5F1F.!\"#*(&F*6$F-F,F2F)F2F5F.F9*&F5F.&F*6$\"\"%F,F2F9F2&%#DxG6#%!GF2% #^~GF2&%#DtG6#FDF2" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "b eta:= homotopy_dH(alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG* &,,*$)&%\"uG6$\"\"!F,\"\"%\"\"\"#\"\"\"F-*&)F)\"\"#F.&F*6$F3F,F0F0*&F) F0)&F*6$F0F,F3F.!\"#*&F8F0&F*6$\"\"$F,F0F:*$)F4F3F.F0F0&%#DtG6#%!GF0" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 35 "Check a Conservation Law: Method 2:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 53 "Construct the entire conservat ion law and check it. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 32 "omega:= (H &mult Dx) &plus beta ;" }}{PARA 0 "euc>" 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG,&*&,&*$)&%\"uG6$\"\"!F-\"\"$\"\"\"#\"\"\"F.*$ )&F+6$F1F-\"\"#F/!\"\"F1&%#DxG6#%!GF1F1*&,,*$)F*\"\"%F/#F1F@*&)F*F6F/& F+6$F6F-F1F1*&F*F1F3F/!\"#*&F4F1&F+6$F.F-F1FG*$)FDF6F/F1F1&%#DtG6#F;F1 F1" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 35 "check:=pullback(P hi_xxx,dH(omega)):" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#**%#0~G \"\"\"&%#DxG6#%!GF%%#^~GF%&%#DtG6#F)F%" }}}{EXCHG {PARA 0 "euc > " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 39 "Check a Generalized Symmetry. Method 1" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 71 "Q:=u[5,0] + 5/3*u[0,0]*u[3,0]+ 10/3*u[1,0]*u[2,0] +5/ 6*u[0,0]^2*u[1,0]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 21 "Y := Q &mult D_u[0,0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG*&&,*&% \"uG6$\"\"&\"\"!\"\"\"*&&F)6$F,F,F-&F)6$\"\"$F,F-#F+F3*&&F)6$\"\"#F,F- &F)6$F-F,F-#\"#5F3*&)F/F8\"\"\"F9F?#F+\"\"'6#%!GF-&&%$D_uG6#7$F,F,6#FC F-" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 17 "Y3:=pr_vect(Y,3):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 34 "LieY3Eq:=Lie_derivati ve(Y3,Delta);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(LieY3EqG,F*&,*&%\" uG6$\"\"&\"\"!\"\"\"*&&F)6$F,F,F-&F)6$\"\"$F,F-#F+F3*&&F)6$\"\"#F,F-&F )6$F-F,F-#\"#5F3*&)F/F8\"\"\"F9F?#F+\"\"'F-F9F?F-*&,.&F)6$FAF,F-*&F/F? &F)6$\"\"%F,F-F4*&F9F?F1F?F+*$)F6F8F?F;*&F>F?F6F?F@*&F/F?)F9F8F?F4F-F/ F?F-&F)6$F+F-!\"\"*&&F)6$F3F-F-F/F?#!\"&F3*&&F)6$F8F-F-F9F?#!#5F3*&&F) 6$F-F-F-F6F?Fen*&FhnF?F>F?#FWFA*&&F)6$F,F-F-F1F?FV*(F]oF?F/F?F9F?FV*&F 9F?F(F-#\"#DF3*$)F1F8F?#\"#NF3*(F1F?F/F?F9F?#\"#?F3&F)6$\"\")F,F-*&F/F ?FDF-F4*&F6F?FOF?F<*&F/F?FLF?F+*&F6F?FGF?#\"#bF3*&FGF?F>F?F@" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc \+ > " 0 "" {MPLTEXT 1 0 46 "Phi:=diffeq_to_transform(pr_diffeq(KdVxxx,6) ):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 30 "expand(pullback(Phi,LieY3Eq));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 261 "" 0 "" {TEXT -1 40 "Check a Generalized Sy mmetry. Method 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 78 "This method uses the generalized Lie bracket of two gen eralized vector fields" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 49 "KdVVect:= (u[3,0] +u [1,0]*u[0,0]) &mult D_u[0,0]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 27 "gen _Lie_bracket(KdVVect,Y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#0~G\" \"\"&%$D_xG6#%!GF%" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 264 "" 0 "" {TEXT -1 9 "Utilities" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 66 "We list a few ulitities which are useful for working on \+ jet spaces" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "coord_init([x,y],[u],euc):" }}}{PARA 261 "" 0 " " {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 21 "pr_Vessiot_frame_data " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 69 "This command prolongs the jet space of the current frame by 1 order." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 19 "frameInformation();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2frame~n ame:~~~eucG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8library~name:~~~vst_e ucG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5Frame~Jet~Variables:G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7'%\"xG%\"yG&%\"uG6$\"\"!F)&F'6$\"\"\" F)&F'6$F)F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Frame~LabelsG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7'&%$D_xG6#%!G&%$D_yG6#%!G&&%$D_uG6#7$ \"\"!F16#%!G&&F.6#7$\"\"\"F16#%!G&&F.6#7$F1F86#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%/CoFrame~LabelsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7'&%#dxG6#%!G&%#dyG6#%!G&&%#duG6#7$\"\"!F16#%!G&&F.6#7$\"\"\"F16#%!G&& F.6#7$F1F86#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Horizontal~Cofram e~LabelsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$&%#DxG6#%!G&%#DyG6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Vertical~Coframe~LabelsG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%&&%#CuG6#7$\"\"!F)6#%!G&&F&6#7$\"\" \"F)6#%!G&&F&6#7$F)F06#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/------ --------G" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 24 "pr_Vessiot _frame_data();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 19 "frameInformation();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2frame~name:~~~eucG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8library~name:~~~vst_eucG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5Frame~Jet~Variables:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*%\"xG%\"yG&%\"uG6$\"\"!F)&F'6$\"\"\"F)&F'6$F)F,&F'6$\"\"#F)&F' 6$F,F,&F'6$F)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Frame~LabelsG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7*&%$D_xG6#%!G&%$D_yG6#%!G&&%$D_uG6#7$ \"\"!F16#%!G&&F.6#7$\"\"\"F16#%!G&&F.6#7$F1F86#%!G&&F.6#7$\"\"#F16#%!G &&F.6#7$F8F86#%!G&&F.6#7$F1FE6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %/CoFrame~LabelsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*&%#dxG6#%!G&%#d yG6#%!G&&%#duG6#7$\"\"!F16#%!G&&F.6#7$\"\"\"F16#%!G&&F.6#7$F1F86#%!G&& F.6#7$\"\"#F16#%!G&&F.6#7$F8F86#%!G&&F.6#7$F1FE6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Horizontal~Coframe~LabelsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$&%#DxG6#%!G&%#DyG6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Vertical~Coframe~LabelsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7( &&%#CuG6#7$\"\"!F)6#%!G&&F&6#7$\"\"\"F)6#%!G&&F&6#7$F)F06#%!G&&F&6#7$ \"\"#F)6#%!G&&F&6#7$F0F06#%!G&&F&6#7$F)F=6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/--------------G" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" } }{PARA 261 "" 0 "" {TEXT -1 17 "frameJetDimension" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 20 "frameJet Dimension();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 261 " " 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 13 "frameJetOrder" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 16 "frameJetOrder();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 17 "frameJetVariables" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 20 "frameJetVariables();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*%\"xG%\"yG&%\"uG6$\"\"!F)&F'6$\"\"\"F)&F'6$F)F,& F'6$\"\"#F)&F'6$F,F,&F'6$F)F1" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }} {PARA 261 "" 0 "" {TEXT -1 15 "frameJetVectors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 25 "Gamma:=f rameJetVectors();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&GammaG7*&%$D_x G6#%!G&%$D_yG6#%!G&&%$D_uG6#7$\"\"!F36#%!G&&F06#7$\"\"\"F36#%!G&&F06#7 $F3F:6#%!G&&F06#7$\"\"#F36#%!G&&F06#7$F:F:6#%!G&&F06#7$F3FG6#%!G" }}} {EXCHG {PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 13 "frameJetForms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " euc > " 0 "" {MPLTEXT 1 0 23 "Omega:=frameJetForms();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&OmegaG7*&%#dxG6#%!G&%#dyG6#%!G&&%#duG6#7$\"\"!F 36#%!G&&F06#7$\"\"\"F36#%!G&&F06#7$F3F:6#%!G&&F06#7$\"\"#F36#%!G&&F06# 7$F:F:6#%!G&&F06#7$F3FG6#%!G" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 0 " " }}{PARA 261 "" 0 "" {TEXT -1 14 "order_function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 23 "f:= u[2 ,3]*u[0,1] +x^4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*&&%\"uG6$ \"\"#\"\"$\"\"\"&F(6$\"\"!F,F,F,*$)%\"xG\"\"%\"\"\"F," }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 15 "objectOrder(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 0 "" } }{PARA 261 "" 0 "" {TEXT -1 11 "objectOrder" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 34 "omega1:=u[1,3 ] &mult form(u[1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'omega1G*&& %\"uG6$\"\"\"\"\"$F)&&%#duG6#7$F)F)6#%!GF)" }}}{EXCHG {PARA 0 "euc > \+ " 0 "" {MPLTEXT 1 0 20 "objectOrder(omega1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 35 "omega2:= u[1,1] &mult form(u[1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%'omega2G*&&%\"uG6$\"\"\"F)F)&&%#duG6#7$F)\"\"#6#%!GF)" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 20 "objectOrder(omega2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }} {PARA 261 "" 0 "" {TEXT -1 23 "Special Transformations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "Id: =identity_transform(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IdG7'/% \"xGF'/%\"yGF)/&%\"uG6$\"\"!F.F+/&F,6$\"\"\"F.F0/&F,6$F.F2F4" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 30 "PI:=projection_transform (2,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#PIG7'/%\"xGF'/%\"yGF)/&% \"uG6$\"\"!F.F+/&F,6$\"\"\"F.F0/&F,6$F.F2F4" }}}{EXCHG {PARA 0 "euc > \+ " 0 "" {MPLTEXT 1 0 20 "transformDomain(PI);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%$eucG\"\"#" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 19 "transformRange(PI);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7$%$eucG\"\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }} }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 263 19 "up dated 01/23/03:IA" }}}{MARK "14 0" 0 }{VIEWOPTS 1 1 0 3 4 1802 }