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0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 261 356 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 "" 0 357 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 " " 0 358 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 "" 0 359 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 "" 11 360 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 "" 261 361 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 " " 261 362 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 99 " \+ Vessiot Tutorial: Moving Frames and Differential Invariants" }}{PARA 260 "" 0 "" {TEXT 257 7 "Purpose" }}{PARA 257 "" 0 "" {TEXT -1 104 "we demostrate the method of moving frames to calculate diffe rential invariants and invariant coframes." }}{PARA 256 "" 0 "" {TEXT -1 7 "Summary" }}{PARA 257 "" 0 "" {TEXT -1 96 "with Vessiot's transfo rmation capabilities, the moving frames algorithm is easily implement ed. " }}{PARA 256 "" 0 "" {TEXT -1 9 "Reference" }}{PARA 257 "" 0 "" {TEXT -1 17 "Fels and Olver : " }{TEXT 260 17 "Moving coframe I " } {TEXT -1 34 "Acta Applicandae Math. 51 161-213." }}{PARA 256 "" 0 "" {TEXT -1 22 "Procedures Illustrated" }}{PARA 257 "" 0 "" {TEXT -1 63 " right_moving_frame, right_moving_frame_matrix, Maurer_Cartan. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 33 "with(Vessiot):with(moving_frame);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&%.Maurer_CartanG%9left_moving_frame_matrixG%3right_mo ving_frameG%:right_moving_frame_matrixG" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 262 "" 0 "" {TEXT -1 52 "The Gen eral Linear Group -projectable action on R^2" }}{PARA 263 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 264 "euc > " 0 "" {MPLTEXT 1 0 24 "coord_ init([x],[u],euc):" }}}{EXCHG {PARA 265 "euc>" 0 "" {MPLTEXT 1 0 29 "c oord_init([a,b,c,d],[],GL2):" }}}{EXCHG {PARA 266 "GL2>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 267 "GL2>" 0 "" {MPLTEXT 1 0 67 "Phi :=transform( euc,euc,[x=(a*x +b)/(c*x +d) ,u[0]= u[0]/(c*x+d)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PhiG7$/%\"xG*&,&*&%\"aG\"\"\"F'F,F, %\"bGF,\"\"\",&*&%\"cGF,F'F.F,%\"dGF,!\"\"/&%\"uG6#\"\"!*&F5F.F/F3" }} }{EXCHG {PARA 268 "euc > " 0 "" {MPLTEXT 1 0 26 "Phi4:=pr_transform(Ph i,4):" }}}{EXCHG {PARA 346 "" 0 "" {TEXT -1 0 "" }}{PARA 347 "" 0 "" {TEXT 256 72 "Check the action on the 2-jet to determine an admissabl e cross-section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 269 "euc > " 0 "" {MPLTEXT 1 0 20 "pullback(Phi4,u[1]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&,(*&&%\"uG6#\"\"!\"\"\"%\"cGF*!\"\"*(&F'6#F*F*F+\" \"\"%\"xGF*F**&F.F0%\"dGF*F*F0,&*&%\"aGF*F3F0F**&%\"bGF*F+F0F,!\"\"" } }}{EXCHG {PARA 270 "euc > " 0 "" {MPLTEXT 1 0 20 "pullback(Phi4,u[2]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*(,(*&)%\"cG\"\"#\"\"\")%\"xGF)F *\"\"\"*(F(F-F,F-%\"dGF-F)*$)F/F)F*F-F-,&*&F(F*F,F*F-F/F-F-&%\"uG6#F)F -F**$),&*&%\"aGF-F/F*F-*&%\"bGF-F(F*!\"\"\"\"#F*!\"\"" }}}{EXCHG {PARA 271 "euc > " 0 "" {MPLTEXT 1 0 19 "_EnvExplicit:=true;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-_EnvExplicitG%%trueG" }}}{EXCHG {PARA 272 "euc > " 0 "" {MPLTEXT 1 0 64 "rho:=right_moving_frame(Phi4,GL2,[x =0,u[0]=1,u[1]=0 ,u[2]=1],2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rh oG7&/%\"aG,$*$-%%sqrtG6#*&&%\"uG6#\"\"#\"\"\"&F/6#\"\"!F2\"\"\"!\"\"/% \"bG*&F*F6%\"xGF2/%\"cG&F/6#F2/%\"dG,&F3F2*&F>F2F;F6F7" }}}{EXCHG {PARA 273 "" 0 "" {TEXT -1 0 "" }}{PARA 274 "" 0 "" {TEXT 256 39 "Chec k that rho(g*x) = rho(x)*g^(-1)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 275 "euc > " 0 "" {MPLTEXT 1 0 25 "A:=matrix([[a,b],[c, d]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$%\"aG% \"bG7$%\"cG%\"dG" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 90 "mov ing_frame_matrixR:=right_moving_frame_matrix(Phi4,GL2,[x=0,u[0]=1,u[1] =0 ,u[2]=1],2,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%5moving_frame_m atrixRG-%'matrixG6#7$7$,$*$-%%sqrtG6#*&&%\"uG6#\"\"#\"\"\"&F16#\"\"!F4 \"\"\"!\"\"*&F,F8%\"xGF47$&F16#F4,&F5F4*&F=F4F;F8F9" }}}{EXCHG {PARA 276 "euc > " 0 "" {MPLTEXT 1 0 62 "LHS:=map(simplify, map2(pullback,Ph i4, moving_frame_matrixR));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$LHSG -%'matrixG6#7$7$,$*$-%%sqrtG6#*&*(&%\"uG6#\"\"#\"\"\"&F26#\"\"!F5),&*& %\"cGF5%\"xGF5F5%\"dGF5F4\"\"\"F?*$),&*&%\"aGF5F>F5F5*&%\"bGF5FF?F5F?FBFI,$*&,(*&F6F?FDF?FG*(FSF?FDF?F=F ?F5*&FSF?FFF?F5F?FBFIFG" }}}{EXCHG {PARA 277 "euc > " 0 "" {MPLTEXT 1 0 56 "RHS:=map(simplify,evalm(moving_frame_matrixR &*A^(-1)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$RHSG-%'matrixG6#7$7$,$*&*&-%%sqrtG6 #*&&%\"uG6#\"\"#\"\"\"&F26#\"\"!F5\"\"\",&*&%\"cGF5%\"xGF5F5%\"dGF5F5F 9,&*&%\"aGF5F>F5F5*&%\"bGF5FF9F5F9F?FE,$*&,( *&F6F9FAF9FD*(FOF9FAF9F=F9F5*&FOF9FCF9F5F9F?FEFD" }}}{EXCHG {PARA 0 "e uc > " 0 "" {MPLTEXT 1 0 38 "map(simplify,evalm(LHS-RHS),symbolic);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"!F(F'" }}}{EXCHG {PARA 278 "" 0 "" {TEXT -1 0 "" }}{PARA 279 "" 0 "" {TEXT 256 79 "Exhi bit the left moving frame. This is the one given in the paper page, 180." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 280 "euc > " 0 "" {MPLTEXT 1 0 30 "inverse(moving_frame_matrixR);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'matrixG6#7$7$*&,&&%\"uG6#\"\"!!\"\"*&&F+6#\"\"\"F2 %\"xGF2F2\"\"\"*&-%%sqrtG6#*&&F+6#\"\"#F2F*F2F4F*\"\"\"!\"\"*&F3F4F*F> 7$*&F0F4*&-F76#F9F4F*\"\"\"F>*&F4F4F*F>" }}}{EXCHG {PARA 348 "" 0 "" {TEXT -1 0 "" }}{PARA 281 "" 0 "" {TEXT 256 54 "Find the 3rd and 4th order differential invariants." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 282 "euc > " 0 "" {MPLTEXT 1 0 24 "u3:=pullback(Phi4,u[3] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u3G*&*&,0*()%\"cG\"\"$\"\"\"& %\"uG6#\"\"#\"\"\")%\"xGF0F,F+**)F*F0F,F-F,F3F1%\"dGF1\"\"'*(F*F1F-F,) F6F0F,F+*(&F.6#F+F1F)F,)F3F+F,F1**F;F,F5F,F2F,F6F,F+**F;F,F*F,F3F,F9F, F+*&F;F,)F6F+F,F1F1),&*&F*F,F3F,F1F6F1F0F,F,*$),&*&%\"aGF1F6F,F1*&%\"b GF1F*F,!\"\"\"\"$F,!\"\"" }}}{EXCHG {PARA 283 "euc > " 0 "" {MPLTEXT 1 0 31 "I1:=simplify(pullback(rho,u3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I1G,$*&,&*&&%\"uG6#\"\"\"F,&F*6#\"\"#F,\"\"$*&&F*6#F0F,&F*6# \"\"!F,F,\"\"\"*&F-\"\"\"-%%sqrtG6#*&F-F7F4F7F7!\"\"!\"\"" }}}{EXCHG {PARA 284 "euc > " 0 "" {MPLTEXT 1 0 24 "u4:=pullback(Phi4,u[4]):" }} {PARA 285 "euc > " 0 "" {MPLTEXT 1 0 31 "I2:=simplify(pullback(rho,u4) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I2G*&,(*&&%\"uG6#\"\"%\"\"\") &F)6#\"\"!\"\"#\"\"\"F,*&)&F)6#F,F1F2&F)6#F1F,\"#7*(F5F,&F)6#\"\"$F,F. F,\"\")F2*&F.\"\"\")F7\"\"#F2!\"\"" }}}{EXCHG {PARA 352 "" 0 "" {TEXT -1 0 "" }}{PARA 355 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 70 "To construct invariant forms on the jet space, proceed as follow s: " }}{PARA 286 "" 0 "" {TEXT 256 0 "" }}{PARA 353 "" 0 "" {TEXT 256 52 "1. First construct the multiplication map on GL(2): " }}{PARA 354 "" 0 "" {TEXT 256 81 "2. Then execute the program Maurer_Cartan to get the invariant forms on the group" }}{PARA 356 "" 0 "" {TEXT 256 71 "3. Pullback by the moving frame to get invariant forms on the jet \+ space" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 29 "G_dot:=proc(X,Y) local A,B,C;" }}{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 17 "A:=matrix(2,2,X);" }}{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 17 "B:=matrix(2,2,Y);" }}{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 15 "C:=evalm(A&*B);" }}{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 33 "map(op,convert(C,listlist)); end;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&G_dotGR6$%\"XG%\"YG6%%\"AG%\"BG%\"CG6\"F-C&>8$-%'m atrixG6%\"\"#F49$>8%-F26%F4F49%>8&-%&evalmG6#-%#&*G6$F0F7-%$mapG6$%#op G-%(convertG6$F<%)listlistGF-F-F-" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 35 "G_d ot([a1,b1,c1,d1],[a2,b2,c2,d2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7& ,&*&%#a1G\"\"\"%#a2GF'F'*&%#b1GF'%#c2GF'F',&*&F&\"\"\"%#b2GF'F'*&F*F.% #d2GF'F',&*&%#c1GF'F(F.F'*&%#d1GF'F+F.F',&*&F4F.F/F.F'*&F6F.F1F.F'" }} }{EXCHG {PARA 287 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 " " {MPLTEXT 1 0 21 "change_frame_to(GL2);" }}}{EXCHG {PARA 0 "GL2 > " 0 "" {MPLTEXT 1 0 41 "Omega:=Maurer_Cartan(G_dot,[1,0,0,1])[4];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%8The~output~is~listed~asG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%fo[right~inv.~vectors,~right~inv.~~forms,~l eft~inv.~vectors,~left~inv.~forms]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%&OmegaG7&,&*&*&%\"dG\"\"\"&%#daG6#%!G\"\"\"F*,&*&%\"aGF/F)F/F/*&%\" bGF/%\"cGF/!\"\"!\"\"F/*&*&F5F*&%#dbG6#%!GF/F*F0F7F6,&*&*&F4F*&F,6#%!G F/F*F0F7F6*&*&F2F*&F;6#%!GF/F*F0F7F/,&*&*&F)F*&%#dcG6#%!GF/F*F0F7F/*&* &F5F*&%#ddG6#%!GF/F*F0F7F6,&*&*&F4F*&FM6#%!GF/F*F0F7F6*&*&F2F*&FS6#%!G F/F*F0F7F/" }}}{PARA 349 "" 0 "" {TEXT -1 0 "" }}{PARA 288 "" 0 "" {TEXT 256 43 "Construct the invariant forms on jet space." }}{PARA 350 "" 0 "" {TEXT 256 1 " " }}{EXCHG {PARA 289 "GL2 > " 0 "" {MPLTEXT 1 0 36 "Inv_forms:=map2(pullback,rho,Omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Inv_formsG7&,(*&*&&%\"uG6#\"\"\"F,&%#dxG6#%!GF,\"\" \"&F*6#\"\"!!\"\"F,*&&&%#duG6#7#F46#%!GF1F2F5#F,\"\"#*&&&F96#7#F?6#%!G F1&F*6#F?F5F>*&*&FGF1&F.6#%!GF,F1*$-%%sqrtG6#*&FGF,F2F,F1F5,(*&*&)F)F? F1&F.6#%!GF,F1*&-FP6#FRF1F2\"\"\"F5!\"\"*&*&F)F1&F86#%!GF,F1*&-FP6#FRF 1F2\"\"\"F5F,*&&&F96#7#F,6#%!GF1*$-FP6#FRF1F5Fhn,&*&*&F)F1&F.6#%!GF,F1 F2F5Fhn*&&F86#%!GF1F2F5F," }}}{PARA 351 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 54 "To construct the operator of invariant diffe rential:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 73 "1. Construct \+ a horizontonal invariant biform and convert back to a form." }}{PARA 0 "" 0 "" {TEXT 256 24 "2. Find the dual vector." }}{PARA 0 "" 0 "" {TEXT 256 35 "3. Convert to a total vector field." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "I" }{TEXT 256 61 "n this e xample we can skip step 1 and begin with Inv_forms[2]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 40 "X:=dual_frame([Inv_forms[2]], [D_x])[1];" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG*&&*&*$-%%sqrtG6#*&&%\"uG6#\"\" #\"\"\"&F.6#\"\"!F1\"\"\"F5F-!\"\"6#%!GF1&%$D_xG6#%!GF1" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 30 "Inv_Diff:=pr_vect(total(X),3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Inv_DiffG,,*&&*&*$-%%sqrtG6#*&& %\"uG6#\"\"#\"\"\"&F/6#\"\"!F2\"\"\"F6F.!\"\"6#%!GF2&%$D_xG6#%!GF2F2*& &*&*&&F/6#F2F2F*F6F6F.F7F8F2&&%$D_uG6#7#F56#%!GF2F2*&F*F6&&FF6#7#F26#% !GF2F2*&&*&*&&F/6#\"\"$F2F*F6F6F.F7F8F2&&FF6#7#F16#%!GF2F2*&&*&*&F*F6& F/6#\"\"%F2F6F.F7F8F2&&FF6#7#FX6#%!GF2F2" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 40 "J2:=expand(Lie_derivative(Inv_Diff,I1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J2G,,*&*&&%\"uG6#\"\"\"F+&F)6#\"\"$ F+\"\"\"*$)&F)6#\"\"#\"\"#F/!\"\"F+*&*$)F(F4F/F/*&F2\"\"\"&F)6#\"\"!\" \"\"F6#F.F4!\"$F+*&*&)F,F4F/F " 0 "" {MPLTEXT 1 0 29 "simplify(J2+I2 -3/2*I1^2 +3);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 290 " " 0 "" {TEXT 256 103 "Check the invariance of the differential invar iants and the invariant forms by infinitesimal methods." }}{PARA 358 " " 0 "" {TEXT -1 0 "" }}{PARA 357 "" 0 "" {TEXT 256 65 "1. Construct th e infinitesimal generators for the GL(2) action. " }}{PARA 359 "" 0 " " {TEXT 256 21 "2. Lie differentiate." }{TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 291 "euc > " 0 "" {MPLTEXT 1 0 59 "Gamm a:=transform_to_vect(Phi,[a,b,c,d], [a=1,b=0,c=0,d=1]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&GammaG7&*&%\"xG\"\"\"&%$D_xG6#%!GF(&F*6#%!G,& *&)F'\"\"#\"\"\"&F*6#%!GF(!\"\"*(&%\"uG6#\"\"!F(F'F4&&%$D_uG6#7#F=6#%! GF(F8,&*&F'F4&F*6#%!GF(F8*&F:F4&F?6#%!GF(F8" }}}{EXCHG {PARA 292 "euc \+ > " 0 "" {MPLTEXT 1 0 29 "Gamma4:=map(pr_vect,Gamma,4):" }}}{EXCHG {PARA 293 "euc > " 0 "" {MPLTEXT 1 0 42 "simplify(map(Lie_derivative, \+ Gamma4, I1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!F$F$F$" }}} {EXCHG {PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 294 "euc > " 0 "" {MPLTEXT 1 0 42 "s implify(map(Lie_derivative, Gamma4, I2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!F$F$F$" }}}{EXCHG {PARA 295 "euc>" 0 "" {MPLTEXT 1 0 52 "f:= (i,j)->Lie_derivative( Gamma4[i], Inv_forms[j]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"iG%\"jG6\"6$%)operator G%&arrowGF)-%/Lie_derivativeG6$&%'Gamma4G6#9$&%*Inv_formsG6#9%F)F)F)" }}}{EXCHG {PARA 296 "euc > " 0 "" {MPLTEXT 1 0 15 "matrix(4,4 ,f);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&*&%#0~G\"\"\"&%#dxG6#% !GF*F(F(F(F'F'F'" }}}}{SECT 1 {PARA 297 "" 0 "" {TEXT -1 39 "The Eucli dean group SE(2) acting on R^2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 256 42 "First we find the differential invariant s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 31 "coord_init([a,b,theta],[],SE2):" }}}{EXCHG {PARA 0 "S E2>" 0 "" {MPLTEXT 1 0 24 "coord_init([x],[u],euc):" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT 256 34 "Define the action and prolong i t. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 101 "Phi:= transform(euc,euc,[x=cos(theta)*x -sin(theta)* u[0] +a ,u[0]=sin(theta)*x +cos(theta)*u[0] +b]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PhiG7$/%\"xG,(*&-%$cosG6#%&thetaG\"\"\"F'F.F.*&-%$si nGF,F.&%\"uG6#\"\"!F.!\"\"%\"aGF./F2,(*&F0\"\"\"F'F;F.*&F*F;F2F;F.%\"b GF." }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "Phi2:=pr_transfo rm(Phi,2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%Phi2G7&/%\"xG,(*&-%$c osG6#%&thetaG\"\"\"F'F.F.*&-%$sinGF,F.&%\"uG6#\"\"!F.!\"\"%\"aGF./F2,( *&F0\"\"\"F'F;F.*&F*F;F2F;F.%\"bGF./&F36#F.*&,&F0F.*&F*F;F?F.F.F;,&F*F .*&F0F;F?F;F6!\"\"/&F36#\"\"#,$*&FHF;,.*$)F*\"\"$F;F6*(F0F;)F*FJF;F?F; FP*&F*F;)F?FJF;!\"$*&FOF;FTF;FP*&)F?FPF;F0F;F.*(FXF;F0F;FRF;F6FFF6" }} }{EXCHG {PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT 256 59 "Construct the moving frame. This is a right moving frame." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 54 "rho:=right_moving_frame(Phi2,SE2,[x=0,u[0]=0,u[1]=0]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG7%/%\"aG,$*&,&%\"xG\"\"\"* &&%\"uG6#F,F,&F/6#\"\"!F,F,\"\"\"*$-%%sqrtG6#,&*$)F.\"\"#F4F,F,F,F4!\" \"!\"\"/%\"bG*&,&*&F.F4F+F,F,F1F>F4*$-F76#F9F4F=/%&thetaG,$-%'arctanG6 #F.F>" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT 256 39 "Construct the differential invariants." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 24 "u2:= pullback(Phi2,u[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u2G,$*&&%\" uG6#\"\"#\"\"\",.*$)-%$cosG6#%&thetaG\"\"$F+!\"\"*(-%$sinGF1\"\"\")F/F *F+&F(6#F8F8F3*&F/F8)F:F*F+!\"$*&F.F+F=F+F3*&)F:F3F+F6F+F8*(FAF+F6F+F9 F+F4!\"\"F4" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 41 "kappa0:= simplify(pullback(rho,u2),trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %'kappa0G*&*&-%%sqrtG6#,&*$)&%\"uG6#\"\"\"\"\"#\"\"\"F0F0F0F2&F.6#F1F0 F2,(*$)F-\"\"%F2F0F+F1F0F0!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 23 "kappa0:=factor(kappa0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'kappa0G*&&%\"uG6#\"\"#\"\"\"*$),&*$)&F'6#\"\"\"F)F*F2F2F2#\" \"$F)F*!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "Phi3:=p r_transform(Phi,3):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 24 " u3:=pullback(Phi3,u[3]):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 33 "kappa1:=factor(pullback(rho,u3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'kappa1G*&,(*&)&%\"uG6#\"\"#F,\"\"\"&F*6#\"\"\"F0!\"$&F*6#\"\" $F0*&F2F0)F.F,F-F0F-*$),&*$F6F-F0F0F0\"\"$F-!\"\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 40 "Construct the Mau rer_Cartan forms on G." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 21 "change_frame_to(SE2);" }}} {EXCHG {PARA 0 "SE2 > " 0 "" {MPLTEXT 1 0 95 "M_rep:= (a,b,theta) ->ma trix([[cos(theta), -sin(theta),a], [sin(theta),cos(theta),b],[0,0,1]]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&M_repGR6%%\"aG%\"bG%&thetaG6\" 6$%)operatorG%&arrowGF*-%'matrixG6#7%7%-%$cosG6#9&,$-%$sinGF5!\"\"9$7% F8F39%7%\"\"!F?\"\"\"F*F*F*" }}}{EXCHG {PARA 0 "G>" 0 "" {MPLTEXT 1 0 16 "G_dot:=proc(X,Y)" }}{PARA 0 "G>" 0 "" {MPLTEXT 1 0 14 "local A,B,C ; " }}{PARA 0 "G>" 0 "" {MPLTEXT 1 0 17 "A:=M_rep(op(X)); " }}{PARA 0 "G>" 0 "" {MPLTEXT 1 0 16 "B:=M_rep(op(Y));" }}{PARA 0 "G>" 0 "" {MPLTEXT 1 0 29 "C:=map(combine,evalm(A&*B)); " }}{PARA 0 "SE2 > " 0 " " {MPLTEXT 1 0 36 "[ C[1,3], C[2,3], X[3] +Y[3] ]; end:" }}}{EXCHG {PARA 0 "SE2 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "SE2 > " 0 " " {MPLTEXT 1 0 30 "G_dot([a,b,theta], [c,d,phi]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,(*&-%$cosG6#%&thetaG\"\"\"%\"cGF*F**&-%$sinGF(F*%\"d GF*!\"\"%\"aGF*,(*&F-\"\"\"F+F4F**&F&F4F/F4F*%\"bGF*,&F)F*%$phiGF*" }} }{EXCHG {PARA 0 "SE2 > " 0 "" {MPLTEXT 1 0 40 "Omega:=Maurer_Cartan(G_ dot, [0,0,0])[4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8The~output~is~l isted~asG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%fo[right~inv.~vectors,~r ight~inv.~~forms,~left~inv.~vectors,~left~inv.~forms]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&OmegaG7%,&&%#daG6#%!G\"\"\"*&%\"bGF+&%'dthetaG6 #F*F+F+,&&%#dbG6#F*F+*&%\"aGF+&F/6#F*F+!\"\"&F/6#F*" }}}{EXCHG {PARA 0 "SE2 > " 0 "" {MPLTEXT 1 0 38 "Inv_forms:=map2(pullback, rho, Omega) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Inv_formsG7%,&*&&%#dxG6#%!G\" \"\"*$-%%sqrtG6#,&*$)&%\"uG6#\"\"\"\"\"#F,F7F7F7F,!\"\"!\"\"*&*&F4F7&& %#duG6#7#\"\"!6#F+F7F,*$-F/6#F1F,F9F:,&*&*&F4F,&F)6#F+F7F,*$-F/6#F1F,F 9F7*&&F>6#F+F,*$-F/6#F1F,F9F:,$*&&&F?6#7#F76#F+F,F1F9F:" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT 256 39 "Comp ute the invariant horizontal form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 42 "sigma:=form_to_biform(In v_forms[1],[1,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaG,$*&-%% sqrtG6#,&*$)&%\"uG6#\"\"\"\"\"#\"\"\"F0F0F0F2&%#DxG6#%!GF0!\"\"" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 " " 0 "" {TEXT 258 47 "The Special Linear Group -- fiber action on R^2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 24 "coord_init([x],[u],euc):" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 27 "coord_init([a,b,c],[],SL2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0frame~name:~SL2G" }}}{EXCHG {PARA 0 "SL2>" 0 "" {MPLTEXT 1 0 84 "Phi:=simplify(transform(euc,euc, [x=x, u[0]= (a*u[0] +b )/(c*u[0] + (1+b*c)/a)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ PhiG7$/%\"xGF'/&%\"uG6#\"\"!*&*&,&*&%\"aG\"\"\"F)F2F2%\"bGF2F2F1\"\"\" F4,(*(F)F4F1F4%\"cGF2F2F2F2*&F7F4F3F2F2!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "Phi3:=pr_transform(Phi,3):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 19 "_EnvExplicit:=true;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-_EnvExplicitG%%trueG" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 61 "rho:=right_moving_frame(Phi3,SL2,[u[0]=0, u[1]=1 , u[2]=0],1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG7%/%\"aG*&\"\" \"F)*$-%%sqrtG6#&%\"uG6#\"\"\"F)!\"\"/%\"bG,$*&&F/6#\"\"!F)*$-F,6#F.F) F2!\"\"/%\"cG,$*&&F/6#\"\"#F)*$)F.#\"\"$FDF)F2#F1FD" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 37 "A:= matrix([[a,b], [c, (1+b*c)/a ]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$%\"aG%\"bG7$%\"cG*&,&\"\"\"F0*&F- F0F+F0F0\"\"\"F*!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 87 "moving_frame_matrixR:=right_moving_frame_matrix(Phi3,SL2,[u[0]=0, \+ u[1]=1, u[2]=0],1,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%5moving_fra me_matrixRG-%'matrixG6#7$7$*&\"\"\"F+*$-%%sqrtG6#&%\"uG6#\"\"\"F+!\"\" ,$*&&F16#\"\"!F+*$-F.6#F0F+F4!\"\"7$,$*&&F16#\"\"#F+*$)F0#\"\"$FCF+F4# F3FC,$*&,&*$)F0FCF+!\"#*&FAF3F7F3F3F+*$)F0#\"\"$FCF+F4#F=FC" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT 256 34 "Fundamenta l Differential Invariant" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 50 "I1:=simplify( pullback(rho, pul lback(Phi3,u[3])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I1G,$*&,&*$) &%\"uG6#\"\"#F-\"\"\"!\"$*&&F+6#\"\"$\"\"\"&F+6#F4F4F-F.*$)F5\"\"#F.! \"\"#F4F-" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT 256 19 "Maurer Carta n Forms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 " " {MPLTEXT 1 0 21 "change_frame_to(SL2);" }}}{EXCHG {PARA 0 "SL2 > " 0 "" {MPLTEXT 1 0 43 "A:= matrix([[a1,b1], [c1, (1 +b1*c1)/a1]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$%#a1G%#b1G7$%#c1 G*&,&\"\"\"F0*&F+F0F-F0F0\"\"\"F*!\"\"" }}}{EXCHG {PARA 0 "SL2 > " 0 " " {MPLTEXT 1 0 43 "B:= matrix([[a2,b2], [c2, (1 +b2*c2)/a2]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7$7$%#a2G%#b2G7$%#c2 G*&,&\"\"\"F0*&F+F0F-F0F0\"\"\"F*!\"\"" }}}{EXCHG {PARA 0 "SL2 > " 0 " " {MPLTEXT 1 0 23 "simplify(evalm(A &*B));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,&*&%#a1G\"\"\"%#a2GF+F+*&%#b1GF+%#c2GF +F+*&,(*(F*\"\"\"%#b2GF+F,F3F+F.F+*(F.F3F4F3F/F3F+F3F,!\"\"7$*&,(*(%#c 1GF+F,F3F*F3F+F/F+*(F/F3F.F3F;F3F+F3F*F6*&,,**F;F3F4F3F*F3F,F3F+F+F+*& F4F3F/F3F+*&F.F3F;F3F+**F.F3F;F3F4F3F/F3F+F3*&F*\"\"\"F,\"\"\"F6" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 256 23 "Define multiplication." }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 41 "G_dot:=proc(X,Y) local a 1,b1,c1,a2,b2,c2;" }}{PARA 0 "SL2 > " 0 "" {MPLTEXT 1 0 58 "a1:=X[1];b 1:=X[2]; c1:=X[3]; a2:=Y[1]; b2:=Y[2]; c2:=Y[3];" }}{PARA 0 "SL2 > " 0 "" {MPLTEXT 1 0 67 "[a1*a2+b1*c2,(a1*b2*a2+b1+b1*b2*c2)/a2,(c1*a2*a1 +c2+c2*b1*c1)/a1 ] " }}{PARA 0 "SL2 > " 0 "" {MPLTEXT 1 0 4 "end:" }}} {EXCHG {PARA 0 "SL2 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "SL2 \+ > " 0 "" {MPLTEXT 1 0 40 "Omega:=Maurer_Cartan(G_dot, [1,0,0])[4];" }} {PARA 0 "SL2 > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8The~output~is~listed~asG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%f o[right~inv.~vectors,~right~inv.~~forms,~left~inv.~vectors,~left~inv.~ forms]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&OmegaG7%,&*&*&,&\"\"\"F* *&%\"cGF*%\"bGF*F*F*&%#daG6#%!GF*\"\"\"%\"aG!\"\"F**&F,F2&%#dbG6#F1F*! \"\",&*&F-F2&F/6#F1F*F9*&F3F*&F76#F1F*F*,(*&*(F,F2F)F2&F/6#F1F*F2*$)F3 \"\"#F2F4F**&*&)F,\"\"#F2&F76#F1F*F2F3F4F9*&&%#dcG6#F1F2F3F4F*" }}} {EXCHG {PARA 0 "SL2 > " 0 "" {MPLTEXT 1 0 36 "Inv_forms:=map2(pullback ,rho,Omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Inv_formsG7%,&*&*&& %\"uG6#\"\"#\"\"\"&&%#duG6#7#\"\"!6#%!GF-\"\"\"*$)&F*6#F-\"\"#F6!\"\"# F-F,*&&&F06#7#F-6#F5F6F9F<#!\"\"F,,$*&&F/6#F5F6F9F " 0 "" {MPLTEXT 1 0 33 "simplify(pu llback(Phi3,I1) - I1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 48 "pullback(Phi3,Inv_forms[ 1]) &minus Inv_forms[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#0~G\" \"\"&%#dxG6#%!GF%" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 48 "pu llback(Phi3,Inv_forms[2]) &minus Inv_forms[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#0~G\"\"\"&%#dxG6#%!GF%" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 48 "pullback(Phi3,Inv_forms[3]) &minus Inv_forms[3]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#0~G\"\"\"&%#dxG6#%!GF%" }}}} {SECT 1 {PARA 298 "" 0 "" {TEXT 259 19 "A Similarity Group" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 102 "This is an in teresting example since the orbit dimension on E and J^1(E) are equal \+ but not on J^2(E)." }}{PARA 0 "" 0 "" {TEXT 256 54 "This is called ps eudo-stablization of orbit dimension." }{TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 24 "coord_ init([x],[u],euc):" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 29 "coo rd_init([a,b,alpha],[],S):" }}}{EXCHG {PARA 0 "S>" 0 "" {MPLTEXT 1 0 62 "Phi:=transform(euc,euc,[x= alpha*x +a, u[0] = alpha*u[0] +b]):" }} }{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "Phi3:=pr_transform(Phi, 3):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 21 "pullback(Phi3, u [1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"uG6#\"\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 20 "pullback(Phi3,u[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"uG6#\"\"#\"\"\"%&alphaG!\"\"" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 53 "rho:=right_moving_frame( Phi3,S,[x=0,u[0]=0, u[2]=1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rh oG7%/%\"aG,$*&&%\"uG6#\"\"#\"\"\"%\"xGF.!\"\"/%\"bG,$*&F*\"\"\"&F+6#\" \"!F.F0/%&alphaGF*" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 38 "I 1:=pullback(rho,pullback(Phi3,u[3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I1G*&&%\"uG6#\"\"$\"\"\"*$)&F'6#\"\"#\"\"#F*!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 299 "" 0 "" {TEXT -1 41 "The General Linear Group GL(2) on R^2 x R" }}{PARA 300 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 301 "euc > " 0 "" {MPLTEXT 1 0 27 "coord_init([x,y], [u],euc):" }}}{EXCHG {PARA 302 "euc>" 0 "" {MPLTEXT 1 0 29 "coord_init([a,b,c,d],[],GL2):" }}}{EXCHG {PARA 303 "G L2>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 304 "GL2>" 0 "" {MPLTEXT 1 0 78 "Phi:=transform(euc,euc, [x= a*x + b*y, y=c*x +d*y, u[0,0] =(a* d-b*c)*u[0,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PhiG7%/%\"xG,&* &%\"aG\"\"\"F'F+F+*&%\"bGF+%\"yGF+F+/F.,&*&%\"cGF+F'\"\"\"F+*&%\"dGF+F .F3F+/&%\"uG6$\"\"!F:*&,&*&F*F3F5F3F+*&F-F3F2F3!\"\"F+F7F+" }}}{EXCHG {PARA 305 "euc > " 0 "" {MPLTEXT 1 0 26 "Phi2:=pr_transform(Phi,2):" } }}{EXCHG {PARA 306 "euc > " 0 "" {MPLTEXT 1 0 66 "rho:=right_moving_fr ame(Phi2,GL2, [x=1, y=0, u[1,0]=1, u[0,1]=0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG7&/%\"aG*&&%\"uG6$\"\"\"\"\"!\"\"\",&*&F)F,%\"xG F,F,*&&F*6$F-F,F,%\"yGF,F,!\"\"/%\"bG*&F3F.F/F6/%\"cG,$*&F5F.F/F6!\"\" /%\"dG*&F1F.F/F6" }}}{EXCHG {PARA 307 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 25 "A:=matrix([[a,b],[c,d]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$%\"aG%\"bG7$%\"c G%\"dG" }}}{EXCHG {PARA 308 "euc > " 0 "" {MPLTEXT 1 0 94 "moving_fram e_matrixR:=right_moving_frame_matrix(Phi2,GL2, [x=1, y=0, u[1,0]=1, u[ 0,1]=0],1,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%5moving_frame_matri xRG-%'matrixG6#7$7$*&&%\"uG6$\"\"\"\"\"!\"\"\",&*&F+F.%\"xGF.F.*&&F,6$ F/F.F.%\"yGF.F.!\"\"*&F5F0F1F87$,$*&F7F0F1F8!\"\"*&F3F0F1F8" }}} {EXCHG {PARA 309 "euc > " 0 "" {MPLTEXT 1 0 50 "I0:=simplify(pullback( rho,pullback(Phi2,u[0,0])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I0G *&&%\"uG6$\"\"!F)\"\"\",&*&&F'6$\"\"\"F)F/%\"xGF/F/*&&F'6$F)F/F/%\"yGF /F/!\"\"" }}}{EXCHG {PARA 310 "euc > " 0 "" {MPLTEXT 1 0 51 "I11:=simp lify(pullback(rho,pullback(Phi2,u[2,0])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$I11G*&,(*&)%\"xG\"\"#\"\"\"&%\"uG6$F*\"\"!\"\"\"F0*( F)F0%\"yGF0&F-6$F0F0F0F**&)F2F*F+&F-6$F/F*F0F0F+,&*&&F-6$F0F/F0F)F+F0* &&F-6$F/F0F0F2F+F0!\"\"" }}}{EXCHG {PARA 311 "euc > " 0 "" {MPLTEXT 1 0 51 "I11:=simplify(pullback(rho,pullback(Phi2,u[1,1])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$I11G*&,**(%\"xG\"\"\"&%\"uG6$\"\"!F)F)&F+6$ \"\"#F-F)!\"\"*(F(\"\"\"&F+6$F)F-F)&F+6$F)F)F)F)*(%\"yGF)F*F3F6F3F1*(F 9F3F4F3&F+6$F-F0F)F)F3,&*&F4F3F(F3F)*&F*F3F9F3F)!\"\"" }}}{EXCHG {PARA 312 "euc > " 0 "" {MPLTEXT 1 0 51 "I22:=simplify(pullback(rho,pu llback(Phi2,u[0,2])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$I22G*&,(* &)&%\"uG6$\"\"!\"\"\"\"\"#\"\"\"&F*6$F.F,F-F-*(F)F-&F*6$F-F,F-&F*6$F-F -F-!\"#*&)F3F.F/&F*6$F,F.F-F-F/,&*&F3F/%\"xGF-F-*&F)F/%\"yGF-F-!\"\"" }}}{EXCHG {PARA 313 "euc > " 0 "" {MPLTEXT 1 0 21 "change_frame_to(GL2 ):" }}}{EXCHG {PARA 314 "GL2 > " 0 "" {MPLTEXT 1 0 29 "G_dot:=proc(X,Y ) local A,B,C;" }}{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 17 "A:=matrix(2,2 ,X);" }}{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 17 "B:=matrix(2,2,Y);" }} {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 15 "C:=evalm(A&*B);" }}{PARA 0 "GL2 > " 0 "" {MPLTEXT 1 0 33 "map(op,convert(C,listlist)); end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&G_dotGR6$%\"XG%\"YG6%%\"AG%\"BG%\"CG6\"F- C&>8$-%'matrixG6%\"\"#F49$>8%-F26%F4F49%>8&-%&evalmG6#-%#&*G6$F0F7-%$m apG6$%#opG-%(convertG6$F<%)listlistGF-F-F-" }}}{EXCHG {PARA 0 "GL2 > \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "GL2 > " 0 "" {MPLTEXT 1 0 35 "G_dot([a1,b1,c1,d1],[a2,b2,c2,d2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&,&*&%#a1G\"\"\"%#a2GF'F'*&%#b1GF'%#c2GF'F',&*&F&\"\"\"%#b2GF'F '*&F*F.%#d2GF'F',&*&%#c1GF'F(F.F'*&%#d1GF'F+F.F',&*&F4F.F/F.F'*&F6F.F1 F.F'" }}}{EXCHG {PARA 360 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "GL2 > " 0 "" {MPLTEXT 1 0 21 "change_frame_to(GL2);" }}}{EXCHG {PARA 0 "G L2 > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "GL2 > " 0 "" {MPLTEXT 1 0 41 "Omega:=Maurer_Cartan(G_dot,[1,0,0,1])[4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8The~output~is~listed~asG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%fo[right~inv.~vectors,~right~inv.~~forms,~left~inv.~ve ctors,~left~inv.~forms]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&OmegaG7 &,&*&*&%\"dG\"\"\"&%#daG6#%!G\"\"\"F*,&*&%\"aGF/F)F/F/*&%\"bGF/%\"cGF/ !\"\"!\"\"F/*&*&F5F*&%#dbG6#%!GF/F*F0F7F6,&*&*&F4F*&F,6#%!GF/F*F0F7F6* &*&F2F*&F;6#%!GF/F*F0F7F/,&*&*&F)F*&%#dcG6#%!GF/F*F0F7F/*&*&F5F*&%#ddG 6#%!GF/F*F0F7F6,&*&*&F4F*&FM6#%!GF/F*F0F7F6*&*&F2F*&FS6#%!GF/F*F0F7F/ " }}}{PARA 361 "" 0 "" {TEXT -1 0 "" }}{PARA 315 "" 0 "" {TEXT 256 43 "Construct the invariant forms on jet space." }}{PARA 362 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 316 "GL2 > " 0 "" {MPLTEXT 1 0 36 "Inv_f orms:=map2(pullback,rho,Omega);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%* Inv_formsG7&,&*&*&&%\"uG6$\"\"\"\"\"!\"\"\"&%#dxG6#%!GF,F.,&*&F)F,%\"x GF,F,*&&F*6$F-F,F,%\"yGF,F,!\"\"!\"\"*&*&F7F.&%#dyG6#%!GF,F.F3F:F;,&*& *&F7F.&&%#duG6#7$F,F-6#%!GF,F.F3F:F;*&*&F)F.&&FG6#7$F-F,6#%!GF,F.F3F:F ,,&*&*&F9F.&F06#%!GF,F.F3F:F,*&*&F5F.&F?6#%!GF,F.F3F:F;,&*&*&F5F.&FF6# %!GF,F.F3F:F;*&*&F9F.&FO6#%!GF,F.F3F:F;" }}}{EXCHG {PARA 317 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 320 "euc > " 0 "" {MPLTEXT 1 0 22 "sigma1:= Inv_forms[1]:" }}}{EXCHG {PARA 323 "euc > " 0 "" {MPLTEXT 1 0 21 "sigma2:=Inv_forms[3]:" }}}{EXCHG {PARA 326 "euc>" 0 " " {MPLTEXT 1 0 36 "invD:= dual_frame([sigma1, sigma2]);" }}{PARA 327 " euc > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%in vDG7$,&*&%\"xG\"\"\"&%$D_xG6#%!GF)!\"\"*&%\"yGF)&%$D_yG6#%!GF)F.,&*&&% \"uG6$\"\"!F)F)&F+6#%!GF)F)*&&F86$F)F:F)&F26#%!GF)F." }}}{EXCHG {PARA 329 "" 0 "" {TEXT 256 17 "Check Invariance:" }}{PARA 330 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 331 "euc > " 0 "" {MPLTEXT 1 0 34 "simpl ify(pullback(Phi2, I0) - I0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"! " }}}{EXCHG {PARA 333 "euc > " 0 "" {MPLTEXT 1 0 36 "simplify(pullback (Phi2, I11) - I11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 335 "euc > " 0 "" {MPLTEXT 1 0 36 "simplify(pullback(Phi2 , I12) - I12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 337 "euc > " 0 "" {MPLTEXT 1 0 36 "simplify(pullback(Phi2, I22) \+ - I22);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 339 "euc > " 0 "" {MPLTEXT 1 0 47 "simplify(pullback(Phi2, sigma1) &minus \+ sigma1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#0~G\"\"\"&%#dxG6#%!GF% " }}}{EXCHG {PARA 341 "euc > " 0 "" {MPLTEXT 1 0 47 "simplify(pullback (Phi2, sigma2) &minus sigma2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%# 0~G\"\"\"&%#dxG6#%!GF%" }}}{EXCHG {PARA 343 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 344 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 345 "" 0 " " {TEXT -1 30 "The Euclidean group on R^2 x R" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "coord_in it([x,y],[u],euc):" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 30 "coo rd_init([a,b,theta],[],E2):" }}}{EXCHG {PARA 0 "E2>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "E2>" 0 "" {MPLTEXT 1 0 108 "Phi:= transform( euc,euc,[x=cos(theta)*x -sin(theta)*y +a ,y=sin(theta)*x +cos(theta)*y +b, u[0,0] =u[0,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PhiG7%/% \"xG,(*&-%$cosG6#%&thetaG\"\"\"F'F.F.*&-%$sinGF,F.%\"yGF.!\"\"%\"aGF./ F2,(*&F0\"\"\"F'F8F.*&F*F8F2F8F.%\"bGF./&%\"uG6$\"\"!F?F<" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "Phi2:=pr_transform(Phi,2):" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 72 "rho:=simplify(right_movi ng_frame(Phi2,E2,[x=0,y=0, u[1,0]=0]),symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG7%/%\"aG,$*&,&*&%\"xG\"\"\"&%\"uG6$\"\"!F-F-F-*& &F/6$F-F1F-%\"yGF-!\"\"\"\"\"*$-%%sqrtG6#,&*$)F3\"\"#F7F-*$)F.F?F7F-F7 !\"\"F6/%\"bG,$*&,&*&F3F7F,F7F-*&F.F7F5F7F-F7*$-F:6#F " 0 "" {MPLTEXT 1 0 59 "J:= simplify(pullback(rho,pullback(Phi2,u[0,1])),symbolic);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG*$-%%sqrtG6#,&*$)&%\"uG6$\"\"\" \"\"!\"\"#\"\"\"F/*$)&F-6$F0F/F1F2F/F2" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 64 "I11:=J^ 2*simplify(pullback(rho,pullback(Phi2,u[2,0])),symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$I11G,(*&)&%\"uG6$\"\"!\"\"\"\"\"#\"\"\"&F)6$ F-F+F,F,*(F(F,&F)6$F,F+F,&F)6$F,F,F,!\"#*&)F2F-F.&F)6$F+F-F,F," }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 55 "I12:=J^2*simplify(pullba ck(rho,pullback(Phi2,u[1,1])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ I12G,**(&%\"uG6$\"\"!\"\"\"F+&F(6$F+F*F+&F(6$\"\"#F*F+F+*&)F'F0\"\"\"& F(6$F+F+F+F+*&F4F3)F,F0F3!\"\"*(F,F3F'F3&F(6$F*F0F+F8" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 55 "I22:=J^2*simplify(pullback(rho, pullback(Phi2,u[0,2])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$I22G,(* &&%\"uG6$\"\"#\"\"!\"\"\")&F(6$F,F+F*\"\"\"F,*(&F(6$F+F,F,F.F,&F(6$F,F ,F,F**&&F(6$F+F*F,)F2F*F0F," }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 58 "Express some familar invariants in terms of I11, I12, I22." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }{MPLTEXT 1 0 25 "simplify((I11 +I22) /J^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 29 "factor((I11*I22 -I12^2) /J^4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&&%\"uG6$\"\"!\"\"#\"\"\" &F&6$F)F(F*F**$)&F&6$F*F*F)\"\"\"!\"\"" }}}}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 264 19 "updated 01/20/03:IA" }}} {MARK "8 0" 63 }{VIEWOPTS 1 1 0 3 4 1802 }