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0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Vess_Title2" -1 269 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 128 0 64 1 2 2 0 0 0 2 0 0 0 }1 0 0 0 4 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Vess_Titlle3" 269 270 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 "Vess_IO" -1 271 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 0 0 0 0 0 0 0 0 1 0 0 } 1 0 0 -1 -1 -1 3 30 0 0 0 0 -1 3 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 1 14 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 "" 0 273 1 {CSTYLE "" -1 -1 "" 1 14 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 "Vessiot_test" -1 274 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 128 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 1 0 -1 0 }{PSTYLE "" 3 275 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 20 "A GUIDED TOUR OF " } {TEXT 258 7 "VESSIOT" }}{PARA 4 "" 0 "" {TEXT 256 12 "Ian Anderson" } {TEXT 259 0 "" }}{PARA 4 "" 0 "" {TEXT 257 23 "Utah State University \+ " }}{PARA 4 "" 0 "" {TEXT 321 84 "Prepared for a talk given at the Fo CM meeting August 14, 2002, Univ. of Minnesota." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Updated 01/23.03." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 " Motivation" }}{PARA 4 "" 0 "" {TEXT -1 158 "* Many years ago a good f riend of mine asked the following question at a workshop on symbolic \+ methods in differential geometry and differential equations." }} {PARA 257 "" 0 "" {TEXT 260 143 "How many people have developed softw are packages which are being used by research groups at institutio ns other than your home institution?" }}{PARA 256 "" 0 "" {TEXT -1 9 " *Answer: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 5 "None." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 129 "Of cour se, this situation has changed in recent years but this little s tory had a major impact on the development of the " }}{PARA 4 "" 0 " " {TEXT -1 47 "software suite Vessiot over the last 15 years. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Design Goals " }}{PARA 4 "" 0 "" {TEXT -1 28 "*In designing the program " }{TEXT 263 9 "Vessiot, " } {TEXT -1 84 "our goal was to create a program which could (and woul d) be used by many people." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 264 34 "1. Helpfiles, Manuals, Tutorials." }}{PARA 272 "" 0 "" {TEXT 309 1 " " }}{PARA 259 "" 0 "" {TEXT 265 32 "2. Lots \+ of Interface Commands." }}{PARA 273 "" 0 "" {TEXT 310 1 " " }}{PARA 260 "" 0 "" {TEXT 266 97 "3. Written in Maple V and continually chec ked for compatiblity with all subsequent releases." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT 267 132 "4. Vessiot was d esigned to do nothing in particular, rather it serves as a environme nt for doing a wide range of computations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT 268 171 "5. The Vessiot environmen t is integrated --- the same commands can be used in a wide range of different contexts and Vessiot understands the context of the comm and." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT 269 135 "6. Vessiot serves a programming language with large array of u litilies for easily writing programs to preform new computations. \+ " }}{PARA 264 "" 0 "" {TEXT -1 3 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "History" }}{PARA 4 "" 0 "" {TEXT -1 16 "1. Bryan Croft: " }}{PARA 4 "" 0 "" {TEXT -1 40 "Developed original package in Macsyma ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 22 "2. C innamon Hillyard: " }}{PARA 4 "" 0 "" {TEXT -1 55 "Converted to Maple, introduced current data structures." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 19 "3. Charles Miller: " }}{PARA 4 "" 0 "" {TEXT -1 58 "Added multiple coordinates functionality, tensors package ." }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 65 "4. F lorin Cantrina, John Stevens, Adam Bowers, Jamie Jorgenson: " }} {PARA 4 "" 0 "" {TEXT -1 32 "Developed Lie algebra routines. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 18 "5. Abbey \+ Bennett. " }}{PARA 4 "" 0 "" {TEXT -1 55 "Create library of differenti al equations and symmetries" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 21 "6. Class of Math 5820" }}{PARA 4 "" 0 "" {TEXT -1 36 "Extended Lie algebra classification." }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 6 "7. IA " }}{PARA 4 "" 0 "" {TEXT -1 15 "Everything else" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Gettin g Vessiot" }}{PARA 4 "" 0 "" {TEXT 270 16 "www.math.usu.edu" }{TEXT -1 1 " " }}{PARA 4 "" 0 "" {TEXT 262 9 "Click on:" }{TEXT -1 0 "" }} {PARA 4 "" 0 "" {TEXT -1 49 "Formal Geometry and Mathematical Physics \+ Group. " }}{PARA 4 "" 0 "" {TEXT -1 16 "Go to Symbolics " }}{PARA 4 " " 0 "" {TEXT -1 43 "Download Vessiot mfiles (in zipped format) " }} {PARA 4 "" 0 "" {TEXT -1 35 "Download (PDF) the Vessiot Manual " }} {PARA 4 "" 0 "" {TEXT -1 38 "Download(PDF) the Koszul manual(PDF) " } }{PARA 4 "" 0 "" {TEXT -1 25 "Browse the tutorial page." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Installatio n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 76 "1.Pu t the mfiles in a directory of you chosing - say C:\\Vessiot\\mfi les " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 93 "2 . The Maple variable libname tells the Maple program where to loo k for user packages. " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 67 "Add the path you chose in 1. to libname with the fol lowing command:" }}{PARA 4 "" 0 "" {TEXT 271 41 "libname:= libname, \" C:\\\\Vessiot\\\\mfiles\";" }{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 54 "Note the use of double quotes and double backslashes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 101 "3. Everytime yo u start Maple, it reads the Maple.ini file located in the bin.wnt directory. " }}{PARA 4 "" 0 "" {TEXT -1 109 "You can put the lib name command into the Maple.ini file and it will be automatically exec uted upon startup." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 17 "4. Use the Maple" }{TEXT 311 6 " with" }{TEXT -1 30 " \+ command to load the package." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(Vessiot);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7jx%'&minusG%(&m minusG%(&mmult1G%(&mmult2G%(&mmult3G%(&mmult4G%'&mplusG%&&multG%(&mwed geG%&&plusG%'&wedgeG%$EL0G%(EL_formG%2EL_form_to_diffeqG%9FrameToCoord ProcedureBFVG% " 0 "" {MPLTEXT 1 0 13 "with(Koszul):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "w ith(tensors):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "A Summary of Vessiot Packages" }}{PARA 4 "" 0 "" {TEXT -1 95 "Here is list of the current softwar e packages which comprise the Vessiot software suite." }}{PARA 4 " " 0 "" {TEXT -1 78 "One of the main features of Vessiot is that all t hese packages are integrated" }}{PARA 4 "" 0 "" {TEXT -1 10 "together. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 10 "1. V essiot" }}{PARA 0 "" 0 "" {TEXT 272 80 "The main package for computa tions with vector fields, differential forms, maps" }}{PARA 0 "" 0 "" {TEXT 273 14 "on jet spaces." }{TEXT -1 1 " " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 9 "2. Koszul" }}{PARA 0 "" 0 "" {TEXT 274 40 "A package for Lie algebra computations" }{TEXT -1 0 " " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 10 "3. te nsors" }}{PARA 0 "" 0 "" {TEXT 275 56 "A standard tensor package (wit h jet space capabilities)" }{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 18 "4. Vessiot_library" }}{PARA 0 "" 0 "" {TEXT 276 169 "Libraries of abstract Lie algebras (Winternitz, Tu rkowski), invariant 4 dimensional metrics (Petrov), differential equa tions and their symmetries (Rogers and Shadwick)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 277 20 "Specialized Packages" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 8 "1. Mubar" }}{PARA 0 "" 0 "" {TEXT 278 67 "A package for the classification of \+ low dimensional Lie algebras." }}{PARA 4 "" 0 "" {TEXT -1 0 "" }} {PARA 4 "" 0 "" {TEXT -1 10 "2. Darboux" }}{PARA 0 "" 0 "" {TEXT 279 56 "A package for the study of Darboux integrable equations." }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 11 "3. de_appls" }} {PARA 0 "" 0 "" {TEXT 280 56 "Noether's Theorem, Reduction of Differe ntial Equations." }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 10 "4. Douglas" }}{PARA 0 "" 0 "" {TEXT 281 65 "A package for the inverse problem of the calculus of variations.*" }}{PARA 4 "" 0 " " {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 18 "5. Gelfand--Dickey" }} {PARA 0 "" 0 "" {TEXT 282 31 "The Gelfand-Dickey transform. " }} {PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 12 "6. Froben ius" }}{PARA 0 "" 0 "" {TEXT 283 78 "A simple package for solving to tal differential equations with Maple dsolve." }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 6 "7. EDS" }}{PARA 0 "" 0 "" {TEXT 284 42 "An exterior differential systems package*" }}{PARA 4 " " 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 20 "8. invariant_metri cs" }}{PARA 4 "" 0 "" {TEXT 285 99 "Curvature tensor calculations for invariant metrics on homogeneous and cohomogeneity 1 spaces. " } {TEXT 312 2 " " }{TEXT -1 1 " " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }} {PARA 4 "" 0 "" {TEXT -1 13 "9. Isometries" }}{PARA 0 "" 0 "" {TEXT 286 87 "Calculation of the isometry algebra (and its isotropy subalg ebra) for a given metric." }}{PARA 4 "" 0 "" {TEXT -1 11 "10. Spencer " }}{PARA 0 "" 0 "" {TEXT 287 44 " Tableau computations and Spencer co homology" }}{PARA 4 "" 0 "" {TEXT -1 17 "11. repere_mobile" }}{PARA 0 "" 0 "" {TEXT 288 62 "Moving frames for group actions and different ial invariants" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 4 "Help" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "?Ves siot;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "?pullback;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "?Koszul; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 15 "Getting Started" }}{PARA 4 "" 0 "" {TEXT -1 83 "Ea ch Vessiot session begins by defining a system of local coordin ates either" }}{PARA 4 "" 0 "" {TEXT -1 33 "on a manifold or a fiber b undle. " }}{PARA 4 "" 0 "" {TEXT -1 73 "Within a given session one can work simultaneously with many manifolds." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "with(Vessiot):with(tensors):" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 26 "coord_init([x,y],[u], R2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%/frame~name:~R2G" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 71 "Many things happen behind the scenes when this command is ex ecuted. " }}{PARA 4 "" 0 "" {TEXT -1 66 "The first order jet space J ^1=(x,y,u,u_x, u_y) is constructed; " }}{PARA 4 "" 0 "" {TEXT -1 97 " the names D_x, D_y , D_u[0,0].. are assigned to the standard frame fo r the tangent space of J^1;" }}{PARA 4 "" 0 "" {TEXT -1 82 "the names \+ dx, dy, du[0,0], du[1,0], du[0,1] are assigned to the standard cofram e;" }}{PARA 4 "" 0 "" {TEXT -1 60 "the name Cu[0,0] is assigned to \+ the standard contact form" }}{PARA 4 "" 0 "" {TEXT -1 112 "structure \+ equation information for this frame is generated and stored in the glo bal table _Vessiot_frame_data." }}{PARA 4 "" 0 "" {TEXT -1 115 "infor mation on the printing of vectors, forms, tensors, etc. is stored \+ in the global table _Vessiot_frame_data." }}{PARA 4 "" 0 "" {TEXT -1 11 "The command" }{TEXT 313 21 " frameInformation() " }{TEXT -1 83 "g ives access to all this information -- in an ordinary Vessiot session \+ you need not" }}{PARA 4 "" 0 "" {TEXT -1 61 " worry about any of this \+ but it is important in programming." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "R2>" 0 "" {MPLTEXT 1 0 21 "frameInformation(R2);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%1frame~name:~~~R2G" }}{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," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%-Frame~LabelsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'&%$D_xG6#%!G&%$D_yG6#F'&&%$D_uG6#7$\"\"!F06#F'&&F-6#7$\"\"\"F06#F '&&F-6#7$F0F66#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/CoFrame~LabelsG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'&%#dxG6#%!G&%#dyG6#F'&&%#duG6#7$ \"\"!F06#F'&&F-6#7$\"\"\"F06#F'&&F-6#7$F0F66#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Horizontal~Coframe~LabelsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$&%#DxG6#%!G&%#DyG6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Vertical~Coframe~LabelsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7% &&%#CuG6#7$\"\"!F)6#%!G&&F&6#7$\"\"\"F)6#F+&&F&6#7$F)F06#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/--------------G" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 210 "One of the key features of \+ Vessiot is the ability to work in many coordinate systems (or dif ferent manifolds) simultaneously. The global variable _Vessiot_current _frame_name defines the active frame." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "P2>" 0 "" {MPLTEXT 1 0 28 "_Vessiot_current_fra me_name;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#R2G" }}}{EXCHG {PARA 0 " R2>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "R2>" 0 "" {MPLTEXT 1 0 29 "coord_init([r,theta],[v],P2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% /frame~name:~P2G" }}}{EXCHG {PARA 0 "P2>" 0 "" {MPLTEXT 1 0 28 "_Vessi ot_current_frame_name;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#P2G" }}} {PARA 4 "" 0 "" {TEXT -1 77 "The change_frame_command changes from one coordinate system to another." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "P2>" 0 "" {MPLTEXT 1 0 20 "change_frame_to(R2);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 28 "_Vessiot_current_frame_name;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%#R2G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 112 "The underlying jet space is automatically prolonged to whate ver order is required by a given calculation. " }}{PARA 4 "" 0 "" {TEXT -1 79 "For example, I define a vector field on (x,y,u) and pro long it to the 2 jet. " }}{PARA 4 "" 0 "" {TEXT -1 63 "Then the under lying jet space will automatically be prolonged." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 20 "frameJetD imension();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 16 "frameJetOrder();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 26 "X:= u[0,0] &mult D_u[0,0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG* &&%\"uG6$\"\"!F)\"\"\"&&%$D_uG6#7$F)F)6#%!GF*" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 17 "X2:=pr_vect(X,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#X2G,.*&&%\"uG6$\"\"!F*\"\"\"&&%$D_uG6#7$F*F*6#%!GF+F +*&&F(6$F+F*F+&&F.6#7$F+F*6#F2F+F+*&&F(6$F*F+F+&&F.6#7$F*F+6#F2F+F+*&& F(6$\"\"#F*F+&&F.6#7$FFF*6#F2F+F+*&&F(6$F+F+F+&&F.6#7$F+F+6#F2F+F+*&&F (6$F*FFF+&&F.6#7$F*FF6#F2F+F+" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 20 "frameJetDimension();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 16 "frameJetOrde r();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 19 "frameInformation();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1frame~name:~~~R2G" }}{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#F'&&%$D_uG6#7$\"\"!F06#F'&&F-6#7 $\"\"\"F06#F'&&F-6#7$F0F66#F'&&F-6#7$\"\"#F06#F'&&F-6#7$F6F66#F'&&F-6# 7$F0FA6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/CoFrame~LabelsG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7*&%#dxG6#%!G&%#dyG6#F'&&%#duG6#7$\"\" !F06#F'&&F-6#7$\"\"\"F06#F'&&F-6#7$F0F66#F'&&F-6#7$\"\"#F06#F'&&F-6#7$ F6F66#F'&&F-6#7$F0FA6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Horizont al~Coframe~LabelsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$&%#DxG6#%!G&%# DyG6#F'" }}{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#F+&&F&6#7$F)F06#F+&&F&6#7$\"\"#F)6#F+&&F&6#7$F0F06#F+&&F&6#7 $F)F;6#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/--------------G" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 314 89 "There ar e many commands for accessing the data associated to a given coordin ate system." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 28 "frameIndependentVariables();" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7$%\"xG%\"yG" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 21 "frameBaseDimension();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 19 "frameBase Vectors();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$&%$D_xG6#%!G&%$D_yG6#F '" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 20 "frameJetVariables() ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*%\"xG%\"yG&%\"uG6$\"\"!F)&F'6$ \"\"\"F)&F'6$F)F,&F'6$\"\"#F)&F'6$F,F,&F'6$F)F1" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 20 "frameJetDimension();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 0 " " }}{PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 16 "frameJetForms();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*&%#dxG6#%!G&%#dyG6#F'&&%#duG6#7$\"\"!F06#F '&&F-6#7$\"\"\"F06#F'&&F-6#7$F0F66#F'&&F-6#7$\"\"#F06#F'&&F-6#7$F6F66# F'&&F-6#7$F0FA6#F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 51 "Creating Vecto r Fields, Forms, Tensors, Transforms." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 274 "" 0 "" {TEXT 314 114 "As Vessiot has evolved we \+ have developed a variety of means for creating vector fields, f orms, etc. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 274 "" 0 "" {TEXT 314 100 "This is still far from ideal --- there are plans t o improve the display format significantly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 61 "The original set of com mands for creating objects were " }{TEXT 315 26 "vect, form, bifor m, tens, " }{TEXT -1 132 " together with the arithmetic operations &ar ith, &mult, &wedge, &tensor and v_zip. This still are very useful f rom time to time. " }}{PARA 4 "" 0 "" {TEXT -1 276 "Now the preferred way to enter vectors, forms etc is to use the pre-assigned names \+ that are created during coordinate initiation and the Vessiot parser . Use ordinary + for addition, - for subtraction, * for scalar mu ltiplication, &w and &t for wedge and tensor. " }}{PARA 4 "" 0 "" {TEXT -1 89 "To see the internal representation of a Vessiot object us e, apply the Maple op command. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "R2>" 0 "" {MPLTEXT 1 0 28 "with(Vessiot):with(tensors) :" }}{PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 26 "coord_init([x,y],[u], R2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/frame~name:~R2G" }}}{EXCHG {PARA 0 "R2>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "R2>" 0 "" {MPLTEXT 1 0 12 "V1:=vect(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V1 G&%$D_xG6#%!G" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 8 "V2:=D_y; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V2G&%$D_yG6#%!G" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 7 "op(V1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%%vectG%#R2G7\"7#7$7#\"\"\"F+" }}}{EXCHG {PARA 274 "" 0 "" {TEXT -1 0 "" }}{PARA 274 "" 0 "" {TEXT 314 146 "Maple doesn' t do a good job of assigning precedent to user defined operations so it is important to use lots of parenthesis with &mult etc." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 49 "V3:= (y &mult vect(x)) &minus ( x &mult vect(y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V3G,&*&%\"yG\"\"\"&%$D_xG6#%!GF(F(*&%\"xGF(& %$D_yG6#F,F(!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 314 48 "Here's the same result using the Vessiot parser. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 25 "V3:=evalV( y*D_x -x*D_y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V3G,&*&%\"yG\"\"\"&%$D_xG6#%!GF(F(*&%\"xGF(&%$D_yG6# F,F(!\"\"" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 314 122 "The Vessiot v_zi p command can also be used to creating vectors, forms etc. It is \+ especially useful in programming.." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 34 "V4:= v_zip([x,y], [D_x,D _y],plus);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V4G,&*&%\"xG\"\"\"&%$ D_xG6#%!GF(F(*&%\"yGF(&%$D_yG6#F,F(F(" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 21 "alpha:= dx &wedge dy;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG*(&%#dxG6#%!G\"\"\"%#^~GF*&%#dyG6#F)F*" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 36 "T:= evalV(dx &t D_y - dy &t D _x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG,&*&&%#dxG6#%!G\"\"\"&%$ D_yG6#F*F+F+*&&%#dyG6#F*F+&%$D_xG6#F*F+!\"\"" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&&%#dxG6#%!G\"\"\"&%$D_yG6#F(F)F)*&&%#dyG6#F(F)&% $D_xG6#F(F)!\"\"" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 13 "vect (u[1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&&%$D_uG6#7$\"\"\"F(6#%!G " }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 22 "beta1:=biform(u[0,0] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&beta1G&&%#CuG6#7$\"\"!F*6#%!G " }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 314 116 "There are lots of Vessiot commands for converting from one data \+ structure to another. Here are just a few examples." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 15 "omega: =Cu[0,0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG&&%#CuG6#7$\"\"! F*6#%!G" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 29 "beta2:=biform _to_form(omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&beta2G,(*&&%\"u G6$\"\"\"\"\"!F*&%#dxG6#%!GF*!\"\"*&&F(6$F+F*F*&%#dyG6#F/F*F0&&%#duG6# 7$F+F+6#F/F*" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 2 "T;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&&%#dxG6#%!G\"\"\"&%$D_yG6#F(F)F)*& &%#dyG6#F(F)&%$D_xG6#F(F)!\"\"" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 20 "A:=tens_to_array(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$\"\"!\"\"\"7$!\"\"F*" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 41 "array_to_tens(A, [[cov_bas,cov_bas],[]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&&%#dyG6#%!G\"\"\"&%#dxG6#F(F)!\"\" *&&F+6#F(F)&F&6#F(F)F)" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 314 32 "Create a simple transforma tion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 98 "phi:=transform(R2,R2,[x=x*cos(theta) -y*sin(theta), y =x*sin(theta) +y*cos(theta), u[0,0]=v[0,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG7%/%\"xG,&*&F'\"\"\"-%$cosG6#%&thetaGF*F**&%\"yG F*-%$sinGF-F*!\"\"/F0,&*&F'\"\"\"F1F7F**&F0F7F+F7F*/&%\"uG6$\"\"!F=&% \"vGF<" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 41 "transform_to_v ect(phi,[theta],[theta=0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#,&*&% \"yG\"\"\"&%$D_xG6#%!GF'!\"\"*&%\"xGF'&%$D_yG6#F+F'F'" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Vessiot Data Structures" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 314 102 "The creation of data structures for each kind of mathematical object is a key feature \+ of Vessiot." }}{PARA 4 "" 0 "" {TEXT 314 73 "In general, every data s tructure is a list consisting of two parts. " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 314 101 "The first part is the \+ attribute list which contains information about the type of Vessiot \+ object." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 314 88 "The second part is the component list which cont ains a list of nonzero components. " }}{PARA 4 "" 0 "" {TEXT 314 97 "There is host of commands which allows one to access the various com ponents of a Vessiot object." }}{PARA 4 "" 0 "" {TEXT -1 100 "Again, t his information isn't really need in most Vessiot sessions but is ofte n used in programming." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 14 "with(Vessiot):" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 25 "coord_init([x,y],[u],R2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/frame~name:~R2G" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 44 "alpha:=evalV( a*dx &w dy + b*dx &w du[0,0]);" }} {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG,&**%\"aG\"\"\"&%#dxG6#%!GF(%#^~GF(&%#dyG6#F,F(F(**%\"bGF( &F*6#F,F(F-F(&&%#duG6#7$\"\"!F:6#F,F(F(" }}}{EXCHG {PARA 0 "R2 > " 0 " " {MPLTEXT 1 0 10 "op(alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%% %formG%#R2G\"\"#7$7$7$\"\"\"F'%\"aG7$7$F+\"\"$%\"bG" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 18 "objectType(alpha);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%%formG" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 19 "objectFrame(alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#R2G" }}} {EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 18 "formDegree(alpha);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 24 "objectComponents(alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7$7$\"\"\"\"\"#%\"aG7$7$F&\"\"$%\"bG" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 22 "coeff_list(alpha,all);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%\"aG%\"bG" }}}{EXCHG {PARA 0 "R2 > " 0 " " {MPLTEXT 1 0 26 "coeff_list(alpha,[[1,2]]);" }}{PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"aG" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 17 "coeff_set(alpha);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<$%\"aG%\"bG" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 75 "phi:=transform(R2,R2,[x=epsilon*x, y=epsilon*y, u[0,0 ]= 1/epsilon*u[0,0]]);" }}{PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG7%/%\"xG*&%(epsilonG\"\"\"F'F*/ %\"yG*&F)\"\"\"F,F*/&%\"uG6$\"\"!F3*&F0F.F)!\"\"" }}}{EXCHG {PARA 0 "R 2 > " 0 "" {MPLTEXT 1 0 26 "phi2:=pr_transform(phi,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%phi2G7*/%\"xG*&%(epsilonG\"\"\"F'F*/%\"yG*&F)\" \"\"F,F*/&%\"uG6$\"\"!F3*&F0F.F)!\"\"/&F16$F*F3*&F7F.*$)F)\"\"#F.F5/&F 16$F3F**&F>F.*$)F)\"\"#F.F5/&F16$\"\"#F3*&FEF.*$)F)\"\"$F.F5/&F16$F*F* *&FMF.*$)F)\"\"$F.F5/&F16$F3FG*&FTF.*$)F)\"\"$F.F5" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 9 "op(phi2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$7%%)helm_mapG7$7$%#R2G\"\"#F'7$%,projectableGF)7*7$*&%(epsilon G\"\"\"%\"xGF0F17$*&F/\"\"\"%\"yGF0F57$*&&%\"uG6$\"\"!F;F4F/!\"\"F87$* &&F96$F0F;F4*$)F/\"\"#F4F " 0 "" {MPLTEXT 1 0 22 "transfo rmDomain(phi2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%#R2G\"\"#" }}} {EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 21 "transformRange(phi2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$%#R2G\"\"#" }}}{EXCHG {PARA 0 "R2 > \+ " 0 "" {MPLTEXT 1 0 20 "transformType(phi2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%,projectableG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 289 1 " " }}{PARA 0 "" 0 "" {TEXT 314 83 "Even Lie al gebras are created and manipulated using these data structures. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 314 127 "Aga in the first component carries properties of the algebra and the seco nd component containts structure constant information." }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 13 "with(Koszul):" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 83 "matrix_list:=[matrix([[1,0],[0,-1]]),matrix([[0,1],[0 ,0]]), matrix([[0,0],[1,0]])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,m atrix_listG7%-%'matrixG6#7$7$\"\"\"\"\"!7$F,!\"\"-F'6#7$7$F,F+7$F,F,-F '6#7$F3F*" }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 66 "Lie_alg_dat a:=matrix_algebra_to_Lie_algebra_data(matrix_list,sl2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%-Lie_alg_dataG7$7%%(Lie_algG%$sl2G7#\"\"$7%7$7 %\"\"\"\"\"#F/F/7$7%F.F*F*!\"#7$7%F/F*F.F." }}}{EXCHG {PARA 0 "R2 > " 0 "" {MPLTEXT 1 0 59 "create_direct_sum_of_algebras([Lie_alg_data,Lie_ alg_data]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%(Lie_algG%'KoszulG 7#\"\"'7(7$7%\"\"\"\"\"#F-F-7$7%F,\"\"$F0!\"#7$7%F-F0F,F,7$7%\"\"%\"\" &F7F-7$7%F6F(F(F17$7%F7F(F6F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Sample Computations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "restart:with(Vessiot):coord_init([x ,y],[u,v]):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 290 28 "The Euler-Lagra nge Operator" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 314 134 "Euler Lagrange expressions can be computed \+ either as ordinary Maple expressions or as source forms in the var iational bicomplex." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 59 "L:= 1/2*( u[1,0] -v[0,1])^2 +1/2*u[0, 0]^2 -1/2*v[0,0]^2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,(*$),& &%\"uG6$\"\"\"\"\"!F,&%\"vG6$F-F,!\"\"\"\"#\"\"\"#F,F2*$)&F*6$F-F-F2F3 F4*$)&F/F8F2F3#F1F2" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 7 " EL0(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,(&%\"uG6$\"\"!F(\"\"\"&F &6$\"\"#F(!\"\"&%\"vG6$F)F)F),(&F/F'F-&F&F0F)&F/6$F(F,F-" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 30 "lambda:= L &mult vol_biform( );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lambdaG**,,*$)&%\"uG6$\"\"\" \"\"!\"\"#\"\"\"#F,F.*&F)F,&%\"vG6$F-F,F,!\"\"*$)F2F.F/F0*$)&F*6$F-F-F .F/F0*$)&F3F;F.F/#F5F.F,&%#DxG6#%!GF,%#^~GF,&%#DyG6#FCF," }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 23 "Delta:=EL_form(lambda);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaG,&*.,(&%\"uG6$\"\"!F+\"\"\"&F )6$\"\"#F+!\"\"&%\"vG6$F,F,F,F,&%#DxG6#%!GF,%#^~GF,&%#DyG6#F7F,F8F,&&% #CuG6#7$F+F+6#F7F,F,*.,(&F2F*F0&F)F3F,&F26$F+F/F0F,&F56#F7F,F8F,&F:6#F 7F,F8F,&&%#CvGF?6#F7F,F," }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 291 27 "Th e Helmholtz Conditions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 " " 0 "" {TEXT 314 134 "Let check that the source form Delta defined in the previous example satisfies the Helmholtz conditions, as of cou rse it should. " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 314 124 "Recall that for a source form Delta, the Helmholtz conditions are expressed in terms of the variational bicomplex as " }}{PARA 4 "" 0 "" {TEXT 314 22 "I_part o dV( Delta) =0" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 6 "D elta;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*.,(&%\"uG6$\"\"!F)\"\"\"&F '6$\"\"#F)!\"\"&%\"vG6$F*F*F*F*&%#DxG6#%!GF*%#^~GF*&%#DyG6#F5F*F6F*&&% #CuG6#7$F)F)6#F5F*F**.,(&F0F(F.&F'F1F*&F06$F)F-F.F*&F36#F5F*F6F*&F86#F 5F*F6F*&&%#CvGF=6#F5F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 19 "Delta_V:=dV(Delta);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Delta_VG,**0&%#DxG6#%!G\"\"\"%#^~GF +&%#DyG6#F*F+F,F+&&%#CuG6#7$\"\"!F56#F*F+F,F+&&F26#7$\"\"#F56#F*F+F+*0 &F(6#F*F+F,F+&F.6#F*F+F,F+&F16#F*F+F,F+&&%#CvG6#7$F+F+6#F*F+!\"\"*0&F( 6#F*F+F,F+&F.6#F*F+F,F+&&FFF36#F*F+F,F+&&F2FG6#F*F+FJ*0&F(6#F*F+F,F+&F .6#F*F+F,F+&FQ6#F*F+F,F+&&FF6#7$F5F;6#F*F+F+" }}}{EXCHG {PARA 0 "eucli d > " 0 "" {MPLTEXT 1 0 17 "I_parts(Delta_V);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*2%#0~G\"\"\"&%#DxG6#%!GF%%#^~GF%&%#DyG6#F)F%F*F%&&%#Cu G6#7$\"\"!F36#F)F%F*F%&&%#CvGF16#F)F%" }}}{PARA 275 "" 0 "" {TEXT -1 0 "" }{TEXT 314 80 "We can then use the vertical homotopy operator t o find a Lagrangian for Delta:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 314 0 "" }}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 23 "mu:= homotopy_dV(Delta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#muG**,.*$)&% \"vG6$\"\"!F,\"\"#\"\"\"#!\"\"F-*&F)\"\"\"&%\"uG6$F2F2F2#F2F-*&F)F.&F* 6$F,F-F2F/*$)&F4F+F-F.F6*&F" 0 "" {MPLTEXT 1 0 32 "check:=EL_form(mu) &minus Delt a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&checkG*.%#0~G\"\"\"&%#DxG6#%! GF'%#^~GF'&%#DyG6#F+F'F,F'&&%#CuG6#7$\"\"!F56#F+F'" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 292 17 "A Null Lagrangian" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 130 "Since the two Lagrangians lambda and mu \+ have the same Euler-Lagrange forms , the difference of this two Lagr angians is a null" }}{PARA 267 "" 0 "" {TEXT 314 172 "Lagrangian . We can now use the horizontal homotopy operator for the variational b icomplex to find a type(1,0) form omega=Adx +Bdy such that dH(omeg a) = lambda - mu" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "e uclid>" 0 "" {MPLTEXT 1 0 22 "chi:=lambda &minus mu;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$chiG**,0*&&%\"vG6$\"\"!F+\"\"\"&%\"uG6$F,F,F,#!\" \"\"\"#*&F(\"\"\"&F)6$F+F2F,#F,F2*&&F.F*F,&F.6$F2F+F,F7*&F9F4&F)F/F,F0 *$)&F.6$F,F+F2F4F7*&F@F,&F)6$F+F,F,F1*$)FCF2F4F7F,&%#DxG6#%!GF,%#^~GF, &%#DyG6#FJF," }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 13 "EL_for m(chi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*.%#0~G\"\"\"&%#DxG6#%!GF%% #^~GF%&%#DyG6#F)F%F*F%&&%#CuG6#7$\"\"!F36#F)F%" }}}{EXCHG {PARA 0 "euc lid > " 0 "" {MPLTEXT 1 0 24 "omega:=homotopy_dH(chi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG,&*&,&*&&%\"vG6$\"\"!F,\"\"\"&F*6$F,F-F-# !\"\"\"\"#*&F)\"\"\"&%\"uG6$F-F,F-#F-F2F-&%#DxG6#%!GF-F-*&,&*&&F6F+F-F .F4F0*&F@F4F5F4F8F-&%#DyG6#F" 0 "" {MPLTEXT 1 0 30 "check:=dH(omega) &minus (chi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&checkG**%#0~G\"\"\"&%#DxG6#%!GF'%#^~GF'&%#DyG6#F+F' " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 298 17 "Jacobi's Identity" }}{PARA 267 " " 0 "" {TEXT 314 77 "Let's check the the Jacobi identity for a set of 3 vector fields in R^3." }}{PARA 267 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 26 "coord_init([x,y,z],[] ,R3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/frame~name:~R3G" }}}{EXCHG {PARA 0 "R3>" 0 "" {MPLTEXT 1 0 28 "X1:=evalV(x^2*D_x+ y*z*D_y);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#X1G,&*&)%\"xG\"\"#\"\"\"&%$D_xG6#%! G\"\"\"F/*(%\"yGF/%\"zGF/&%$D_yG6#F.F/F/" }}}{EXCHG {PARA 0 "R3 > " 0 "" {MPLTEXT 1 0 28 "X2:=evalV(x*z*D_x +y^2*D_z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#X2G,&*(%\"xG\"\"\"%\"zGF(&%$D_xG6#%!GF(F(*&)%\"yG\" \"#\"\"\"&%$D_zG6#F-F(F(" }}}{EXCHG {PARA 0 "R3 > " 0 "" {MPLTEXT 1 0 30 "X3:=evalV(z^2*D_y + y*z*D_z); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#X3G,&*&)%\"zG\"\"#\"\"\"&%$D_yG6#%!G\"\"\"F/*(%\"yGF/F(F/&%$D_zG6#F .F/F/" }}}{EXCHG {PARA 0 "R3>" 0 "" {MPLTEXT 1 0 40 "J1:=Lie_bracket(X 1, Lie_bracket(X2,X3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J1G,(*&, &*()%\"xG\"\"#\"\"\"%\"yG\"\"\"%\"zGF.F.*(F-F,)F/F+F,F*F.!\"\"F.&%$D_x G6#%!GF.F.*&,&*&)F-F+F,F1F,\"\"%*$)F-F;F,F2F.&%$D_yG6#F6F.F.**F-F,F/F, ,&*$F1F,!\"#*$F:F,\"\"$F.&%$D_zG6#F6F.F." }}}{EXCHG {PARA 0 "R3 > " 0 "" {MPLTEXT 1 0 40 "J2:=Lie_bracket(X3, Lie_bracket(X1,X2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J2G,(**)%\"xG\"\"#\"\"\"%\"yG\"\"\"%\"zGF ,&%$D_xG6#%!GF,!\"\"*()F+F)F*)F-F)F*&%$D_yG6#F1F,!\"(*&,&*&)F+\"\"$F*F -F*F,*&)F-F>F*F+F*\"\"%F,&%$D_zG6#F1F,F," }}}{EXCHG {PARA 0 "R3 > " 0 "" {MPLTEXT 1 0 40 "J3:=Lie_bracket(X2, Lie_bracket(X3,X1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J3G,(**%\"yG\"\"\")%\"zG\"\"#\"\"\"%\"xGF (&%$D_xG6#%!GF(F(*()F'F+F,,&*$F)F,\"\"$*$F3F,F(F(&%$D_yG6#F1F(F(*&,&*& )F*F6F,F'F,!\"#*&)F'F6F,F*F(!\"%F(&%$D_zG6#F1F(F(" }}}{EXCHG {PARA 0 " R3 > " 0 "" {MPLTEXT 1 0 21 "J1 &plus J2 &plus J3;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&%#0~G\"\"\"&%$D_xG6#%!GF%" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 297 15 "Cartans Formula" }}{PARA 267 "" 0 "" {TEXT 314 189 "We check the Cartan formula for the Lie derivative of a differen tial form in terms of the exterior derivative to illustrate some co mputations with forms and vector field in Vessiot." }}{PARA 19 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "R3 > " 0 "" {MPLTEXT 1 0 26 "coord_i nit([x,y,z],[],R3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/frame~name:~R 3G" }}}{EXCHG {PARA 0 "R3>" 0 "" {MPLTEXT 1 0 36 "X:= evalV( x*D_x + \+ y*D_y + z*D_z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG,(*&%\"xG\" \"\"&%$D_xG6#%!GF(F(*&%\"yGF(&%$D_yG6#F,F(F(*&%\"zGF(&%$D_zG6#F,F(F(" }}}{EXCHG {PARA 0 "R3>" 0 "" {MPLTEXT 1 0 57 "omega:=evalV(x^2*dy &w d z -y^2*dx &w dz + z^2*dx &w dy);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %&omegaG,(**)%\"zG\"\"#\"\"\"&%#dxG6#%!G\"\"\"%#^~GF/&%#dyG6#F.F/F/**) %\"yGF)F*&F,6#F.F/F0F/&%#dzG6#F.F/!\"\"**)%\"xGF)F*&F26#F.F/F0F/&F:6#F .F/F/" }}}{EXCHG {PARA 0 "R3>" 0 "" {MPLTEXT 1 0 32 "omega1:=Lie_deriv ative(X,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'omega1G,(**)%\"z G\"\"#\"\"\"&%#dxG6#%!G\"\"\"%#^~GF/&%#dyG6#F.F/\"\"%**)%\"yGF)F*&F,6# F.F/F0F/&%#dzG6#F.F/!\"%**)%\"xGF)F*&F26#F.F/F0F/&F;6#F.F/F4" }}} {EXCHG {PARA 0 "R3>" 0 "" {MPLTEXT 1 0 59 "omega2:= ext_d(hook(X,omega )) &plus hook(X, ext_d(omega));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% 'omega2G,(**)%\"zG\"\"#\"\"\"&%#dxG6#%!G\"\"\"%#^~GF/&%#dyG6#F.F/\"\"% **)%\"yGF)F*&F,6#F.F/F0F/&%#dzG6#F.F/!\"%**)%\"xGF)F*&F26#F.F/F0F/&F;6 #F.F/F4" }}}{EXCHG {PARA 0 "R3>" 0 "" {MPLTEXT 1 0 23 "evalV(omega1 - \+ omega2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 267 "" 0 " " {TEXT 314 75 "NOTE: Vessiot does not use this formula for compu ting Lie derivatives." }}{PARA 267 "" 0 "" {TEXT 314 124 "Instead it \+ used the fact that the Lie derivative is the unique derivation which \+ computes with the exterior derivative. " }}{PARA 267 "" 0 "" {TEXT 314 101 "Vessiot contains general routines for constructing derivati ons, anti-derivations and homomorphisms." }}}{SECT 1 {PARA 265 "" 0 " " {TEXT -1 53 "Vector Fields and One Parameter Transformation Groups" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 239 "In \+ elementary differential geometry one learns that every vector f ield X on an manifold determines a one-parameter group of transform ations phi_t and, conversely, every such one parameter group de termines a vector field. " }}{PARA 267 "" 0 "" {TEXT 314 55 "These c orrespondences are easily computed in Vessiot." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 20 "coord_i nit([x],[u]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3frame~name:~euclidG " }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 32 "X:= evalV(u[0]*D_x - x*D_u[0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG,&*&&%\"uG6#\" \"!\"\"\"&%$D_xG6#%!GF+F+*&%\"xGF+&&%$D_uG6#7#F*6#F/F+!\"\"" }}} {EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 28 "phi:=vect_to_transfor m(X,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG7$/%\"xG,&*&-%$cosG6 #%\"tG\"\"\"F'F.F.*&-%$sinGF,F.&%\"uG6#\"\"!F.F./F2,&*&F0\"\"\"F'F9!\" \"*&F*F9F2F9F." }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 28 "pt:= transform_pt(phi,[1,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ptG7$-% $cosG6#%\"tG,$-%$sinGF(!\"\"" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 46 "plot([op(pt), t=0..2*Pi],scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$ $\"\"\"\"\"!F*7$$\"1#4hRPij!**!#;$!1kwb#=y_O\"F.7$$\"18J#))4-Qn*F.$!1D \\ff*)GLDF.7$$\"1N5')yke[#*F.$!1Mj&[K5J!QF.7$$\"1goz=42`')F.$!1:NXR4U7 ]F.7$$\"1b](G._U!zF.$!1`H-qceDhF.7$$\"1]$R'oTd#3(F.$!1jB*3qU&fqF.7$$\" 1H$>jubk6'F.$!1:7_\\!>8\"zF.7$$\"1GuIc:x5]F.$!1$>p(Qh-a')F.7$$\"1C3nW' R,#QF.$!1L+16bcT#*F.7$$\"1n9AwiQDDF.$!1'Q\"[M\"oen*F.7$$\"1B^hAo8X8F.$ !1)fVPO<\"4**F.7$$!1qB/u(p5(f!#>$!08;t@)******!#:7$$!1)\\T#fB[i8F.$!1q %>wGZn!**F.7$$!1crN]@=YEF.$!1A>=\\D`V'*F.7$$!1zV1J*3Jx$F.$!1zF#e`m3E*F .7$$!1(\\AOF:B/&F.$!1V'yI2&oN')F.7$$!1igNw%*[RgF.$!1j\")RQ+BqzF.7$$!1X UwYjJ*3(F.$!1fvOg?x_qF.7$$!1#e_b?/U!zF.$!14HvXQF.7$$!12RS4Nij'*F.$!1%p9m# Q%=d#F.7$$!1xpT>+.2**F.$!1V?00]Ug8F.7$$!1gKG4>&*****F.$!1DA\\O%485$!#= 7$$!1`Is=!Q%3**F.$\"1$4*>c=8]8F.7$$!1_NDna1u'*F.$\"1NbFGIGKDF.7$$!1_J. 8AEj#*F.$\"15&=j`Bsw$F.7$$!1v#)4Kh=u')F.$\"1FNX')3zv\\F.7$$!1B*z7xng%z F.$\"1W(H!pUCrgF.7$$!1D84Pfc5rF.$\"1?#o?\"yMJqF.7$$!1Wy6Ike[gF.$\"1Jw! yeGL'zF.7$$!1AUD=%)o!*\\F.$\"1_Mh6Mil')F.7$$!1,1s\"[\"ysPF.$\"10`8S*** 4E*F.7$$!1yCH\"el'3EF.$\"1V>(fp[Pl*F.7$$!177Cvnc\"H\"F.$\"1$RoI*>C;**F .7$$!1zyOHMQdIF`s$\"11O#>E`*****F.7$$\"1IAj`Ks(G\"F.$\"1ZZ3X=u;**F.7$$ \"1KRqX8/bDF.$\"1$[o%H'z!o'*F.7$$\"1%)yb&Q)zNQF.$\"1.2<#>x]B*F.7$$\"17 w4w0I.]F.$\"1Igb5wMe')F.7$$\"1\\#)[*R]$4hF.$\"1a*[=H2o\"zF.7$$\"1/wY'Q +$*4(F.$\"1r&)eg?sUqF.7$$\"1f=s$Ry,!zF.$\"11$H2ndo*fQF.7$$\"1mGAp))oa'*F.$\" 1g]nRQ=0EF.7$$\"1zb'f`bp!**F.$\"1up91t'4O\"F.7$F($!1YKhSr8/#)!#D-%'COL OURG6&%$RGBG$\"#5!\"\"F*F*-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$% !GFf[l-%%VIEWG6$%(DEFAULTGFj[l" 1 2 0 1 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 36 "Y:=transform_to_vect(phi,[t],[t=0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG7#,&*&&%\"uG6#\"\"!\"\"\"&%$D_xG6#%!GF,F,*&%\"xGF,&&%$D_uG 6#7#F+6#F0F,!\"\"" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 293 40 "Prolongat ions of Vector Fields and Maps" }{TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 127 "A fundemental fact fact about jet space is that tra nsformations and vectors field on the underlying fiber bundle can b e " }}{PARA 267 "" 0 "" {TEXT 314 142 " uniquely lifted or prolonged \+ to the jet space (by requiring that the prolonged transformations/vec tor fields preserve the contact ideal). " }{TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 70 "We show how \+ to prolong transformations and vector field in Vessiot." }}{PARA 19 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "R3>" 0 "" {MPLTEXT 1 0 14 "with (Vessiot):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "coord_init([x ],[u]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3frame~name:~euclidG" }}} {EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 31 "X:= evalV(u[0]* D_x -x* D_u[0]):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 17 "X3:=pr_v ect(X,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#X3G,,*&&%\"uG6#\"\"!\" \"\"&%$D_xG6#%!GF+F+*&%\"xGF+&&%$D_uG6#7#F*6#F/F+!\"\"*&,&*$)&F(6#F+\" \"#\"\"\"F8F8F+F+&&F46#7#F+6#F/F+F+*(F=F+&F(6#F?F+&&F46#7#F?6#F/F+!\"$ *&,&*$)FGF?F@FN*&F=F@&F(6#\"\"$F+!\"%F+&&F46#7#FV6#F/F+F+" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 28 "phi:=vect_to_transform(X,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG7$/%\"xG,&*&-%$cosG6#%\"tG\" \"\"F'F.F.*&-%$sinGF,F.&%\"uG6#\"\"!F.F./F2,&*&F0\"\"\"F'F9!\"\"*&F*F9 F2F9F." }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 26 "phi3:=pr_t ransform(phi,3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%phi3G7'/%\"xG,& *&-%$cosG6#%\"tG\"\"\"F'F.F.*&-%$sinGF,F.&%\"uG6#\"\"!F.F./F2,&*&F0\" \"\"F'F9!\"\"*&F*F9F2F9F./&F36#F.,$*&,&F0F.*&F*F9F=F.F:F9,&F*F.*&F0F9F =F9F.!\"\"F:/&F36#\"\"#,$*&FGF9,.*$)F*\"\"$F9F:*(F0F9)F*FIF9F=F9!\"$*& F*F9)F=FIF9FR*&FNF9FTF9FO*&)F=FOF9F0F9F:*(FWF9F0F9FQF9F.FEF:/&F36#FO*& ,(*&)FGFIF9F0F9FR*&FZF.F*F9F.*(FZF9F0F9F=F9F.F9,:*$)F*\"\"&F9F.*(F0F9) F*\"\"%F9F=F9F_oFU\"#5*&F^oF9FTF9!#5FXFco*(FWF9FaoF9F0F9Feo*&F*F9)F=Fb oF9F_o*&FNF9FhoF9Feo*&F^oF9FhoF9F_o*&)F=F_oF9F0F9F.*(F\\pF9F0F9FQF9!\" #*(F\\pF9FaoF9F0F9F.FE" }}}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 91 "Note the infinitesimal generator for this one parameter group of transformations is X3:" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 40 "Z:=tr ansform_to_vect(phi3,[t],[t=0])[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"ZG,,*&&%\"uG6#\"\"!\"\"\"&%$D_xG6#%!GF+F+*&%\"xGF+&&%$D_uG6#7#F*6 #F/F+!\"\"*&,&*$)&F(6#F+\"\"#\"\"\"F8F8F+F+&&F46#7#F+6#F/F+F+*(F=F+&F( 6#F?F+&&F46#7#F?6#F/F+!\"$*&,&*$)FGF?F@FN*&F=F@&F(6#\"\"$F+!\"%F+&&F46 #7#FV6#F/F+F+" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 12 "X3 \+ &minus Z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#0~G\"\"\"&%$D_xG6#%!G F%" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 294 39 "Point Symmetries of the \+ heat equation" }{TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 88 "Lets check that the heat equation has a 6 dimensional Lie algebra of symmetries. " }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc lid > " 0 "" {MPLTEXT 1 0 40 "with(Vessiot):coord_init([x,t],[u],euc); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0frame~name:~eucG" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 23 "Delta:=u[2,0] - u[0,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaG,&&%\"uG6$\"\"#\"\"!\"\"\"&F'6$F*F+ !\"\"" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 8 "X1:=D_x:" }}} {EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 21 "prX1:=pr_vect(D_x,2):" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc \+ > " 0 "" {MPLTEXT 1 0 8 "X2:=D_t:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 21 "prX2:=pr_vect(D_t,2):" }}}{EXCHG {PARA 0 "euc > " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 29 "X 3:= (u[0,0]) &mult D_u[0,0]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 20 "prX3:=pr_vect(X3,2):" }}}{EXCHG {PARA 0 "euc > " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 34 "X 4:=v_zip([x,2*t],[D_x,D_t],plus):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 20 "prX4:=pr_vect(X4,2):" }}}{EXCHG {PARA 0 "euc > " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 47 "X 5:=v_zip([2*t,-x*u[0,0]],[D_x,D_u[0,0]],plus):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 20 "prX5:=pr_vect(X5,2):" }}}{EXCHG {PARA 0 "eu c > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 70 "X6:=v_zip([4*t*x,4*t^2, -(x^2 +2*t)*u[0,0]], [D_x,D_t,D_u[0,0] ],plus):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 20 "prX6:=pr_ve ct(X6,2):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 27 "Gamma:=[X1,X2,X3,X4,X5,X 6];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&GammaG7(&%$D_xG6#%!G&%$D_tG6 #F)*&&%\"uG6$\"\"!F1\"\"\"&&%$D_uG6#7$F1F16#F)F2,&*&%\"xGF2&F'6#F)F2F2 *&%\"tGF2&F+6#F)F2\"\"#,&*&F?\"\"\"&F'6#F)F2FB*(F;FEF.FE&F46#F)F2!\"\" ,(*(F?FEF;FE&F'6#F)F2\"\"%*&)F?FBFE&F+6#F)F2FP*(,&*$)F;FBFEF2F?FBF2F.F E&F46#F)F2FK" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 28 "Lie_derivative(prX1, Delta);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "euc > " 0 " " {MPLTEXT 1 0 28 "Lie_derivative(prX2, Delta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 28 "Lie_derivative(prX3, Delta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&&% \"uG6$\"\"#\"\"!\"\"\"&F%6$F(F)!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 28 "Lie_derivative(prX4, Delta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%\"uG6$\"\"!\"\"\"\"\"#&F%6$F)F'!\"#" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 28 "Lie_derivative(prX5, Delta);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&&%\"uG6$\"\"!\"\"\"F)%\"xGF)F)*&F *\"\"\"&F&6$\"\"#F(F)!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\",&& %\"uG6$\"\"!F%F%&F(6$\"\"#F*!\"\"F%" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 28 "Lie_derivative(prX6, Delta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&&%\"uG6$\"\"!\"\"\"F))%\"xG\"\"#\"\"\"F)*&F%F-%\"tG F)\"#5*&&F&6$F,F(F)F*F-!\"\"*&F/F-F2F-!#5" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*& ,&&%\"uG6$\"\"!\"\"\"F)&F&6$\"\"#F(!\"\"F),&*$)%\"xGF,\"\"\"F)%\"tG\"# 5F)" }}}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 104 "For calculations like this use the map and map2 commands in \+ Maple to execute all the steps at once." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 31 "pr_Gamma:=map(pr_ vect,Gamma,2):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 63 "Sym:=subs( u[0,1] = u[2, 0],map(Lie_derivative,pr_Gamma,Delta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SymG7(\"\"!F&F&F&F&F&" }}}{PARA 267 "" 0 "" {TEXT -1 0 "" }} {PARA 267 "" 0 "" {TEXT 314 135 "See the Vessiot helpfile for the co mmands diffeq and pr_diffeq for other Vessiot tools for working \+ with differential equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 299 29 "Lie algebras of vector fields" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 121 "The \+ Lie algebra of vector fields in the previous example can be conve rted into a 6 dimensional abstract Lie algebra." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 13 "with(Kos zul):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 6 "Gamma;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(&%$D_xG6#%!G&%$D_tG6#F'*&&%\"uG6$\"\"!F/\"\"\"&&%$D_u G6#7$F/F/6#F'F0,&*&%\"xGF0&F%6#F'F0F0*&%\"tGF0&F)6#F'F0\"\"#,&*&F=\"\" \"&F%6#F'F0F@*(F9FCF,FC&F26#F'F0!\"\",(*(F=FCF9FC&F%6#F'F0\"\"%*&)F=F@ FC&F)6#F'F0FN*(,&*$)F9F@FCF0F=F@F0F,FC&F26#F'F0FI" }}}{EXCHG {PARA 0 " euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 35 "L:=vect_to_Lie_alg(Gamma,heat_alg);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"LG7$7%%(Lie_algG%)heat_algG7#\"\"'7+7$7%\"\"\"\" \"%F.F.7$7%F.\"\"&\"\"$!\"\"7$7%F.F*F2\"\"#7$7%F7F/F7F77$7%F7F2F.F77$7 %F7F*F3!\"#7$7%F7F*F/F/7$7%F/F2F2F.7$7%F/F*F*F7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 314 85 "Now take this Lie algeb ra data structure and use it to initialize a Lie algebra." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 314 81 "The comm and Lie alg_init does much the same thing as the command coord_ini t. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 314 103 " In particular, the structure constant information is stored in the g lobal table _Vessiot_frame_data." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%6Lie~alge bra:~heat_algG" }}}{EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 25 " Lie_bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrix G6#7*7*%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F(&%#e4G6#F(&%#e5G6#F(&%# e6G6#F(7*F(%$---G%%----GF>F>F>F>F>7*&F+6#F(F)\"\"!FBFB&F+6#F(,$&F16#F( !\"\",$&F76#F(\"\"#7*&F.6#F(F)FBFBFB,$&F.6#F(FL,$&F+6#F(FL,&&F16#F(!\" #&F46#F(\"\"%7*&F16#F(F)FBFBFBFBFBFB7*&F46#F(F),$&F+6#F(FH,$&F.6#F(FYF BFB&F76#F(,$&F:6#F(FL7*&F76#F(F)&F16#F(,$&F+6#F(FYFB,$&F76#F(FHFBFB7*& F:6#F(F),$&F76#F(FY,&&F16#F(FL&F46#F(!\"%FB,$&F:6#F(FYFBFB" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 314 74 "The ve ctors and dual covectors (or forms) are defined by ei and thetai." } }{PARA 0 "" 0 "" {TEXT 314 0 "" }}{PARA 0 "" 0 "" {TEXT 314 103 "This \+ are the default labels and can be changed using optional arguments \+ of the command Lie_alg_init" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 7 "op(e1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%%vectG%)heat_algG7\"7#7$7#\"\"\"F+" }}} {EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 11 "op(theta1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%%formG%)heat_algG\"\"\"7#7$7#F'F' " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 314 90 "One of the nice features of Vessiot is that the various packages are tig htly integrated. " }}{PARA 0 "" 0 "" {TEXT 314 122 "The same commands that are used for calculus of vectors fields, forms and tensors on manifolds work for Lie algebras." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 19 "Lie_bracket(e2,e6); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%#e3G6#%!G!\"#&%#e4G6#F'\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "heat_alg > \+ " 0 "" {MPLTEXT 1 0 14 "ext_d(theta1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(&%'theta1G6#%!G\"\"\"%#^~GF)&%'theta4G6#F(F)!\"\"*(&%'theta2 G6#F(F)F*F)&%'theta5G6#F(F)!\"#" }}}{EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 14 "with(tensors):" }}}{EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 40 "T:= evalV( e1 &t theta2 - e5 &t theta3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG,&*&&%#e1G6#%!G\"\"\"&%'theta2G6#F*F+F+*& &%#e5G6#F*F+&%'theta3G6#F*F+!\"\"" }}}{EXCHG {PARA 0 "heat_alg > " 0 " " {MPLTEXT 1 0 21 "Lie_derivative(e2,T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&&%#e1G6#%!G\"\"\"&%'theta3G6#F(F)!\"#*&&F&6#F(F)&%'theta4G6# F(F)F-*&&%#e5G6#F(F)&%'theta1G6#F(F)!\"\"*&&F66#F(F)&%'theta6G6#F(F)F- " }}}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 162 "A very important invariant of a Lie algebra of vector field s is the istropy subalgebra -- the subalgebra of vector field which \+ vanish at a given point. " }}{PARA 267 "" 0 "" {TEXT 314 112 "The c ommand isotropy_subalgebra computes this subalgebra and the correspon ding linear isotropy representation." }}{PARA 19 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 52 "IS:=isotropy_subalgebra(Ga mma, [x=0,t=0 ,u[0,0]=1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ISG7% 7$7(\"\"!F(F(\"\"\"F(F(-%'matrixG6#7%7%F)F(F(7%F(\"\"#F(7%F(F(F(7$7(F( F(F(F(F)F(-F+6#7%F/F17%!\"\"F(F(7$7(F(F(F(F(F(F)-F+6#7%F1F17%F(!\"#F( " }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 6 "IS[1];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7$7(\"\"!F%F%\"\"\"F%F%-%'matrixG6#7%7%F&F%F%7%F %\"\"#F%7%F%F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 314 220 "The first element of this list is are the compon ents of the vector in the isotropy subalgera and the second is the li near isotropy representation of the tangent space of the space of inde pendent and dependent variables." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 27 "v_zip(IS[1][1],Gamma,plu s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\"&%$D_xG6#%!GF&F& *&%\"tGF&&%$D_tG6#F*F&\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 266 "" 0 "" {TEXT -1 43 "Transformations of Differentia l Equations" }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 134 "As a simple example of a transformation of a differ ential equation we rewrite the two dimensional Laplace in polar co ordinates" }{TEXT -1 1 "." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "coord_init([x,y],[u],euc):" }}} {EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 32 "coord_init([r,theta],[v],p olar):" }}}{EXCHG {PARA 0 "polar>" 0 "" {MPLTEXT 1 0 73 "phi:=transfor m(polar,euc,[x=r*cos(theta), y=r*sin(theta),u[0,0]=v[0,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG7%/%\"xG*&%\"rG\"\"\"-%$cosG6#%&theta GF*/%\"yG*&F)\"\"\"-%$sinGF-F*/&%\"uG6$\"\"!F9&%\"vGF8" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 27 "phi_2:=pr_transform(phi,2):" }} }{EXCHG {PARA 0 "polar > " 0 "" {MPLTEXT 1 0 40 "answer:=pullback(phi_ 2,u[2,0] + u[0,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG*&,(* &&%\"vG6$\"\"\"\"\"!F+%\"rGF+F+&F)6$F,\"\"#F+*&&F)6$F0F,F+)F-F0\"\"\"F +F5*$)F-\"\"#F5!\"\"" }}}{EXCHG {PARA 0 "polar > " 0 "" {MPLTEXT 1 0 15 "expand(answer);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&&%\"vG6$\" \"\"\"\"!\"\"\"%\"rG!\"\"F(*&&F&6$F)\"\"#F**$)F+\"\"#F*F,F(&F&6$F0F)F( " }}}{PARA 5 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 295 32 "Some simple tensor calculations " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 78 "We check the isometries and \+ scalar curvature for the Poincare half-plane." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "polar > " 0 "" {MPLTEXT 1 0 30 "coord_ init([x,y],[],Poincare);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5frame~na me:~PoincareG" }}}{EXCHG {PARA 0 "Poincare>" 0 "" {MPLTEXT 1 0 14 "wit h(tensors):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 314 97 "Another handy way to input tensors (like a metric tensor) \+ is with the command array_to_tens." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "Poincare>" 0 "" {MPLTEXT 1 0 33 "A:=matrix([[1/y^2,0] ,[0,1/y^2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7 $*&\"\"\"F+*$)%\"yG\"\"#F+!\"\"\"\"!7$F1F*" }}}{EXCHG {PARA 0 "Poincar e>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "Poincare>" 0 "" {MPLTEXT 1 0 45 "g:= array_to_tens(A,[[cov_bas,cov_bas],[]]);:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,&*(&*&\"\"\"F)*$)%\"yG\"\"#F)! \"\"6#%!G\"\"\"&%#dyG6#F0F1&F36#F0F1F1*(F'F)&%#dxG6#F0F1&F96#F0F1F1" } }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 26 "Ch eck the Killing vectors" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "Poincare>" 0 "" {MPLTEXT 1 0 8 "X1:=D_x:" }}}{EXCHG {PARA 0 " Poincare > " 0 "" {MPLTEXT 1 0 27 "X2:=evalV( x*D_x + y*D_y):" }}} {EXCHG {PARA 0 "Poincare > " 0 "" {MPLTEXT 1 0 42 "X3:= evalV( (x^2 -y ^2)*D_x + 2*x*y*D_y):" }}}{EXCHG {PARA 0 "Poincare > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "Poincare > " 0 "" {MPLTEXT 1 0 18 "Gamma:=[X1,X2,X3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&GammaG7%& %$D_xG6#%!G,&*&%\"xG\"\"\"&F'6#F)F-F-*&%\"yGF-&%$D_yG6#F)F-F-,&*&,&*$) F,\"\"#\"\"\"F-*$)F1F:F;!\"\"F-&F'6#F)F-F-*(F,F;F1F;&F36#F)F-F:" }}} {EXCHG {PARA 0 "Poincare>" 0 "" {MPLTEXT 1 0 38 "answer:=map(Lie_deriv ative, Gamma, g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG7%*(%#0 ~G\"\"\"&%#dxG6#%!GF(&F*6#F,F(F&F&" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 314 146 "Instead of using the Lie derivative to check that X3 is a symmetry of g, we coulds also use the covariant derivative and Killi ng's equation. " }}{PARA 267 "" 0 "" {TEXT 314 74 "It is necessary to \+ convert X3 to a tensor using the vect_to_tens command. " }}{PARA 267 " " 0 "" {TEXT 314 73 "The command offel2 computes the Christoffel symb ol (of the second kind)." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "Poincare>" 0 "" {MPLTEXT 1 0 42 " Z:=lower_indices(g,vect_to_ tens(X3),[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"ZG,&*&&*&,&*$)% \"xG\"\"#\"\"\"\"\"\"*$)%\"yGF-F.!\"\"F.*$)F2\"\"#F.!\"\"6#%!GF/&%#dxG 6#F9F/F/*&&,$*&F,F.F2F7F-F8F/&%#dyG6#F9F/F/" }}}{EXCHG {PARA 0 "Poinca re>" 0 "" {MPLTEXT 1 0 13 "C:=offel2(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG,***&,$*&\"\"\"F*%\"yG!\"\"!\"\"6#%!G\"\"\"&%#dxG6#F/F0&%$ D_xG6#F/F0&%#dyG6#F/F0F0**&F)F.F0&F26#F/F0&%$D_yG6#F/F0&F26#F/F0F0**F' F*&F86#F/F0&F56#F/F0&F26#F/F0F0**F'F*&F86#F/F0&F?6#F/F0&F86#F/F0F0" }} }{EXCHG {PARA 0 "Poincare>" 0 "" {MPLTEXT 1 0 35 "covderZ:=covariant_d erivative(Z,C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(covderZG,&*(&,$* &,&*$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF.F/F0F/*$)F3\"\"$F/!\"\"!\"\"6#%! GF0&%#dxG6#F:F0&%#dyG6#F:F0F0*(&F)F9F0&F?6#F:F0&F<6#F:F0F0" }}}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 47 "Note that \+ this is skew-symmetric as required" }}{PARA 267 "" 0 "" {TEXT -1 1 "." }}{EXCHG {PARA 0 "Poincare>" 0 "" {MPLTEXT 1 0 39 "Killing_eq:=sym metrize(covderZ, [1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+Killing _eqG*(%#0~G\"\"\"&%#dxG6#%!GF'&F)6#F+F'" }}}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 37 "Finally compute the curvatur e tensor." }}{PARA 267 "" 0 "" {TEXT -1 1 "." }}{EXCHG {PARA 0 "Poinca re > " 0 "" {MPLTEXT 1 0 26 "Riem:=curvature_tensor(C);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%RiemG,**,&,$*&\"\"\"F**$)%\"yG\"\"#F*!\"\"!\" \"6#%!G\"\"\"&%#dxG6#F2F3&%$D_yG6#F2F3&F56#F2F3&%#dyG6#F2F3F3*,&F)F1F3 &F56#F2F3&F86#F2F3&F=6#F2F3&F56#F2F3F3*,F@F*&F=6#F2F3&%$D_xG6#F2F3&F56 #F2F3&F=6#F2F3F3*,F'F*&F=6#F2F3&FM6#F2F3&F=6#F2F3&F56#F2F3F3" }}} {EXCHG {PARA 0 "Poincare > " 0 "" {MPLTEXT 1 0 21 "h:=inverse_metric(g );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG,&*()%\"yG\"\"#\"\"\"&%$D_ xG6#%!G\"\"\"&F,6#F.F/F/*(F'F*&%$D_yG6#F.F/&F46#F.F/F/" }}}{EXCHG {PARA 0 "Poincare > " 0 "" {MPLTEXT 1 0 24 "R:=Ricci_scalar(h,Riem);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG!\"#" }}}{EXCHG {PARA 0 "Poinc are>" 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 296 38 "Some simple Lie algebra calculat ions. " }}{PARA 267 "" 0 "" {TEXT 314 46 "The Lie algebra package i s very extensive. " }}{PARA 267 "" 0 "" {TEXT 314 43 "Here we just \+ illustrate a few commands. " }}{PARA 267 "" 0 "" {TEXT 314 113 "Let u s return to the the Lie algebra of symmetries of the heat equatio n. (be sure to execute the sections " }{TEXT 316 39 "Point Symmetries of the heat equation " }{TEXT 314 5 " and " }{TEXT 317 31 " Lie alge bras of vector fields." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 26 "change_frame_to(heat_alg);" }} }{EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_t able();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7*7*%!G%\"|grG& %#e1G6#F(&%#e2G6#F(&%#e3G6#F(&%#e4G6#F(&%#e5G6#F(&%#e6G6#F(7*F(%$---G% %----GF>F>F>F>F>7*&F+6#F(F)\"\"!FBFB&F+6#F(,$&F16#F(!\"\",$&F76#F(\"\" #7*&F.6#F(F)FBFBFB,$&F.6#F(FL,$&F+6#F(FL,&&F16#F(!\"#&F46#F(\"\"%7*&F1 6#F(F)FBFBFBFBFBFB7*&F46#F(F),$&F+6#F(FH,$&F.6#F(FYFBFB&F76#F(,$&F:6#F (FL7*&F76#F(F)&F16#F(,$&F+6#F(FYFB,$&F76#F(FHFBFB7*&F:6#F(F),$&F76#F(F Y,&&F16#F(FL&F46#F(!\"%FB,$&F:6#F(FYFBFB" }}}{EXCHG {PARA 0 "heat_alg \+ > " 0 "" {MPLTEXT 1 0 13 "A:=Killing();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7(7(\"\"!F*F*F*F*F*7(F*F*F*F*F*!#?F)7(F*F*F*\" #5F*F*F)7(F*F,F*F*F*F*" }}}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 144 "Again to illustrate the interplay between \+ the various packages, we show that the Killing form is an invarian t tensor with respect to the" }}{PARA 267 "" 0 "" {TEXT 314 17 "adj oint action. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "heat _alg > " 0 "" {MPLTEXT 1 0 45 "K:=array_to_tens(A, [[cov_bas,cov_bas], []]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG-%.array_to_tensG6$%\" AG7$7$%(cov_basGF+7\"" }}}{EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 50 "check:=map(Lie_derivative,[e1,e2,e3,e4,e5,e6], K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&checkG7(\"\"!F&F&F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 12 "C:=cent er();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG7#&%#e3G6#%!G" }}} {EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 21 "DS:=derived_series( );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!7(&%#e1G6#%!G&%#e2G6#F(&%#e 3G6#F(&%#e4G6#F(&%#e5G6#F(&%#e6G6#F(\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%)infinityG7(&%#e1G6#%!G,$&%#e3G6#F(!\"\",$&%#e5G6#F(\" \"#,$&%#e2G6#F(F2,&&F+6#F(!\"#&%#e4G6#F(\"\"%,$&%#e6G6#F(F2\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DSG7$7(&%#e1G6#%!G&%#e2G6#F*&%#e3G6 #F*&%#e4G6#F*&%#e5G6#F*&%#e6G6#F*7(&F(6#F*,$&F/6#F*!\"\",$&F56#F*\"\"# ,$&F,6#F*FD,&&F/6#F*!\"#&F26#F*\"\"%,$&F86#F*FD" }}}{EXCHG {PARA 0 "he at_alg > " 0 "" {MPLTEXT 1 0 13 "R:=radical();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG7%&%#e3G6#%!G&%#e1G6#F)&%#e5G6#F)" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 35 "Of \+ course the radical is an ideal:" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 20 "check_subalgebra(R) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "heat_ alg > " 0 "" {MPLTEXT 1 0 15 "check_ideal(R);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 88 "The package Mubar contains commands for decomposing and classifying Lie algebras." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 12 "with(Mubar):" }}} {EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 2 "L;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%(Lie_algG%)heat_algG7#\"\"'7+7$7%\"\"\"\"\"%F,F ,7$7%F,\"\"&\"\"$!\"\"7$7%F,F(F0\"\"#7$7%F5F-F5F57$7%F5F0F,F57$7%F5F(F 1!\"#7$7%F5F(F-F-7$7%F-F0F0F,7$7%F-F(F(F5" }}}{EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 24 "check_indecomposable(L);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 314 143 "The command Levi decomposition computes the \+ Levi decomposition of a Lie algebra using the method described in t he paper by Winternitz. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 314 66 "This program displays the intermediate steps in \+ the calculation. " }}{PARA 0 "" 0 "" {TEXT 314 99 " To suppress this \+ outline set the global variable _Vessiot_show_intermediate_steps_flag \+ to false." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 25 "LD:=Levi_decomposition();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~Step~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% >Computing~Radical~of~heat_algG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/B egin~~Step~2~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9Computing~Derived~ SeriesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!7(&%#e1G6#%!G&%#e2G6#F (&%#e3G6#F(&%#e4G6#F(&%#e5G6#F(&%#e6G6#F(\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%)infinityG7(&%#e1G6#%!G,$&%#e3G6#F(!\"\",$&%#e5G6#F(\" \"#,$&%#e2G6#F(F2,&&F+6#F(!\"#&%#e4G6#F(\"\"%,$&%#e6G6#F(F2\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%/Begin~~Step~2~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%gnThe~basis~for~the~terminal~algebra~in~the~derived~se ries~is:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(&%#e1G6#%!G&%#e2G6#F'&% #e3G6#F'&%#e4G6#F'&%#e5G6#F'&%#e6G6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%_pThe~terminal~algebra~in~the~~derived~series~is~initialized~as ~Lie~algebra~heat_alg_PG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG 6#7*7*%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F(&%#e4G6#F(&%#e5G6#F(&%#e 6G6#F(7*F(%$---G%%----GF>F>F>F>F>7*&F+6#F(F)\"\"!FBFB&F+6#F(,$&F16#F(! \"\",$&F76#F(\"\"#7*&F.6#F(F)FBFBFB,$&F.6#F(FL,$&F+6#F(FL,&&F16#F(!\"# &F46#F(\"\"%7*&F16#F(F)FBFBFBFBFBFB7*&F46#F(F),$&F+6#F(FH,$&F.6#F(FYFB FB&F76#F(,$&F:6#F(FL7*&F76#F(F)&F16#F(,$&F+6#F(FYFB,$&F76#F(FHFBFB7*&F :6#F(F),$&F76#F(FY,&&F16#F(FL&F46#F(!\"%FB,$&F:6#F(FYFBFB" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@Computing~Radical~of~heat_alg_PG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,End~~Step~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~Step~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%RThe~~radical~o f~the~subalgebra~L_0~is~not~AbelianG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%YInitializing~subalgebra~L_0~as~Lie~algebra~heat_alg_S4_0G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%AThe~radical~of~heat_alg_S4_0~is:G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7%&%#e5G6#%!G&%#e1G6#F'&%#e3G6#F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%UThe~derived~algebra~of~~radical~of~h eat_alg_S4_0~is:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#&%#e3G6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%LThe~~complementary~basis~to~the~radi cal~is:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'&%#e1G6#%!G&%#e2G6#F'&%# e4G6#F'&%#e5G6#F'&%#e6G6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\\oThe ~Factor~Algebra~of~heat_alg_S4_0~by~Radical~is~heat_alg_S4_0_QG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7)7)%!G%\"|grG&%#e1G6#F(&% #e2G6#F(&%#e3G6#F(&%#e4G6#F(&%#e5G6#F(7)F(%$---G%%----GF;F;F;F;7)&F+6# F(F)\"\"!F?&F+6#F(F?,$&F46#F(\"\"#7)&F.6#F(F)F?F?,$&F.6#F(FE,$&F+6#F(F E,$&F16#F(\"\"%7)&F16#F(F),$&F+6#F(!\"\",$&F.6#F(!\"#F?&F46#F(,$&F76#F (FE7)&F46#F(F)F?,$&F+6#F(Fgn,$&F46#F(FYF?F?7)&F76#F(F),$&F46#F(Fgn,$&F 16#F(!\"%,$&F76#F(FgnF?F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%MCompute ~Levi~decomposition~of~Factor~AlgebraG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DThe~~radical~of~heat_alg_S4_0_Q~is:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$&%#e4G6#%!G&%#e1G6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%RThe~semi-simple~subalgebra~of~heat_alg_S4_0_Q~is:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%&%#e2G6#%!G&%#e3G6#F'&%#e5G6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%6The~subalgebra~L_1~isG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7&&%#e3G6#%!G&%#e2G6#F'&%#e4G6#F'&%#e6G6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~Step~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ERadical~of~alg_name2~is~~now~AbelianG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Begin~Step~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%QComputing~Semi-Simple~Part~of~Levi~DecompositionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&Done!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#LDG7$ 7%&%#e5G6#%!G&%#e1G6#F*&%#e3G6#F*7%&%#e2G6#F*,&*&&#!\"\"\"\"#6#F*\"\" \"&F/6#F*F " 0 "" {MPLTEXT 1 0 12 "Show(LD[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%Sh owG6#&-%3Levi_decompositionG6\"6#\"\"\"" }}}{EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 12 "Show(LD[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%&%#e2G6#%!G,&*&&#!\"\"\"\"#6#F'\"\"\"&%#e3G6#F'F/F/&%#e4G6#F'F/&% #e6G6#F'" }}}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 89 "We can initialize the semi-simple part as a Lie algebr a in its own right and study it." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "heat_alg>" 0 "" {MPLTEXT 1 0 41 "subalgebra_to_Lie_alg ebra_data(LD[2],SS);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%(Lie_algG %#SSG7#\"\"$7%7$7%\"\"\"\"\"#F,F-7$7%F,F(F-\"\"%7$7%F-F(F(F-" }}} {EXCHG {PARA 0 "heat_alg > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%0Lie~algebra:~SSG" }}}{EXCHG {PARA 0 "SS > " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7'%!G%\"|grG&%#e1G6#F(&%#e2G6#F(& %#e3G6#F(7'F(%$---G%%----GF5F57'&F+6#F(F)\"\"!,$&F+6#F(\"\"#,$&F.6#F( \"\"%7'&F.6#F(F),$&F+6#F(!\"#F9,$&F16#F(F=7'&F16#F(F),$&F.6#F(!\"%,$&F 16#F(FHF9" }}}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 124 "We can classify this Lie algebra according to a list o f all Lie algebras of dim <6 compiled by P. Winternitz. (It is sl2) " }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "SS > " 0 "" {MPLTEXT 1 0 23 "classify_Lie_algebra();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%BChecking~indecomposability~of:~SSG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>Classifying~the~~algebra:~SS0G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$-%'matrixG6#7%7%\"\"!F)\"\"\"7%F)!\"#F)7%!\"%F)F)7#7$ %+winternitzG7$\"\"$\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Vessiot Libraries" }}{PARA 267 "" 0 "" {TEXT 314 93 "Vessiot contains a number of libraries of Lie \+ algebras, infinitesimal group actions, etc. " }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "SS > " 0 "" {MPLTEXT 1 0 14 "with(Vess iot):" }}}{EXCHG {PARA 0 "SS > " 0 "" {MPLTEXT 1 0 13 "with(Koszul):" }}}{EXCHG {PARA 0 "SS > " 0 "" {MPLTEXT 1 0 22 "with(Vessiot_library): " }}}{EXCHG {PARA 0 "SS > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " SS > " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 300 21 "Lie algebra libraries" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "SS > " 0 "" {MPLTEXT 1 0 30 "Vessiot_lib(winternitz,[5,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%(Lie_algG%$w54G7#\"\"&7$7$7%\" \"#F(\"\"\"F-7$7%\"\"$\"\"%F-F-" }}}{EXCHG {PARA 0 "w54>" 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% 1Lie~algebra:~w54G" }}}{EXCHG {PARA 0 "w54>" 0 "" {MPLTEXT 1 0 25 "Lie _bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6# 7)7)%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F(&%#e4G6#F(&%#e5G6#F(7)F(%$ ---G%%----GF;F;F;F;7)&F+6#F(F)\"\"!F?F?F?F?7)&F.6#F(F)F?F?F?F?&F+6#F(7 )&F16#F(F)F?F?F?&F+6#F(F?7)&F46#F(F)F?F?,$FH!\"\"F?F?7)&F76#F(F)F?,$FC FNF?F?F?" }}}{EXCHG {PARA 0 "w54>" 0 "" {MPLTEXT 1 0 29 "Vessiot_lib(T urkowski,[7,4]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$7%%(Lie_algG%&T_ 7_4G7#\"\"(7-7$7%\"\"\"\"\"#F-F-7$7%F,\"\"$F0!\"#7$7%F-F0F,F,7$7%F,\" \"%F6F,7$7%F,\"\"&F9!\"\"7$7%F-F9F6F,7$7%F0F6F9F,7$7%F6F9\"\"'F,7$7%F6 F(F6F,7$7%F9F(F9F,7$7%FAF(FAF-" }}}{EXCHG {PARA 0 "T_7_4>" 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% 3Lie~algebra:~T_7_4G" }}}{EXCHG {PARA 0 "T_7_4>" 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matri xG6#7+7+%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F(&%#e4G6#F(&%#e5G6#F(&% #e6G6#F(&%#e7G6#F(7+F(%$---G%%----GFAFAFAFAFAFA7+&F+6#F(F)\"\"!,$&F.6# F(\"\"#,$&F16#F(!\"#&F46#F(,$&F76#F(!\"\"FEFE7+&F.6#F(F),$FGFMFE&F+6#F (FE&F46#F(FEFE7+&F16#F(F),$FKFI,$FXFSFE&F76#F(FEFEFE7+&F46#F(F),$FNFSF E,$F[oFSFE&F:6#F(FE&F46#F(7+&F76#F(F)FQ,$FZFSFE,$FboFSFEFE&F76#F(7+&F: 6#F(F)FEFEFEFEFEFE,$&F:6#F(FI7+&F=6#F(F)FEFEFE,$FdoFS,$F[pFS,$FapFMFE " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 301 38 "Petrov's library of inva riant metric" }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 109 "This library is based upon Petrov's classification o f metric with symmetry in the book Einstein Spaces." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 314 74 "Give the section n umber,equation number and the list of the coordinates. " }}{PARA 19 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "T_7_4>" 0 "" {MPLTEXT 1 0 29 "c oord_init([x1,x2,x3,x4],[]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3fram e~name:~euclidG" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 49 "Gam ma:=Vessiot_lib(Petrov, [32,4],[x1,x2,x3,x4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%gq*This~group~action~is~identical~to~Petrov~32.20,~via ~the~transformation~x1=y2~~~~~x2=y3~~~~~x3=-y1~~~~~x4=y4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&GammaG7&&%%D_x2G6#%!G&%%D_x3G6#F),&&%%D_x1G6 #F)!\"\"*&%#x3G\"\"\"&F'6#F)F4F4,(*&F3\"\"\"&F/6#F)F4F1*&&,&*$)F3\"\"# F9#F4FA*$)%#x1GFAF9#F1FA6#F)F4&F'6#F)F4F4*&FEF4&F+6#F)F4F4" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 318 30 "Lie algebras of vector fields" }} {PARA 267 "" 0 "" {TEXT 314 120 "This library is based upon the real \+ classification of Lie algebras of vector fields in the plane found in \+ Olver's book:" }}{PARA 267 "" 0 "" {TEXT 319 36 "Equivalence, Invarian ts and Symmetry" }{TEXT 314 38 ". Use the labeling given in the book. " }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 24 "coord_init([x,u],[],R2):" }}}{EXCHG {PARA 0 "R2>" 0 " " {MPLTEXT 1 0 39 "Gamma:=Vessiot_lib(Olver, [2,3],[x,u]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&GammaG7*&%$D_xG6#%!G&%$D_uG6#F)*&%\"xG\" \"\"&F'6#F)F/*&%\"uGF/&F'6#F)F/*&F.\"\"\"&F+6#F)F/*&F3F7&F+6#F)F/,&*&) F.\"\"#F7&F'6#F)F/F/*(F.F7F3F7&F+6#F)F/F/,&*(F.F7F3F7&F'6#F)F/F/*&)F3F @F7&F+6#F)F/F/" }}}}{SECT 1 {PARA 268 "" 0 "" {TEXT -1 43 "Symmetry al gebras of differential equations" }}{PARA 267 "" 0 "" {TEXT 314 118 "T his library is based upon the symmetry algebras for differential eq uations found in C. Rogers and W. Shadwick. \n" }{TEXT 320 49 "B\344 cklund transformations and their applications. " }{TEXT 314 46 " Mathe matics in Science and Engineering, 161. " }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "R2>" 0 "" {MPLTEXT 1 0 29 "data:=DE_lib(Roge rs,[A,VII]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%dataG7&7%7$%\"tG%\" xG7$%\"uG%\"vG%,schrodingerG7$,(&F+6$\"\"\"\"\"!F2&F,6$F3\"\"#F2*&,&*$ )&F+6$F3F3F6\"\"\"F2*$)&F,F " 0 "" {MPLTEXT 1 0 11 "nops(data);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT 314 53 "The first componet is data for a coordinate \+ system." }}{PARA 0 "" 0 "" {TEXT 314 53 "The second component is a l ist of the equations. " }}{PARA 0 "" 0 "" {TEXT 314 45 "The third co mponet is the list of symmetries." }}{PARA 0 "" 0 "" {TEXT 314 88 "The fourth component is the equations set, as a Vessiot differential e quation object." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "sch rodinger > " 0 "" {MPLTEXT 1 0 8 "data[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7$%\"tG%\"xG7$%\"uG%\"vG%,schrodingerG" }}}{PARA 0 " " 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "schrodinger > " 0 "" {MPLTEXT 1 0 8 "data[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,(&%\"uG 6$\"\"\"\"\"!F(&%\"vG6$F)\"\"#F(*&,&*$)&F&6$F)F)F-\"\"\"F(*$)&F+F3F-F4 F(F(F7F(F-,(&F+F'!\"\"&F&F,F(*&F/F4F2F(F-" }}}{EXCHG {PARA 0 "schrodin ger > " 0 "" {MPLTEXT 1 0 8 "data[3];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7'&%$D_tG6#%!G&%$D_xG6#F',&*&&%\"vG6$\"\"!F0\"\"\"&&%$D_uG6#7$F0F06 #F'F1F1*&&%\"uGF/F1&&%$D_vGF56#F'F1!\"\",(*&%\"tGF1&F)6#F'F1!\"#*(%\"x GF1F-\"\"\"&F36#F'F1F1*(FGFHF9FH&F<6#F'F1F?,**&FBFH&F%6#F'F1\"\"#*&FGF H&F)6#F'F1F1*&F9FH&F36#F'F1F?*&F-FH&F<6#F'F1F?" }}}{EXCHG {PARA 0 "sch rodinger > " 0 "" {MPLTEXT 1 0 12 "op(data[4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7'%'diffeqG%(_diffeqG7$%%evolG\"\"!<$&%\"uG6$F)\"\"#& %\"vGF-%,schrodingerG7$7$7#\"\"&,(&F,6$\"\"\"F)F9F/F9*&,&*$)&F,6$F)F)F .\"\"\"F9*$)&F0F?F.F@F9F9FCF9F.7$7#\"\"',(&F0F8!\"\"F+F9*&F;F@F>F9F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 314 52 "Let's check that the given vectors are symmetries. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "schrod inger > " 0 "" {MPLTEXT 1 0 34 "psi:=diffeq_to_transform(data[4]);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%$psiG70/%\"tGF'/%\"xGF)/&%\"uG6$\"\" !F.F+/&%\"vGF-F0/&F,6$\"\"\"F.F3/&F,6$F.F5F7/&F1F4F:/&F1F8F/&F,6$F5F5FB/&F,6$F.F@,(F:F5*$)F+\"\"$\"\"\"!\"#*&F+F5)F0F@FKFL/& F1F?FP/&F1FCFR/&F1FF,(F3!\"\"*&F0F5)F+F@FKFL*$)F0FJFKFL" }}}{EXCHG {PARA 0 "schrodinger > " 0 "" {MPLTEXT 1 0 126 "f:=proc(i,j) local che ck,X,X2; X:=data[3][i]; X2:=pr_vect(X,2);check:=Lie_derivative(X2,data [2][j]); pullback(psi,check); end;" }}{PARA 0 "schrodinger > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"iG%\"j G6%%&checkG%\"XG%#X2G6\"F-C&>8%&&%%dataG6#\"\"$6#9$>8&-%(pr_vectG6$F0 \"\"#>8$-%/Lie_derivativeG6$F9&&F36#F=6#9%-%)pullbackG6$%$psiGF?F-F-F- " }}}{EXCHG {PARA 0 "schrodinger > " 0 "" {MPLTEXT 1 0 14 "matrix(5,2, f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7$\"\"!F(F'F'F'F' " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Moving Frames" }}{PARA 267 "" 0 "" {TEXT 314 69 "Thus far all \+ our calculations have been done in a coordinate frame." }}{PARA 267 " " 0 "" {TEXT 314 84 " It is a simple matter to introduce moving fram es into the Vessiot environment !. " }}{PARA 267 "" 0 "" {TEXT -1 0 " " }}{PARA 267 "" 0 "" {TEXT 314 42 "Step 1: Create the underlying coor dinates." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "schroding er > " 0 "" {MPLTEXT 1 0 53 "with(Vessiot):with(tensors):coord_init([x ,y],[],fr1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0frame~name:~fr1G" }} }{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 43 "Ste p 2: Define the coframe and its dual:" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "fr1>" 0 "" {MPLTEXT 1 0 44 "coframe:=[(1/y) &mu lt dx, (1/y) &mult dy]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(cofram eG7$*&&%#dxG6#%!G\"\"\"%\"yG!\"\"*&&%#dyG6#F*F+F,F-" }}}{EXCHG {PARA 0 "fr1 > " 0 "" {MPLTEXT 1 0 27 "frame:=dual_frame(coframe);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&frameG7$*&%\"yG\"\"\"&%$D_xG6#%!GF(*&F'\" \"\"&%$D_yG6#F,F(" }}}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 91 "Step 3: The program coframe_data computings the \+ structure constants for the new frame." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "fr1 > " 0 "" {MPLTEXT 1 0 42 "data:=coframe_dat a([x,y],coframe,Dilbert);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dataG7 %7'%-moving_frameG%(DilbertG7#\"\"#%$fr1G7$%\"xG%\"yG7$7$7$\"\"\"F2F.7 $7$F*F*F.7#7$7%F2F*F2!\"\"" }}}{EXCHG {PARA 0 "fr1 > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 86 "Step 4: Initiatize the moving frame. Chose symbols for th e new frame and coframe." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "fr1 > " 0 "" {MPLTEXT 1 0 29 "frame_init(data,[E],[omega]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/frame:~DilbertG" }}}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 76 "Now all computati ons are done relative to this coframe and its dual frame." }}{PARA 267 "" 0 "" {TEXT 314 1 " " }}{EXCHG {PARA 0 "Dilbert > " 0 "" {MPLTEXT 1 0 29 "f:=(x*y) &mult scalar_form();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&%\"xG\"\"\"%\"yGF'" }}}{EXCHG {PARA 0 "Dilbert \+ > " 0 "" {MPLTEXT 1 0 9 "ext_d(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,&*&)%\"yG\"\"#\"\"\"&%'omega1G6#%!G\"\"\"F-*(%\"xGF-F&F-&%'omega2G6#F ,F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "Dilbert > " 0 "" {MPLTEXT 1 0 26 "X:= evalV(y *E1 -x *E2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG,&*&%\"yG\"\"\"&%#E1G6#%!GF(F(*&%\"xGF(&%#E2G6#F, F(!\"\"" }}}{EXCHG {PARA 0 "Dilbert > " 0 "" {MPLTEXT 1 0 31 "beta:=Li e_derivative(X,omega1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG,&* &%\"xG\"\"\"&%'omega1G6#%!GF(F(*&%\"yGF(&%'omega2G6#F,F(F(" }}}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 127 "The com mand change_frame_basis allows one to convert from the moving frame back to the coordinate frame (and conversely)." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "Dilbert > " 0 "" {MPLTEXT 1 0 39 "chan ge_frame_basis(beta,frame,coframe);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,&*&*&%\"xG\"\"\"&%#dxG6#%!GF'\"\"\"%\"yG!\"\"F'&%#dyG6#F+F'" }}} {EXCHG {PARA 0 "fr1 > " 0 "" {MPLTEXT 1 0 48 "g:= evalV( omega1 &t ome ga1 + omega2 &t omega2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,&*& &%'omega1G6#%!G\"\"\"&F(6#F*F+F+*&&%'omega2G6#F*F+&F06#F*F+F+" }}} {EXCHG {PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "Dilbert > " 0 "" {MPLTEXT 1 0 21 "h:=inverse_metric(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG,&*&&%#E1G6#%!G\"\"\"&F(6#F*F+F+*&&%#E2G6#F*F+&F0 6#F*F+F+" }}}{EXCHG {PARA 0 "Dilbert > " 0 "" {MPLTEXT 1 0 13 "C:=offe l2(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG,&*(&%'omega1G6#%!G\" \"\"&%#E2G6#F*F+&F(6#F*F+F+*(&%'omega2G6#F*F+&%#E1G6#F*F+&F(6#F*F+!\" \"" }}}{EXCHG {PARA 0 "Dilbert > " 0 "" {MPLTEXT 1 0 26 "Riem:=curvatu re_tensor(C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%RiemG,***&%'omega1 G6#%!G\"\"\"&%#E2G6#F*F+&F(6#F*F+&%'omega2G6#F*F+!\"\"**&F(6#F*F+&F-6# F*F+&F26#F*F+&F(6#F*F+F+**&F26#F*F+&%#E1G6#F*F+&F(6#F*F+&F26#F*F+F+**& F26#F*F+&FB6#F*F+&F26#F*F+&F(6#F*F+F4" }}}{EXCHG {PARA 0 "Dilbert > " 0 "" {MPLTEXT 1 0 24 "S:=Ricci_scalar(h,Riem);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG!\"#" }}}{PARA 19 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "Sample Applications" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 40 "Here are a few sim ple applications. " }}{PARA 267 "" 0 "" {TEXT 314 111 "We want to e mphasize how easy it easy to write the code for these applications on \+ top of the Vessiot software." }{TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 117 "For further examples and more applications see the \+ Tutorial Page under the Symbolics Page of the FG_MP website." }} {PARA 19 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 306 17 "Noether's Theorem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "Dilbert > " 0 "" {MPLTEXT 1 0 29 "with (Vessiot):with(de_appls);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&%.Lie_reductionG%)Noether1G%)Noether2G%+inver se_tdG" }}}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 99 "Here is the code for Noether's theorem in terms of the oper ations of the variational bicomplex." }}{PARA 19 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "w35>" 0 "" {MPLTEXT 1 0 13 "op(Noether1);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#R6$%'vectorG%+LagrangianG6$%%tempG%.bo undary_formG6\"F*C%>8$-%&&plusG6$-%#dVG6#9%-%(EL_formGF3>8%-%,homotopy _dHG6#F--%'&minusG6$-%*vert_hookG6$-%(pr_vectG6$-%%evolG6#9$-%+order_f ormGF3F8-%+total_hookG6$-%&totalGFGF4F*F*F*" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 63 "Let's compute a few conservation laws for Laplace's \+ equation>" }}}{EXCHG {PARA 0 "w35>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "w35>" 0 "" {MPLTEXT 1 0 23 "coord_frame([x,y],[u]):" }}} {EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 53 "lamba:=(1/2*(u[1,0]^2 + u[0,1]^2))&mult vol_biform():" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 12 "show(lamba);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&* $)&%\"uG6$\"\"\"\"\"!\"\"#\"\"\"#F*F,*$)&F(6$F+F*F,F-F.F*&%#DxG6#%!GF* %#^~GF*&%#DyG6#F6F*" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 19 "Rotation al symmetry" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 41 "X:= (y \+ &mult D_x) &minus (x &mult D_y):" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 8 "show(X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"yG \"\"\"&%$D_xG6#%!GF&F&*&%\"xGF&&%$D_yG6#F*F&!\"\"" }}}{EXCHG {PARA 0 " euclid>" 0 "" {MPLTEXT 1 0 17 "X1:=pr_vect(X,1):" }}}{EXCHG {PARA 0 "e uclid>" 0 "" {MPLTEXT 1 0 9 "show(X1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&%\"yG\"\"\"&%$D_xG6#%!GF&F&*&%\"xGF&&%$D_yG6#F*F&!\"\"*&&%\" uG6$\"\"!F&F&&&%$D_uG6#7$F&F56#F*F&F&*&&F36$F&F5F&&&F86#7$F5F&6#F*F&F0 " }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 18 "Lie_der(X1,lamba); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%'biformG%'euclidG7$\"\"#\"\"! 7#7$7$\"\"\"F(F)" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 25 "om ega:=Noether1(X,lamba):" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 12 "show(omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,(*&%\"xG\" \"\")&%\"uG6$F(\"\"!\"\"#\"\"\"#!\"\"F.*&F'F/)&F+6$F-F(F.F/#F(F.*(F4F( F*F(%\"yGF(F1F(&%#DxG6#%!GF(F(*&,(*&F8F/F)F/F6*&F8F/F3F/F0*(F*F/F4F/F' F/F1F(&%#DyG6#F" 0 "" {MPLTEXT 1 0 17 "sigma:=dH(omega):" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 20 "show(factor(sigma));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,,&&%\"uG 6$\"\"#\"\"!\"\"\"&F&6$F)F(F*F*,&*&&F&6$F)F*F*%\"xGF*!\"\"*&&F&6$F*F)F *%\"yGF*F*F*&%#DxG6#%!GF*%#^~GF*&%#DyG6#F:F*" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 14 "Scale symmetry" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 25 "coord_frame([x,y,z],[u]):" }}}{EXCHG {PARA 0 "euclid> " 0 "" {MPLTEXT 1 0 53 "lambda:=(1/2*( u[1,0,0]^2 + u[0,1,0]^2 + u[0,0 ,1]^2))" }}{PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 19 "&mult vol_biform(): " }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "eucl id>" 0 "" {MPLTEXT 1 0 13 "show(lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*.,(*$)&%\"uG6%\"\"\"\"\"!F+\"\"#\"\"\"#F*F,*$)&F(6%F+F*F+F,F-F. *$)&F(6%F+F+F*F,F-F.F*&%#DxG6#%!GF*%#^~GF*&%#DyG6#F:F*F;F*&%#DzG6#F:F* " }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 38 "Y:= (x &mult D_x) &plus (y&mult D_y) " }}{PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 20 "&plus (z &mult D_z) " }} {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 43 "&plus (((-1/2)*u[0,0,0]) &mult D_u[0,0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG7$7%%%vectG%'eu clidG7\"7&7$7#\"\"\"%\"xG7$7#\"\"#%\"yG7$7#\"\"$%\"zG7$7#\"\"%,$&%\"uG 6%\"\"!F>F>#!\"\"F1" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 17 "Y1:=pr_vect(Y,1):" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 19 " Lie_der(Y1,lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%'biformG%' euclidG7$\"\"$\"\"!7#7$7%\"\"\"\"\"#F(F)" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 26 "omega:=Noether1(Y,lambda):" }}}{EXCHG {PARA 0 "e uclid>" 0 "" {MPLTEXT 1 0 27 "coeff_list(omega, [[1,2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&%\"zG\"\"\")&%\"uG6%F&\"\"!F+\"\"#\"\"\"#! \"\"F,*&F%F-)&F)6%F+F&F+F,F-F.*&F%F-)&F)6%F+F+F&F,F-#F&F,*&F6F&&F)6%F+ F+F+F&F8*(F6F-F2F&%\"yGF&F&*(F6F-F(F&%\"xGF&F&" }}}{EXCHG {PARA 0 "euc lid>" 0 "" {MPLTEXT 1 0 15 "eta:=dH(omega):" }}}{EXCHG {PARA 0 "euclid >" 0 "" {MPLTEXT 1 0 18 "show(factor(eta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*0,*&%\"uG6%\"\"!F)F)\"\"\"*&&F'6%F)F)F*F*%\"zGF*\"\" #*&&F'6%F)F*F)F*%\"yGF*F/*&&F'6%F*F)F)F*%\"xGF*F/F*,(&F'6%F/F)F)F*&F'6 %F)F/F)F*&F'6%F)F)F/F*F*&%#DxG6#%!GF*%#^~GF*&%#DyG6#FBF*FCF*&%#DzG6#FB F*#F*F/" }}}{PARA 0 "" 0 "" {TEXT 307 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 305 67 "Recursion Operators and Higher Order Sym metries of the KdV equation" }}{PARA 267 "" 0 "" {TEXT -1 136 "We wri te a little program which implements the recursion operator for KdV and we use it to generate the higher order symmetries" }}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 14 "with(Vessiot):" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 15 "with(de_appls):" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 130 "The key idea here is to use the horizontal homotopy \+ operator to implement the formal inverse of the total derivative opera tor." }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 15 "op(inverse_td) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#R6#%$expG6#%\"LG6\"F(C%@$0-%3fram eBaseDimensionGF(\"\"\"-%'ERRRORG6#%fnNumber~of~Independent~Variables~ in~Current~Frame~~Must~Be~1G>8$-%&&multG6$9$-%+vol_biformGF(-%#opG6#-% *coeff_setG6#-%,homotopy_dHG6#F4F(F(F(" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 22 "coord_frame([x], [u]):" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 27 "KdVFlow:= u[3] + u[0]*u[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %(KdVFlowG,&&%\"uG6#\"\"$\"\"\"*&&F'6#\"\"!F*&F'6#F*F*F*" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 73 "Here's the procedure which implement s the recursion operator for KdV." }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 84 "KdV_Rec:= exp -> expand(multi_td(exp,[2]) +2/3*u[0]* exp +1/3*u[1]*inverse_td(exp));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%( KdV_RecGR6#%$expG6\"6$%)operatorG%&arrowGF(-%'expandG6#,(-%)multi_tdG6 $9$7#\"\"#\"\"\"*&&%\"uG6#\"\"!F6F3F6#F5\"\"$*&&F96#F6F6-%+inverse_tdG 6#F3F6#F6F=F(F(F(" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 10 "Q _0:=u[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Q_0G&%\"uG6#\"\"\"" }} }{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 18 "Q_1:=KdV_Rec(Q_0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Q_1G,&&%\"uG6#\"\"$\"\"\"*&&F'6#\" \"!F*&F'6#F*F*F*" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 18 "Q_ 2:=KdV_Rec(Q_1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Q_2G,**&&%\"uG6 #\"\"\"F*&F(6#\"\"#F*#\"#5\"\"$&F(6#\"\"&F**&&F(6#\"\"!F*&F(6#F0F*#F3F 0*&)F5F-\"\"\"F'F=#F3\"\"'" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 18 "Q_3:=KdV_Rec(Q_2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Q_3G, 2*(&%\"uG6#\"\"#\"\"\"&F(6#\"\"!F+&F(6#F+F+#\"#q\"\"**&F'\"\"\"&F(6#\" \"$F+#\"#NF8*$)F/F8F5#F:\"#=*&F/F5&F(6#\"\"%F+\"\"(&F(6#FCF+*&F,F5&F(6 #\"\"&F+#FCF8*&F6F5)F,F*F5F=*&)F,F8F5F/F5#F:\"#a" }}}{EXCHG {PARA 0 "e uclid>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 25 "CHECK THAT FLOWS COMMUTE:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "gen_Lie_bracket( KdVFlow &mult vect(u[0]), Q_0 &mult vect(u[0])) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%%vectG%'euclidG7\"7#7$7#\"\" \"\"\"!" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 65 "gen_Lie_bra cket( KdVFlow &mult vect(u[0]), Q_1 &mult vect(u[0]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%%vectG%'euclidG7\"7#7$7#\"\"\"\"\"!" }}} {EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 65 "gen_Lie_bracket( KdVFlo w &mult vect(u[0]), Q_2 &mult vect(u[0]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%%vectG%'euclidG7\"7#7$7#\"\"\"\"\"!" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 65 "gen_Lie_bracket( KdVFlow &mult vect(u[0]), Q_3 &mult vect(u[0]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7$7%%%vectG%'euclidG7\"7#7$7#\"\"\"\"\"!" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT 304 24 "Differential Invariants" }}{PARA 267 "" 0 "" {TEXT -1 234 "With Vessiot capabilities for prolonging transformations o n jet spaces, it is easy to use the the moving frames of Fels and \+ Olver to calculate differential invariants. We do so for simple case of the Euclidean group on R^2." }}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 14 "with( Vessiot):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "with(moving_fr ame):" }}}{EXCHG {PARA 0 "SE2>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 24 "coord_init([x],[u],euc):" }}} {EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 31 "coord_init([a,b,theta],[], SE2):" }}}{EXCHG {PARA 0 "SE2>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT -1 40 "Define the group action and prolong it ." }}{PARA 270 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 102 "Phi:= transform(euc,euc,[x=cos(theta)*x -sin(theta)* u[0] +a ,u[0]=sin(theta)*x +cos(theta)*u[0] +b]):\n" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 10 "show(Phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@Domain~and~~Range:~euc~--->~eucG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=Differential~Order:~0~--->~0G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG,(*&-%$cosG6#%&thetaG\"\"\"F$F+F+*&-%$sinGF)F+&% \"uG6#\"\"!F+!\"\"%\"aGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"uG6# \"\"!,(*&-%$sinG6#%&thetaG\"\"\"%\"xGF.F.*&-%$cosGF,F.F$F.F.%\"bGF." } }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "Phi2:=pr_transform(Phi ,2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%Phi2G7$7%%)helm_mapG7$7$%$e ucG\"\"#F)7$%&pointGF+7&7$,(*&-%$cosG6#%&thetaG\"\"\"%\"xGF6F6*&-%$sin GF4F6&%\"uG6#\"\"!F6!\"\"%\"aGF6F77$,(*&F9\"\"\"F7FDF6*&F2FDF;FDF6%\"b GF6F;7$,$*&,&F9F6*&F2FD&F<6#F6FDF6FD,&F2F?*&F9FDFLF6F6!\"\"F?FL7$*&&F< 6#F+FD,.*$)F2\"\"$FDF6*(F9FD)F2F+FDFLFD!\"$*&F2FD)FLF+FDFX*&FWFDFgnFDF en*&)FLFXFDF9FDF?*(FjnFDF9FDFZFDF6FPFS" }}}{EXCHG {PARA 270 "" 0 "" {TEXT -1 62 "Construct the moving frame as a map from R^2 into the g roup." }}{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 ]):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 10 "show(rho);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%@Domain~and~~Range:~euc~--->~SE2G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%=Differential~Order:~1~--->~0G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"aG,$*&,&%\"xG\"\"\"*&&%\"uG6#F)F)& F,6#\"\"!F)F)\"\"\"*$-%%sqrtG6#,&*$)F+\"\"#F1F)F)F)F1!\"\"!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"bG*&,&*&&%\"uG6#\"\"\"F+%\"xGF+F+& F)6#\"\"!!\"\"\"\"\"*$-%%sqrtG6#,&*$)F(\"\"#F1F+F+F+F1!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&thetaG,$-%'arctanG6#&%\"uG6#\"\"\"!\"\"" }}}{EXCHG {PARA 270 "" 0 "" {TEXT -1 39 "Construct the differential i nvariants." }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 24 "u2:=pullb ack(Phi2,u[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u2G*&&%\"uG6#\" \"#\"\"\",.*$)-%$cosG6#%&thetaG\"\"$F*\"\"\"*(-%$sinGF0F3)F.F)F*&F'6#F 3F3!\"$*&F.F3)F8F)F*F2*&F-F*F " 0 "" {MPLTEXT 1 0 41 "kappa0:= simpli fy(pullback(rho,u2),trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'kappa 0G*&*&&%\"uG6#\"\"#\"\"\"),&*$)&F(6#F+F*\"\"\"F+F+F+#\"\"$F*F2F2,*F+F+ *$)F0\"\"%F2F4F.F4*$)F0\"\"'F2F+!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 " " {MPLTEXT 1 0 23 "kappa0:=factor(kappa0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'kappa0G*&&%\"uG6#\"\"#\"\"\"*$),&*$)&F'6#\"\"\"F)F*F 2F2F2#\"\"$F)F*!\"\"" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 61 "This is the standard formula for the curvature of a curve." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 308 23 "Lie algebra cohomology" }{TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 46 "Construct the Lie algebra pair (so(4) , so(3))." }}{PARA 267 "" 0 "" {TEXT -1 45 "Compute the relative Lie \+ algebra cohomology." }}{PARA 267 "" 0 "" {TEXT -1 46 "Check that (so( 4), so(3)) is symmetry pair. " }}{EXCHG {PARA 0 "E4>" 0 "" {MPLTEXT 1 0 14 "with(Vessiot):" }}}{EXCHG {PARA 0 "E4>" 0 "" {MPLTEXT 1 0 13 " with(Koszul):" }}}{EXCHG {PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 14 "with(te nsors):" }}}{EXCHG {PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 33 "coord_frame([ x1,x2,x3,x4],[],E4):" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 55 "We cons truct so(4) and so(3) as subalgebras of gl(4)." }}}{EXCHG {PARA 0 "E 4>" 0 "" {MPLTEXT 1 0 18 "gl4:=create_gl(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&&%\"eG6#%#ijG%%$gl4G6%7$7%%(Lie_algG%%gl4RG7# \"#;7hn7$7%\"\"\"\"\"#F0F/7$7%F/\"\"$F3F/7$7%F/\"\"%F6F/7$7%F/\"\"&F9! \"\"7$7%F/\"\"*F=F:7$7%F/\"#8F@F:7$7%F0F9F/F/7$7%F0F9\"\"'F:7$7%F0FEF0 F/7$7%F0\"\"(F3F/7$7%F0\"\")F6F/7$7%F0F=\"#5F:7$7%F0F@\"#9F:7$7%F3F9FJ F:7$7%F3F=F/F/7$7%F3F=\"#6F:7$7%F3FPF0F/7$7%F3FZF3F/7$7%F3\"#7F6F/7$7% F3F@\"#:F:7$7%F6F9FMF:7$7%F6F=F[oF:7$7%F6F@F/F/7$7%F6F@F+F:7$7%F6FSF0F /7$7%F6F^oF3F/7$7%F6F+F6F/7$7%F9FEF9F:7$7%F9FPF=F:7$7%F9FSF@F:7$7%FEFJ FJF/7$7%FEFMFMF/7$7%FEFPFPF:7$7%FEFSFSF:7$7%FJF=F9F/7$7%FJFPFEF/7$7%FJ FPFZF:7$7%FJFZFJF/7$7%FJF[oFMF/7$7%FJFSF^oF:7$7%FMFPF[oF:7$7%FMF@F9F/7 $7%FMFSFEF/7$7%FMFSF+F:7$7%FMF^oFJF/7$7%FMF+FMF/7$7%F=FZF=F:7$7%F=F^oF @F:7$7%FPFZFPF:7$7%FPF^oFSF:7$7%FZF[oF[oF/7$7%FZF^oF^oF:7$7%F[oF@F=F/7 $7%F[oFSFPF/7$7%F[oF^oFZF/7$7%F[oF^oF+F:7$7%F[oF+F[oF/7$7%F@F+F@F:7$7% FSF+FSF:7$7%F^oF+F^oF:72%$e11G%$e12G%$e13G%$e14G%$e21G%$e22G%$e23G%$e2 4G%$e31G%$e32G%$e33G%$e34G%$e41G%$e42G%$e43G%$e44G72%*epsilon11G%*epsi lon12G%*epsilon13G%*epsilon14G%*epsilon21G%*epsilon22G%*epsilon23G%*ep silon24G%*epsilon31G%*epsilon32G%*epsilon33G%*epsilon34G%*epsilon41G%* epsilon42G%*epsilon43G%*epsilon44G" }}}{EXCHG {PARA 0 "E4>" 0 "" {MPLTEXT 1 0 19 " Lie_alg_init(gl4):" }}}{EXCHG {PARA 0 "gl4R>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "gl4R>" 0 "" {MPLTEXT 1 0 20 "chan ge_frame_to(E4):" }}}{EXCHG {PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 31 "g:=c anonical_flat_metric(4,0):\n" }}}{EXCHG {PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 8 "show(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&&%$dx1G6#%!G\" \"\"&F&6#F(F)F)*&&%$dx2G6#F(F)&F.6#F(F)F)*&&%$dx3G6#F(F)&F46#F(F)F)*&& %$dx4G6#F(F)&F:6#F(F)F)" }}}{EXCHG {PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 12 "V:=vect(x4):" }}}{EXCHG {PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 43 "so4_subalg:=create_gl_sub algebra(gl4R,[g]):" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 45 " so3_subalg:=create_gl_subalgebra(gl4R,[g,V]):" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 132 "The command subalgebra_pair_to_Lie_algebra_data_p air initialized so4 as a Lie algebra in its own right with so3 as a \+ subalgebra." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "gl4R \+ > " 0 "" {MPLTEXT 1 0 83 "so4_so3data:=subalgebra_pair_to_Lie_algebra_ data_pair(so4_subalg,[so3_subalg],so4):" }}}{EXCHG {PARA 0 "so4>" 0 " " {MPLTEXT 1 0 20 "so3:=so4_so3data[2]:" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 65 "Now compute the cohomology -- first compute the relative chains." }}}{EXCHG {PARA 0 "so4>" 0 "" {MPLTEXT 1 0 24 "C:=relative_c hains(so3):" }}}{EXCHG {PARA 0 "so4>" 0 "" {MPLTEXT 1 0 12 "map(Show,C );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7#\"\"\"7#*&%#0~GF%&%'theta1G6 #%!GF%7#**F(\"\"\"&F*6#F,F%%#^~GF%&%'theta2G6#F,F%7#*,&%'theta3G6#F,F% F2F%&%'theta5G6#F,F%F2F%&%'theta6G6#F,F%" }}}{EXCHG {PARA 0 "so4>" 0 " " {MPLTEXT 1 0 29 "H:=Lie_algebra_cohomology(C):" }}}{EXCHG {PARA 0 "s o4>" 0 "" {MPLTEXT 1 0 12 "map(Show,H);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7#*&%#0~G\"\"\"&%'theta1G6#%!GF'7#**F&\"\"\"&F)6#F+F'%#^~GF'&% 'theta2G6#F+F'7#*,&%'theta3G6#F+F'F1F'&%'theta5G6#F+F'F1F'&%'theta6G6# F+F'" }}}{EXCHG {PARA 0 "so4>" 0 "" {MPLTEXT 1 0 34 "M:=reductive_comp lement(so3,[mu]):" }}}{EXCHG {PARA 0 "so4>" 0 "" {MPLTEXT 1 0 8 "Show( M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&&%#e3G6#%!G\"\"\"*&%#muGF)& %#e4G6#F(F)F),&*&F+\"\"\"&%#e2G6#F(F)!\"\"&%#e5G6#F(F),&*&F+F1&%#e1G6# F(F)F)&%#e6G6#F(F)" }}}{EXCHG {PARA 0 "so4>" 0 "" {MPLTEXT 1 0 28 "che ck_symmetric_pair(so3,M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG " }}}{EXCHG {PARA 0 "so4>" 0 "" {MPLTEXT 1 0 31 "M0:=map(helmsimp,subs (mu=0,M)):" }}}{EXCHG {PARA 0 "so4>" 0 "" {MPLTEXT 1 0 9 "Show(M0);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7%&%#e3G6#%!G&%#e5G6#F'&%#e6G6#F'" }} }{EXCHG {PARA 0 "so4>" 0 "" {MPLTEXT 1 0 29 "check_symmetric_pair(so3, M0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 267 " " 0 "" {TEXT -1 127 "This example illustrates a beautiful theorem tha t in symmetric space the relative chains are all cohomology represen tatives." }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "How Vessiot Works " }}{PARA 267 "" 0 "" {TEXT 314 78 "We briefly mention two key des ign features of the software suite Vessiot." }}{SECT 1 {PARA 3 "" 0 " " {TEXT 302 31 "The _Vessiot_frame_data table" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 146 "Every time a new c oordinate system is introduced, a moving frame or a Lie algebra defin ed, a entry is created in the table_Vessiot_frame_data." }}{PARA 267 " " 0 "" {TEXT 314 144 " This table contains information around the coo rdinate and frame labels, the order of prolongation, and the structur e equations for the frame." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 ">" 0 "" {MPLTEXT 1 0 14 "with(Vessiot):" }}}{EXCHG {PARA 0 ">" 0 "" {MPLTEXT 1 0 13 "with(Koszul):" }}}{EXCHG {PARA 0 ">" 0 "" {MPLTEXT 1 0 22 "with(Vessiot_library):" }}}{EXCHG {PARA 0 ">" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 ">" 0 "" {MPLTEXT 1 0 20 "_Vessiot_frame_data; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4_Vessiot_frame_dataG" }}}{EXCHG {PARA 0 ">" 0 "" {MPLTEXT 1 0 26 "coord_init([x],[u],Gauss);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2frame~name:~GaussG" }}}{EXCHG {PARA 0 "Gau ss>" 0 "" {MPLTEXT 1 0 27 "_Vessiot_frame_data[Gauss];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7'7&%&frameG%(vst_eucG%&GaussG7*7#%\"xG7#%\"uG\"\" \"7%F*&F,6#\"\"!&F,6#F-7%7$Q$D_x6\"7\"7$Q$D_uF77#7#F17$F:7#7#F-7%7$Q#d xF7F87$Q#duF7F;7$FDF>7#7$Q#DxF7F87$7$Q#CuF7F;7$FKF>7#F8FM-%'matrixG6#F ;FN" }}}{EXCHG {PARA 0 "Gauss>" 0 "" {MPLTEXT 1 0 11 "form(u[2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#&&%#duG6#7#\"\"#6#%!G" }}}{PARA 267 " " 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 73 "Note that the \+ table entry for Gauss now contains 2-jet information" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "Gauss > " 0 "" {MPLTEXT 1 0 27 "_Vessiot_frame_data[Gauss];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7 '7&%&frameG%(vst_eucG%&GaussG7*7#%\"xG7#%\"uG\"\"#7&F*&F,6#\"\"!&F,6# \"\"\"&F,6#F-7&7$Q$D_x6\"7\"7$Q$D_uF:7#7#F17$F=7#7#F47$F=7#7#F-7&7$Q#d xF:F;7$Q#duF:F>7$FJFA7$FJFD7#7$Q#DxF:F;7%7$Q#CuF:F>7$FRFA7$FRFD7#F;FU- %'matrixG6#F>FV" }}}{EXCHG {PARA 0 "Gauss > " 0 "" {MPLTEXT 1 0 30 "Ve ssiot_lib(winternitz,[3,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%( Lie_algG%$w35G7#\"\"$7%7$7%\"\"\"\"\"#F,F,7$7%F,F(F-!\"#7$7%F-F(F(F," }}}{EXCHG {PARA 0 "Gauss > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%1Lie~algebra:~w35G" }}}{EXCHG {PARA 0 "w35 > " 0 "" {MPLTEXT 1 0 25 "_Vessiot_frame_data[w35];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7'7&%&frameG%-Koszul_frameG%$w35G7*7%%#z1G%# z2G%#z3G7\"\"\"\"F)7%7$Q#e16\"F-7$Q#e2F2F-7$Q#e3F2F-7%7$Q'theta1F2F-7$ Q'theta2F2F-7$Q'theta3F2F-F-F-7#F-7#7%,$*(&%'theta1G6#%!GF.%#^~GF.&%'t heta2G6#FFF.!\"\",$*(&FD6#FFF.FGF.&%'theta3G6#FFF.\"\"#,$*(&FI6#FFF.FG F.&FQ6#FFF.FK-%'matrixG6#7#7#\"\"!-Fen6#7%7%*&%#0~GF.&%#e1G6#FFF.&Fao6 #FF,$&%#e2G6#FF!\"#7%,$&Fao6#FFFK*&F_o\"\"\"&Fao6#FFF.&%#e3G6#FF7%,$&F go6#FFFS,$&Fcp6#FFFKF^p" }}}{PARA 267 "" 0 "" {TEXT 314 1 " " }}{PARA 267 "" 0 "" {TEXT 314 58 "Attached to each frame is a certain fra me protocol. " }}{PARA 267 "" 0 "" {TEXT 314 81 "Currently there are \+ 3 frame protocols vst_euc, Koszul_frame and repere_mobile. " }} {PARA 267 "" 0 "" {TEXT 314 109 "Each frame protocol specifies a se t of rules for determining computational rules within the given fra me" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 0 "" } }}{SECT 1 {PARA 3 "" 0 "" {TEXT 303 47 "Derivations, Anti-Derivations \+ and Homomorphism." }{TEXT -1 1 " " }}{PARA 267 "" 0 "" {TEXT 314 116 " Vessiot has powerful capabilities for defining operations on f orms, vector, tensors, etc. as derivations. " }}{PARA 267 "" 0 "" {TEXT 314 50 " Here are the code for computing Lie derivatives." }} {PARA 267 "" 0 "" {TEXT 314 58 "A different procedure is called for \+ each Vessiot object." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "w35 > " 0 "" {MPLTEXT 1 0 19 "op(Lie_derivative);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#R6$'%&vectrG%-Vessiot_vectG%\"BG6&%$ansG%$vecG%(objT ypeG%(vecTypeG6\"F.C0-%4optionalFrameChangeG6$7#9\"\"\"#-%2VessiotFram eCheckG6#9$>8%-%1assign_vect_typeGF8@%/-%#opG6$\"\"!9%%)_VESSIOTGF.C$> 8$-%%Lie0G6$F;FD-%'RETURNG6#FH>8&-%+objectTypeG6#FD>8'&-%+vectorTypeG6 #F;6#\"\"\"@$/FP%%vectGC$>FH-%,Lie_bracketGFKFL@$/FP%%formGC$>FH-%)Lie _formGFKFL@$3/FU%,projectableG/FP%'biformGC$>FH-%+Lie_biformGFKFL@$3/F U%%evolGFhoC$>FH-%)Lie_vertGFKFL@$3/FU%$totGFhoC$>FH-%*Lie_totalGFKFL@ $/FP%'tensorGC$>FH-%)Lie_tensGFKFL@$/FP%+connectionGC$>FH-%)Lie_connGF KFL-%&ERRORG6#%BVESSIOT~ERROR:~Invalid~arguementsGF.F.F." }}}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 81 "The key \+ line in the following program is the call to the derivation program ." }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "w35>" 0 "" {MPLTEXT 1 0 13 "op(Lie_form);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#R6$' %&vectrG%-Vessiot_vectG'%&formaG%-Vessiot_formG6$%$ansG%&frameG6\"F.C( -%2VessiotFrameCheckG6#9$@$0-%,objectFrameGF2-F76#9%-%&ERRORG6&%?Vessi ot~Error:~frame~of~vectorGF6%9must~equal~frame~of~formGF8>8%-%.framePr otocolGF.>8$-&%+derivationG6#%+formBiformG6'&FA6#%)Lie_formG%&derivGF: -%*zero_formG6#-%+formDegreeGF9F3-%.VessiotPromptGF.FEF.F.F." }}} {PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 90 "Here is the code where the Lie derivative of a form gets computed in a coo rdinate frame. " }}{PARA 267 "" 0 "" {TEXT 314 15 "Pretty simple. " } }{PARA 19 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "w35 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "w35>" 0 "" {MPLTEXT 1 0 25 "print (vst_euc[Lie_form]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#R6%%%exprG%%fl agG%$vecG6(%&vectrG%\"iG%$ansG%%varsG%\"vG%&Lie_fG6\"F/C%>8$&-%#opG6#9 &6#\"\"#@'/9%.%%bftvGC%>8%\"\"\"?(F/FBFBF/32FA-%%nopsG6#F22&&&F26#FA6# FBFN9$>FA,&FAFBFBFB@%/FJFOC&>8&7\">8'-%/indet_helmlistG6#&FLF8?&8(FY%% trueG>FV7$-F56#FV7$7#-%/tuple_to_indexG6#Fin-%%diffG6$FgnFin>8)-%)_VES SIOTG6#7$7%%%formG%<_Vessiot_current_frame_nameGFBFV>Fho-%*zero_formGF N/F<.%&coeffGC$>FV7#7$FW-&%(vst_eucG6#%%Lie0G6$F7FO>Fho-Fjo6#7$7%%'bif ormGF_p\"\"!FV-%&ERRORG6#%BVESSIOT~ERROR:~insufficient~inputGFhoF/F/F/ " }}}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 314 86 "Here is the code where the Lie derivative of a form gets computed in \+ the Koszul_frame." }{TEXT -1 2 " " }}{PARA 19 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "w35 > " 0 "" {MPLTEXT 1 0 30 "print(Koszul_frame[Lie_ form]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#R6%%%exprG%%flagG%\"vG6$%&d _ansG%#frG6\"F+C&>8%%<_Vessiot_current_frame_nameG@$/9%.%&coeffG>8$\" \"!@$/F2.%%bftvG>F6-%%hookG6$9&&&&&%4_Vessiot_frame_dataG6#F.6#\"\"$6# \"\"\"6#9$F6F+F+F+" }}}{EXCHG {PARA 0 "w35>" 0 "" {MPLTEXT 1 0 0 "" }} }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{MARK "14" 0 }{VIEWOPTS 1 1 0 3 4 1802 }