{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 Input" 2 19 "" 0 1 255 0 0 1 0 0 0 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 } {CSTYLE "" -1 256 "" 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 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 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 24 "Moment of Inertia Tensor " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 200 "In t his worksheet we calculate the moment of inertia matrix in a given coo rdinate system and show that a coordinate system can be found in which the inertia tensor is represented by a diagonal matrix." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "This worksheet req uires " }{TEXT 19 6 "maple8" }{TEXT -1 178 " to work properly (integra l I2 below won't evaluate in earlier versions, and even in maple8 it i ntegrates with a wrong sign). For a workaround that involves some extr as: consult " }{TEXT 19 18 "MomentInertia7.mws" }{TEXT -1 20 " in the \+ same folder." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "restart; with(LinearAlgebra):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plots): with(plottools):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }} {PARA 7 "" 1 "" {TEXT -1 43 "Warning, the name arrow has been redefine d\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 237 "We start with an example \+ for which the calculation of the integrals is relatively straightforwa rd. We assume a homogeneous material for which the mass density is con stant, and given as the ratio of total mass divided by the total volum e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "rho:=M/V;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG*&%\"MG\"\"\"%\"VG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 173 "To practice our computational skills we \+ work on an ellipsoid which is described by a quadratic form. This will be a cigar-shaped body, which has an ellipse as cross section." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "A:=Matrix([[1,1/2,1/3],[1/2, 4,1],[1/3,1,4]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6 %\"))=#p>-%'MATRIXG6#7%7%\"\"\"#F.\"\"##F.\"\"$7%F/\"\"%F.7%F1F.F4%'Ma trixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A-Transpose(A);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")Sxz?-%'MATRIXG6#7%7%\" \"!F,F,F+F+%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "The above null-result shows that A was chosen to be real and symmetric." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "EVals:=evalf(Eigenvalues(A)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&EValsG-%'RTABLEG6%\")S!GH\"-%' MATRIXG6#7%7#^$$\"+?`8&3&!\"*$\"\"!F37#^$$\"*%>&\\3*F1$!\"\"!#57#^$$\" +'[pj+$F1F8&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 207 "The numerical noise in the eigenvalues (non-zero imaginary parts) comes from the evalf calculation to 10 digits (verify this by repeati ng the calculation with Digits set higher than the default value of 10 )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 196 "Th e matrix has positive eigenvalues which are related to the semimajor a xes of the ellipsoid. If one (or two) of them were negative our quadra tic surface would be made up of hyperboloidal sheets. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "We anticipate a res ult to be derived below: in a principal-axis frame, the quadratic form will be given as:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 19 47 "QF = EVals[1]*x^2 + EVals[2]*y^2 + EVals[3]*z^2" }}{PARA 0 "" 0 "" {TEXT -1 136 "We can set QF =1 (to chose a scale), and recog nize that the eigenvalues are equal to EV[i] = 1/a_i^2, where a_i are \+ the semi-major axes." }}{PARA 0 "" 0 "" {TEXT -1 55 "Therefore the vol ume of the ellipsoid is calculated as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "V0:=evalf(4/3*Pi*mul(sqrt(1/EVals[i]),i=1..3));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V0G^$$\"+d.(R7\"!\"*$\"+nb?b!)!#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "seq(Re(sqrt(1/EVals[i])), i=1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+(pTXV%!#5$\"+_E:\\5!\" *$\"+%R$QndF%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Now we calculate the quadratic form." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Rv: =Vector([x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#RvG-%'RTABLEG6 %\")#\\=L\"-%'MATRIXG6#7%7#%\"xG7#%\"yG7#%\"zG&%'VectorG6#%'columnG" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "QF:=Transpose(Rv) . A . Rv ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#QFG,(*&%\"xG\"\"\",(F'F(*&\"\" #!\"\"%\"yGF(F(*&\"\"$F,%\"zGF(F(F(F(*&F-F(,(*&F+F,F'F(F(*&\"\"%F(F-F( F(F0F(F(F(*&F0F(,(*&F/F,F'F(F(F-F(*&F5F(F0F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "QF:=simplify(QF);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#QFG,.*$)%\"xG\"\"#\"\"\"F**&F(F*%\"yGF*F**&#F)\"\"$F **&F(F*%\"zGF*F*F**&\"\"%F*)F,F)F*F**(F)F*F,F*F1F*F**&F3F*)F1F)F*F*" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "surf:=solve(QF-1,z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%surfG6$,(*&\"\"%!\"\"%\"yG\"\"\"F)* &\"#7F)%\"xGF+F)*&F-F),**&\"$N\"F+)F*\"\"#F+F)*(\"#IF+F.F+F*F+F)*&\"#N F+)F.F4F+F)\"#OF+#F+F4F+,(*&F(F)F*F+F)*&F-F)F.F+F)*&F-F)F0F;F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "plot3d(surf[1],x=-1.1..1.1, y=-1.1..1.1,axes=boxed,grid=[30,30],style=patchcontour,shading=zhue,sc aling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6(-%%GRIDG6%;$!#6!\"\"$\"#6F)F&7@7@%*undefinedGF.F.F.F.F. 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E2v;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3kpYrJu<(4#!#F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "E1v . E1v;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"2s`b R*********!#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "E1v . E3v; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!37n$4HbziE#!#E" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "E2v . E3v;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!3W-1Bk77\\A!#E" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Rotate \+ the above graph to obtain a clear understanding of the geometry." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 67 "Change the matrix in the above qua dratic form and repeat the steps." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "We proceed with constructing an o rthogonal matrix out of the normalized eigenvectors. We then transform the matrix to demonstrate that it is diagonalized by this transformat ion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "R:=Matrix([E1v,E2v, E3v]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG-%'RTABLEG6%\"()=SH-%' MATRIXG6#7%7%$\"3E>')Q)3+*H9!#=$!39gMYH,_#))*F0$\"3[T6W#QDhR&!#>7%$\"3 '******\\o>`0(F0$\"3))*****fzi,S\"F0$\"3;+++'H'3ZpF07%$\"3NV<6ol-TpF0$ \"3(y9aT_+l7'F5$!3y1=F*=VE<(F0%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Ri:=MatrixInverse(R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#RiG-%'RTABLEG6%\")wF-7-%'MATRIXG6#7%7%$\"3')p$[D4+*H9!#=$\"3' H'\\r/(>`0(F0$\"3\"H_BfbE5%pF07%$!3Cg@[B,_#))*F0$\"3kVsU4G;+9F0$\"3s;Y 8X.]Eh!#>7%$\"3_yvq\">DhR&F<$\"3#GR>SJ'3ZpF0$!3sr\"35 " 0 "" {MPLTEXT 1 0 11 "Ri . A . R;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")!e>8\"-%'MATRIXG6#7%7%$\"3M!3v%>` 8&3&!#<$\"3_Ei+J'RgC%!#G$\"3)G7%zd;>L_F17%$!3gV(\\0f^XL)!#F$\"31Mb>X>& \\3*!#=$\"3Y'*ph65JrY!#E7%$\"3Y%G()>7'RjZF=$!3x:#>D$fE1ZF1$\"3el`+'[pj +$F.%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "The transformed matrix has the eigenvalues on the diagonal, and zeroes (approximately ) for all non-diagonal entries." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "R . A . Ri;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\") K1d6-%'MATRIXG6#7%7%$\"3mf6*3sSep$!#<$!3qy?U`CXu7F.$\"3(yrLk#[#)4a!#>7 %$!3mD`&4X_WF\"F.$\"3;?$34Ngn7$F.$!3u$)HQ:%[ZY\"F.7%$\"3^%R8rtC)4aF3$! 3OV7*3U[ZY\"F.$\"3/@0?G*)Rx@F.%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 226 "Calculate the quadratic form using the diagonaliz ed matrix. Truncate the near-zero entries to zero, i.e., make up a dia gonal matrix from the eigenvalues and observe the equation for the qua dratic form (absence of cross terms)." }}{PARA 0 "" 0 "" {TEXT -1 129 "Carry out the transformation to the principal axes frame (as done abo ve) for some other matrix A with three positive eigenvalues." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 19 "Volum e calculation." }}{PARA 0 "" 0 "" {TEXT -1 28 "Can we calculate the vo lume " }{TEXT 258 1 "V" }{TEXT -1 76 " in Cartesian coordinates? We ch oose the quadratic form with scale factor 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Let us do the repeated integral s one at a time. 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3_n,q#ecu@\"F-F_\\z-Fb\\z6&Fd\\zFh\\zFh\\zFe\\z-%(SCALINGG6#%,CONSTRAI NEDG-%+AXESLABELSG6$Q\"x6\"Q!F^^cl-%%VIEWG6$;$!#6!\"\"$\"#6Ff^cl%(DEFA ULTG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Following the z-in tegration we need to integrate " }{TEXT 19 2 "I1" }{TEXT -1 36 " over \+ an area bounded by the curves " }{TEXT 19 5 "fy[1]" }{TEXT -1 5 " and \+ " }{TEXT 19 5 "fy[2]" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 157 " We will integrate over y between the lower (blue) and upper (red) boun ding curves. Then we will integrate over x between fixed limits. What \+ are these limits?" }}{PARA 0 "" 0 "" {TEXT -1 63 "They are the points \+ where the derivative y(x) goes to infinity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x0:=solve(1/diff(fy[1],x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G6$,$*(\"\"$\"\"\"\"\"&!\"\"F(#F)\"\"#F+,$*(F(F)F* F+F(F,F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "The direct attempt of definite integration is not successful:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "I2:=int(I1,y=fy[1]..fy[2]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#I2G,***^##\"\"&\"#\")\"\"\"\"#:#F+\"\"#)%\"xGF.F+-%# lnG6#*(,&*&\"$D\"F+F/F+!\"\"\"$N\"F+F-,$*&\"\"$F+F5F-F8#F8F.^#F+F+F+F+ **^##!\"&F*F+F,F-F/F+-F26#*(^#F8F+F5F-F:F=F+F+*(^##F8F,F+F,F-F1F+F+*(^ ##F+F,F+F,F-FCF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "I3idf :=int(expand(I2),x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&I3idfG-%$in tG6$,***^##\"\"&\"#\")\"\"\"\"#:#F.\"\"#)%\"xGF1F.-%#lnG6#*(,&*&\"$D\" F.F2F.!\"\"\"$N\"F.F0,$*&\"\"$F.F8F0F;#F;F1^#F.F.F.F.**^##!\"&F-F.F/F0 F2F.-F56#*(^#F;F.F8F0F=F@F.F.*(^##F;F/F.F/F0F4F.F.*(^##F.F/F.F/F0FFF.F .F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "We should be able to do th is integral at least numerically. We need to know the range of x:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(x0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+&[I#R5!\"*$\"+&[I#R5F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(I2,x=x0[1]..x0[2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!+$[I#R5!\"*$!+3 +zBB!#=7$$!+*[f#R**!#5$!+!o(>=p!#67$$!+gr1X&*F1$!+#f%oo7F17$$!+fmv,\"* F1$!+bTa*)=F17$$!+3w]b')F1$!+8Vq%[#F17$$!+-&z8@)F1$!+]`LZIF17$$!+%*eh* z(F1$!+Zn\\UNF17$$!+H&eKP(F1$!+E!p$GSF17$$!+K\">B$pF1$!+0g8-XF17$$!+LQ z#\\'F1$!+8#=`%\\F17$$!+Q!)oSgF1$!+y!44P&F17$$!+4IZUcF1$!+>xL?dF17$$!+ -s<%>&F1$!+]%4_3'F17$$!+]1/WZF1$!+6[?@kF17$$!+u1D5VF1$!+\"o*>;nF17$$!+ jO7uF17$$!+h \"G&*e#F1$!+cb\"zg(F17$$!+f6&4=#F1$!+ZsIaxF17$$!+KunKF17$$\"*Qe/_*F*$!+[K#RI\"F 17$$\"*&*)pS**F*$!+_Zq'*oF47$$\"+$[I#R5F*F+-%'COLOURG6&%$RGBG$\"#5!\" \"$\"\"!F`[lF_[l-%+AXESLABELSG6$Q\"x6\"Q!Fe[l-%%VIEWG6$;$!+&[I#R5F*$\" +&[I#R5F*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 190 "Someh ow maple8 made a mistake: it chose a branch cut for the ln function wh ich turned our results upside down. We need to integrate this function , and do it numerically and correct the sign:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "V:=evalf(Int(-I2,x=x0[1]..x0[2]));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"VG^$$\"+d.(R7\"!\"*$\"\"!F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "V0:=evalf(4/3*Pi*mul(sqrt(1/Eigenva lues(A)[i]),i=1..3),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V0G^$$ \"0^mpNqR7\"!#9$\"0381%)>`[$!#H" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "The volume integration worked! We carried out two of the three in tegrations analytically, and then the third one numerically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Now we calculat e the " }{TEXT 264 25 "moments of inertia matrix" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "M:=1; rho;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"M G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"+#zJq*))!#5$!\"!\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "MI:=Matrix(3,3):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "for i from 1 to 3 do: for j from 1 \+ to i do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "if i=1 and j=1 then w: =Rv[2]^2+Rv[3]^2; elif i=2 and j=2 then w:=Rv[1]^2+Rv[3]^2; elif i=3 a nd j=3 then w:=Rv[1]^2+Rv[2]^2; else w:=-Rv[i]*Rv[j]; fi:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 30 "I1:=int(w*rho,z=fz[1]..fz[2]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "I2idf:=unapply(int(I1,y),y);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "I2:=expand(I2idf(fy[2])-I2idf(fy[1]));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "I2:=simplify(I2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "MI[i,j]:=evalf(Int(I2,x=x0[1]..x0[2]));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "od: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "print(MI);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")k T=7-%'MATRIXG6#7%7%$\"+*******4\"!#5\"\"!F/7%$\"+++++C!#6$\"+,+++FF.F/ 7%$\"+++++7F3F7$\"++++?FF.%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Now fill the complete matrix by symmetry:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "for i from 1 to 3 do: for j from i to 3 do: M I[i,j]:=MI[j,i]: od: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "print(MI);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")kT=7-%'M ATRIXG6#7%7%$\"+*******4\"!#5$\"+++++C!#6$\"+++++7F17%F/$\"+,+++FF.F27 %F2F2$\"++++?FF.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Eigenvalues(MI);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\" )?;&)>-%'MATRIXG6#7%7#^$$\"3GF8[Cube5!#=$\"\"!F17#^$$\"3;M`@*z'pmGF/F0 7#^$$\"3CVLIwdu%f#F/F0&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 43 "These are the principal moments of inertia." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "EVecsMI:=Eigenvectors(MI);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%(EVecsMIG6$-%'RTABLEG6%\")std?-%'MAT RIXG6#7%7#^$$\"3GF8[Cube5!#=$\"\"!F47#^$$\"3;M`@*z'pmGF2F37#^$$\"3CVLI wdu%f#F2F3&%'VectorG6#%'columnG-F'6%\")'>.4#-F+6#7%7%^$$!3-.\\F37%^$$\"3$*Qs5%zi,S\"F2F3 ^$$\"3w@9G:(>`0(F2F3^$$!3'4:-gG'3ZpF2F37%^$$\"3TMN3=0]EhFQF3^$$\"3)*z* 35aE5%pF2F3^$$\"3GTvv*>VE<(F2F3%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "We are interested in a graph of the three eigenvectors." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "E1vMI:=convert(SubMatrix( EVecsMI[2],1..3,1..1),Vector);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&E 1vMIG-%'RTABLEG6%\")?iv?-%'MATRIXG6#7%7#^$$!3-.\\F2&%'VectorG6#%'columnG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "E1vMI:=Normalize(map(Re,E1vM I),Euclidean);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&E1vMIG-%'RTABLEG6 %\")?,B@-%'MATRIXG6#7%7#$!3-.\\&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "E2vMI:=Normalize(map(Re,convert(SubMatrix(EVecsMI[2],1..3,2..2 ),Vector)),Euclidean);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&E2vMIG-%' RTABLEG6%\")+nM@-%'MATRIXG6#7%7#$\"3#HJu;4+*H9!#=7#$\"3w@9G:(>`0(F07#$ \"3)*z*35aE5%pF0&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "E3vMI:=Normalize(map(Re,convert(SubMatrix(EVecsMI[2], 1..3,3..3),Vector)),Euclidean);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%& E3vMIG-%'RTABLEG6%\")oIP@-%'MATRIXG6#7%7#$!3#)*)*Rl7DhR&!#>7#$!3'4:-gG '3Zp!#=7#$\"3GTvv*>VE<(F4&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 33 "RMI:=Matrix([E1vMI,E2vMI,E3vMI]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$RMIG-%'RTABLEG6%\");5p@-%'MATRIXG6#7%7%$!3-. \\7%$\"3$*Qs5%zi,S\"F0$\" 3w@9G:(>`0(F0$!3'4:-gG'3ZpF07%$\"3TMN3=0]EhF5$\"3)*z*35aE5%pF0$\"3GTvv *>VE<(F0%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "R;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"()=SH-%'MATRIXG6#7%7%$\" 3E>')Q)3+*H9!#=$!39gMYH,_#))*F.$\"3[T6W#QDhR&!#>7%$\"3'******\\o>`0(F. $\"3))*****fzi,S\"F.$\"3;+++'H'3ZpF.7%$\"3NV<6ol-TpF.$\"3(y9aT_+l7'F3$ !3y1=F*=VE<(F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 " E1MI := plottools[arrow]([0, 0, 0], E1vMI, .2, .4, .1, color=red):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "E2MI := plottools[arrow]([0 , 0, 0], E2vMI, .2, .4, .1, color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "E3MI := plottools[arrow]([0, 0, 0], E3vMI, .2, .4, .1 , color=green):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "The eigenvect ors of the MI matrix provide those axes about which rotation of the bo dy without deviation moments is possible." }}{PARA 0 "" 0 "" {TEXT -1 254 "Note that as in the 'spinning textbook' problem the motion about \+ the axes with extreme eigenvalues will be truly stable, while free rot ation about the axis associated with the intermediate eigenvalue resul ts in a wobble (instability in Euler's equation)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 411 "The remarkable thing abo ut rotational mechanics is that for any shape (e.g., a potato) three a xes can be found about which pure rotation is possible without deviati on moments. The ellipsoid is as close a mathematical form that we can \+ handle that describes such a potato. The diagram below illustrates tha t the stable rotation axes do coincide with the axes that are perpendi cular to the elliptic cross sections." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "display(E1MI,E2MI,E3MI, QFp,axes=boxed);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6)-%)POLYGONSG6&7&7%$!+(3@f1'!#7$\"+h?C%f)!#8$!+_`@\")**!#67%$\"+(3 @f1'F*$!+h?C%f)F-$\"+_`@\")**F07%$!+.*3O$))!#5$\"+tAb^7F;$\"+#e1&\\:F; 7%$!+Dt#\\&*)F;$\"+d2uo7F;$!+')[OnWF07%7%$!+#p\\Hx)F;$\"+J!eHC\"F;$\"+ <\"Gwa#F;7%$!+F,_#))*F;$\"+%zi,S\"F;$\"+=0]EhF07%$!+Ole:!*F;$\"+**\\Lx 7F;$!+B!e[W\"F;-%&STYLEG6#%,PATCHNOGRIDG-%'COLOURG6&%$RGBG$\"*++++\"! \")$\"\"!FcoFbo-F$6&7&7%$\"+cgqy8F0$\"+X9s-oF0$!+\\.w)>(F07%$!+cgqy8F0 $!+X9s-oF0$\"+\\.w)>(F07%$\"+x%R!\\6F;$\"++c^pcF;$\"+A**zmpF;7%$\"+*o! yC9F;$\"+)))f+.(F;$\"+_y/FbF;7%7%$\"+s)o6,\"F;$\"+bMC*)\\F;$\"+dfn'o(F ;7%$\"+#4+*H9F;$\"+:(>`0(F;$\"+Tl-TpF;7%$\"+%H^Ec\"F;$\"+L?L5xF;$\"+<= 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"Curve 4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 221 "The eigenve ctors of the moment of inertia matrix look the same as the eigenvector s that diagonalize the quadratic form. Here is an example of two matri ces that share a set of eigenvectors, but have different eigenvalues. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 385 "We o bserve that vectors can be found which are at the same time eigenvecto rs of matrix A (which defines the geometric shape), and of the moment \+ of inertia matrix MI. These two matrices are distinct and have distinc t eigenvalues. According to a theorem in linear algebra a common set o f eigenvectors can be found for two matrices if they commute with each other, i.e., wenn the product " }{TEXT 19 3 "A.B" }{TEXT -1 16 " is t he same as " }{TEXT 19 3 "B.A" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Let us explore the proper ties:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 " First, the orthogonality of the eigenvectors of the MI matrix:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "E1vMI . E2vMI;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$!3/PVMUjLjT!#M" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "E1vMI .E1vMI;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3 A+++++++5!#<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Of course, the e igenvectors of the MI matrix are perpendicular. Try the others by edit ing the above lines." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 121 "The following three lines indicate which eigenvectors \+ of the MI matrix correspond to the principal axes of the ellipsoid:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E1v . E1vMI;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"3[2R$)f!>ln'!#G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E2v . E1vMI;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3Y )Qc-++++\"!#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E3v . E1vM I;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!3kqkG68sCB!#E" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "The first eigenvector of the MI matrix co rresponds to the second eigenvector of the quadratic form." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E1v . E2vMI;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"2'ox(p*********!#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E2v . E2vMI;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3m WwR4^_H9!#F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E3v . E2vMI; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3g1E\"4Z6\"*z\"!#E" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "The second eigenvector of the MI matrix c orresponds to the first eigenvector of the quadratic form." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E1v . E3vMI;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3Sh/ro/RlS!#E" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E2v . E3vMI;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!3u1Beg%)**fv !#G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E3v . 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MI - MI . A;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'RTABLEG6%\")g0B@-%'MATRIXG6#7%7%$\"\"!F-$\"2%[h%=0 )******!#F$\"3y]L\"p*\\LLL!#G7%$!2%[h%=0)******F0F,$!3+r.u#3+++\"F07%$ !3y]L\"p*\\LLLF3$\"3+r.u#3+++\"F0F,%'MatrixG" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 20 "Indeed, the product " }{TEXT 19 6 "A . MI" }{TEXT -1 54 " is within numerical accuracy the same as the product " }{TEXT 19 6 "MI . A" }{TEXT -1 2 " !" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 11 "Exercise 3:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 144 "Explore the relationship between the pri ncipal axes of the quadratic form, and the principal axes of the momen t of inertia matrix. Try matrices " }{TEXT 19 1 "A" }{TEXT -1 85 " for which the form is close to a nearly circular cross section, i.e., fin d matrices " }{TEXT 19 1 "A" }{TEXT -1 50 " such that two of the eigen values are very close. 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