{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 95 " This Maple worksheet \+ is basically used to perform some algebra needed to better understand " }}{PARA 0 "" 0 "" {TEXT -1 88 " the normal modes of some sys tems of coupled oscillators, in several dimensions." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 " The command just below tells Maple that it should get itself prepared to receive commands " }}{PARA 0 "" 0 "" {TEXT -1 102 " involving matrices. It i s quite an old form of that command, the newer editions of Maple" }} {PARA 0 "" 0 "" {TEXT -1 99 " having newer forms in additi on to this one, which is being done in Maple 8. However, " }}{PARA 0 "" 0 "" {TEXT -1 51 " it is sufficient for our purpose s." }}{PARA 0 "" 0 "" {TEXT -1 105 " When invoked in this way, i.e. , with a semi-colon, it lists all the different programs in this block ." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have been red efined and unprotected\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7^r%.Block DiagonalG%,GramSchmidtG%,JordanBlockG%)LUdecompG%)QRdecompG%*Wronskian G%'addcolG%'addrowG%$adjG%(adjointG%&angleG%(augmentG%(backsubG%%bandG %&basisG%'bezoutG%,blockmatrixG%(charmatG%)charpolyG%)choleskyG%$colG% 'coldimG%)colspaceG%(colspanG%*companionG%'concatG%%condG%)copyintoG%* crossprodG%%curlG%)definiteG%(delcolsG%(delrowsG%$detG%%diagG%(diverge G%(dotprodG%*eigenvalsG%,eigenvaluesG%-eigenvectorsG%+eigenvectsG%,ent ermatrixG%&equalG%,exponentialG%'extendG%,ffgausselimG%*fibonacciG%+fo rwardsubG%*frobeniusG%*gausselimG%*gaussjordG%(geneqnsG%*genmatrixG%%g radG%)hadamardG%(hermiteG%(hessianG%(hilbertG%+htransposeG%)ihermiteG% *indexfuncG%*innerprodG%)intbasisG%(inverseG%'ismithG%*issimilarG%'isz eroG%)jacobianG%'jordanG%'kernelG%*laplacianG%*leastsqrsG%)linsolveG%' mataddG%'matrixG%&minorG%(minpolyG%'mulcolG%'mulrowG%)multiplyG%%normG %*normalizeG%*nullspaceG%'orthogG%*permanentG%&pivotG%*potentialG%+ran dmatrixG%+randvectorG%%rankG%(ratformG%$rowG%'rowdimG%)rowspaceG%(rows panG%%rrefG%*scalarmulG%-singularvalsG%&smithG%,stackmatrixG%*submatri xG%*subvectorG%)sumbasisG%(swapcolG%(swaprowG%*sylvesterG%)toeplitzG%& traceG%*transposeG%,vandermondeG%*vecpotentG%(vectdimG%'vectorG%*wrons kianG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 " We intend to look at \+ varying numbers of carts, all of equal mass m, and moving on a frictio nless, horizontal" }}{PARA 0 "" 0 "" {TEXT -1 117 " surface, but \+ each adjacent ones being attached either to each other or to an adjace nt, fixed wall, if they are " }}{PARA 0 "" 0 "" {TEXT -1 100 " on t he ends of the \"caravan\" of carts. Each of these springs all have e qual spring constants k." }}{PARA 0 "" 0 "" {TEXT -1 110 " \+ Therefore, we may suppose that k/m is taken as omega0, squared, an d use it to scale all other " }}{PARA 0 "" 0 "" {TEXT -1 111 " fre quencies involved in the problem. Therefore, the matrix given below, \+ which seems to have all numbers, " }}{PARA 0 "" 0 "" {TEXT -1 63 "only , is really such that every term is multiplied by omega0^2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "For the first \+ case, I just consider two carts, so that we should expect to receive t he same information as already " }}{PARA 0 "" 0 "" {TEXT -1 49 "discus sed in detail in the text and in the class:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "K2:=matrix([[2,-1],[-1,2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#K2G-%'matrixG6#7$7$\"\"#!\"\"7$F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eigenvectors(K2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%\"\"$\"\"\"<#-%'vectorG6#7$!\"\"F%7%F%F%<#-F(6#7$F %F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 " The request for the eig envectors produces a sequence of \"things\" within brackets---in this \+ case two of them." }}{PARA 0 "" 0 "" {TEXT -1 117 " Considering the first one in this sequence, that bracket first tells us that the eige nvalue is 3---remember that " }}{PARA 0 "" 0 "" {TEXT -1 118 " this \+ means that the solution for the problem of what are the values of the \+ normal mode frequencies, omega, is that " }}{PARA 0 "" 0 "" {TEXT -1 24 " omega^2 = 3*omega0^2." }}{PARA 0 "" 0 "" {TEXT -1 123 " \+ Next in that bracket is the integer 1, which is telling us tha t in the list of all the eigenvalues of this " }}{PARA 0 "" 0 "" {TEXT -1 109 "matrix this value, 3, occurs just once; i.e., it it the \+ multiplicity of this value in the set of all of them." }}{PARA 0 "" 0 "" {TEXT -1 108 " Lastly, inside a brace, is the vector which is t he eigenvector associated with this eigenvalue. It is " }}{PARA 0 "" 0 "" {TEXT -1 108 "actually undetermined modulo some arbitrary constan t; therefore, that constant has been chosen by the system" }}{PARA 0 " " 0 "" {TEXT -1 75 "so as to make the various components of the vector as simple as possible. " }}{PARA 0 "" 0 "" {TEXT -1 46 " The \" physical view\" of this eigenvector," }}{PARA 0 "" 0 "" {TEXT -1 111 " or normal mode, is that the first component multiplies that arbitrary \+ constant to give us the amplitude of the " }}{PARA 0 "" 0 "" {TEXT -1 105 "motion of the first cart, while the second component does the sam e thing for the second cart. This one, " }}{PARA 0 "" 0 "" {TEXT -1 108 "namely [-1,1], tells us that the two carts are oscillating in opp osite directions, but with otherwise equal " }}{PARA 0 "" 0 "" {TEXT -1 36 "amplitudes, in this particular mode." }}{PARA 0 "" 0 "" {TEXT -1 108 " We also consider, briefly, the second entry above, which te lls us that the second allow frequency, i.e., " }}{PARA 0 "" 0 "" {TEXT -1 98 "the frequency of the second normal mode, is just 1*omega0 , that it occurs once in our list of all " }}{PARA 0 "" 0 "" {TEXT -1 106 "allowed frequencies, and that its associated mode vector, [1,1], \+ means that the two carts move \"together,\"" }}{PARA 0 "" 0 "" {TEXT -1 105 "i.e., with the same amplitude and in the same direction, when \+ they are moving (only) with this frequency." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Now, let us go ahead and look \+ at the situation with 3 carts; here, Maple is of considerable help gen erating" }}{PARA 0 "" 0 "" {TEXT -1 33 "the determinants that are need ed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "K3:=matrix([[2,-1,0] ,[-1,2,-1],[0,-1,2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#K3G-%'mat rixG6#7%7%\"\"#!\"\"\"\"!7%F+F*F+7%F,F+F*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "eigenvectors(K3);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6%7%\"\"#\"\"\"<#-%'vectorG6#7%!\"\"\"\"!F%7%,&F$F%*$F$#F%F$F%F%<#-F(6 #7%F%,$F/F+F%7%,&F$F%F/F+F%<#-F(6#7%F%F/F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 " Here the 3 eigenvalues are all different, i.e., there \+ is no degeneracy, and" }}{PARA 0 "" 0 "" {TEXT -1 121 " a) the fir st normal mode involves only the first and last carts, moving in oppos ite directions but equal magnitudes," }}{PARA 0 "" 0 "" {TEXT -1 116 " b) the second and third modes involve all three carts moving, with different directions between the two for the " }}{PARA 0 "" 0 "" {TEXT -1 35 " middle one." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 " Now, let's go ahead yet further and look at the situation with \+ 4 carts." }}{PARA 0 "" 0 "" {TEXT -1 113 " Do remember that all this is for the situation where all the masses are the same and the spring s are the same." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "K4:=matrix([[2,-1,0,0],[-1,2,-1,0],[0,-1,2,-1],[0,0,- 1,2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#K4G-%'matrixG6#7&7&\"\"# !\"\"\"\"!F,7&F+F*F+F,7&F,F+F*F+7&F,F,F+F*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "eigenvectors(K4);" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6&7%,&#\"\"$\"\"#\"\"\"*&F'!\"\"\"\"&#F(F'F(F(<#-%'vectorG6#7&F(,&F,F( *&F'F*F+F,F*F2F(7%,&F%F(*&F'F*F+F,F*F(<#-F/6#7&F(,&F,F(*&F'F*F+F,F(F;F (7%,&#F+F'F(*&F'F*F+F,F(F(<#-F/6#7&F*F;,&#F(F'F**&F'F*F+F,F*F(7%,&F?F( *&F'F*F+F,F*F(<#-F/6#7&F*F2,&#F(F'F**&F'F*F+F,F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 " The 4 eigenvalues now come in pairs, as can be seen ea sily; none are simple. The eigenvectors all " }}{PARA 0 "" 0 "" {TEXT -1 111 "involve motions of all 4 carts, although it is clear tha t the ones on each end always have the same amplitudes," }}{PARA 0 "" 0 "" {TEXT -1 90 "and the pair in the middle also have equal amplitude s, different from the ones on the end." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Here the problem, from Tues. \+ night, " }}{PARA 0 "" 0 "" {TEXT -1 97 " with the three pendula connec ted together by a flexible steel bar from which they are suspended." } }{PARA 0 "" 0 "" {TEXT -1 53 " As the algebra is a bit messy, I pre sent it here." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "pen:=matrix([[1,-e,-e],[-e,1,-e],[-e,-e,1]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$penG-%'matrixG6#7%7%\"\"\",$%\"eG! \"\"F+7%F+F*F+7%F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "e igenvectors(pen);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%,&%\"eG\"\"\"F& F&\"\"#<$-%'vectorG6#7%!\"\"\"\"!F&-F*6#7%F-F&F.7%,&*&F'F&F%F&F-F&F&F& <#-F*6#7%F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "So here we hav e a degeneracy in the normal mode frequencies, with one frequency occu ring twice;" }}{PARA 0 "" 0 "" {TEXT -1 110 " nonetheless, there are two independent mode vectors (eigenvectors) associated with that repe ated frequency." }}{PARA 0 "" 0 "" {TEXT -1 101 "However, as those two vectors correspond to the same frequency any other pair of linear com binations " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 41 " \+ of them is equally valid." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 " \+ For good measure, let us consider one with 5 carts." }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "MK:=matrix([[2,-1,0,0,0 ],[-1,2,-1,0,0],[0,-1,2,-1,0],[0,0,-1,2,-1],[0,0,0,-1,2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MKG-%'matrixG6#7'7'\"\"#!\"\"\"\"!F,F,7'F +F*F+F,F,7'F,F+F*F+F,7'F,F,F+F*F+7'F,F,F,F+F*" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "MKI:= matrix([[2-eta,-1,0,0,0],[-1,2-eta,-1,0,0],[0,-1,2-eta,-1,0],[0,0,-1,2 -eta,-1],[0,0,0,-1,2-eta]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$MKI G-%'matrixG6#7'7',&\"\"#\"\"\"%$etaG!\"\"F.\"\"!F/F/7'F.F*F.F/F/7'F/F. F*F.F/7'F/F/F.F*F.7'F/F/F/F.F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "factor(det(MKI));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**,&%$e taG\"\"\"F'!\"\"F',&\"\"#F(F&F'F',&F&F'\"\"$F(F',(*$)F&F*F'F'*&\"\"%F' F&F'F(F'F'F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eigenvect ors(MK);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'7%,&\"\"#\"\"\"*$\"\"$#F&F %F&F&<#-%'vectorG6#7'F&,$F'!\"\"F%F/F&7%,&F%F&F'F0F&<#-F,6#7'F&F'F%F'F &7%F%F&<#-F,6#7'F&\"\"!F0F " 0 "" {MPLTEXT 1 0 1 ";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "10 1 0 " 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }