{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 } {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 "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 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 -1 1 " " }{XPPEDIT 19 1 "Phase Pl ane Plots" "6#*(%&PhaseG\"\"\"%&PlaneGF%%&PlotsGF%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 " Below plotting phase plane plots for harmonic \+ oscillators: I scale them all so that the amplitude is used to scale \+ the distances," }}{PARA 0 "" 0 "" {TEXT -1 78 " but have p ut in reasonably arbitrary values for omega0 and delta." }}{PARA 0 "" 0 "" {TEXT -1 125 " However, do note that by choosing omega0 less th an 1 I cause the plots to be more horizontal than vertical, which is g ood " }}{PARA 0 "" 0 "" {TEXT -1 40 " for presentations on \+ \"paper.\"" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 " We first look at some undamped plots" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "xun:=cos(omega0*t-delta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xunG-%$cosG6#,&*&%'omega0G\"\"\"%\"tGF+!\"\"%&deltaG F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "vun:=diff(xun,t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$vunG*&-%$sinG6#,&*&%'omega0G\"\"\"% \"tGF,!\"\"%&deltaGF,F,F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "omega0:=1/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'omega0G#\"\"\" \"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "delta:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deltaG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 44 "plot([xun,vun,t=0..20],scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 682 682 682 {PLOTDATA 2 "6&-%'CURVESG6$7]q7 $$\"3l(R\"oeI-.a!#=$\"3_#[RS#\\N2UF*7$$\"3G'p@:uNiG'F*$\"3?h\\!*y&[&)) QF*7$$\"3G^w9M]&[4(F*$\"34([_\"\\,gBNF*7$$\"3VZQJC'\\-t(F*$\"3%yN+?-y= <$F*7$$\"3%*)f1aF$='H)F*$\"3I()4RkYl\"z#F*7$$\"3v<(z55]L%))F*$\"3n*GtR GkUL#F*7$$\"3-LS>Bu.!H*F*$\"3Q1g&Qj^.&=F*7$$\"3=XV)ompIj*F*$\"31r)3g]5 ?M\"F*7$$\"3G(*\\oP]>l)*F*$\"3!y#*RFC'=#=)!#>7$$\"3;-_[7*[M)**F*$\"3#R oiZ\"R`vGFU7$$\"3I!\\DL!4&y)**F*$!3i%RvT=6RY#FU7$$\"3?v@-Q%e,*)*F*$!3% 3wF.5x/R(FU7$$\"3]yy?bU]&p*F*$!3a\\>?#yeWA\"F*7$$\"3[8V1Vt\"QR*F*$!3`_ kCgaO9h;% F*7$$\"3qb%4z@gAf%F*$!3%\\$p2%H(fTWF*7$$\"3&H@'*3uM:s$F*$!3I:t(4/c3k%F *7$$\"3033+qNo;GF*$!3A)3z-6fvz%F*7$$\"3?5esMkbnDQF@\\F*7$$ \"3enQ_WD#*ypFU$!3/9v[(z3y)\\F*7$$!3'GAU\"=d0VQFU$!3q\"*QRmjI'*\\F*7$$ !3'p/j^[,?Y\"F*$!3_OLab^FY\\F*7$$!3oPa[y%*\\%[#F*$!3))p%*)pnBK%[F*7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 50 " So now lets make the damping noticeably larger:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "betalist:=[beta \+ = omega0/5];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)betalistG7#/%%betaG #\"\"\"\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "plot(subs(be talist,[xunder,vunder,t=0..30]),scaling=constrained,numpoints=500);" } }{PARA 13 "" 1 "" {GLPLOT2D 682 682 682 {PLOTDATA 2 "6&-%'CURVESG6$7`j l7$$\"3l(R\"oeI-.a!#=$\"3Ck]S'yY?e$F*7$$\"3q%G`oZIUi&F*$\"3c[Hf604^MF* 7$$\"3SB%>AQe*4eF*$\"3,2,d#38dL$F*7$$\"3*)*y20'4C6gF*$\"3+r\"H1&o\\/KF *7$$\"3Dhq*\\:cc?'F*$\"3W)\\kDK')42$F*7$$\"3-Hv]IS*3R'F*$\"3[rttiY$o$H F*7$$\"3_PX;RH@blF*$\"3ys2vfiW6GF*7$$\"34.SB%euxr'F*$\"3IuJ.!e52o#F*7$ $\"3/44s'\\Hx(oF*$\"3$**Ggc-oYa#F*7$$\"3njL)*=/')GqF*$\"3Q=(Qg*4N3CF*7 $$\"334q%Rd=c<(F*$\"3ai'RW&Q`nAF*7$$\"3op6\"4()\\vH(F*$\"3hdf`GS6V@F*7 $$\"3%*y4+0ZdEuF*$\"3aiXFU+u-?F*7$$\"3#*)p_UaJta(F*$\"3]jTSlxfh=F*7$$ \"3'eu^j]a`l(F*$\"3?V^Pm\"Hbs\"F*7$$\"3?-)e6#4NYxF*$\"3<#4)f@m--;F*7$$ \"3ck+*\\rod%yF*$\"3uz'*zvbPb9F*7$$\"32\\['RLMD#zF*$\"3^8HqaNUJ8F*7$$ \"3#p$>3U@D.!)F*$\"3;$p4T!*)o(=\"F*7$$\"3aY6&)4,.n!)F*$\"3qv--d8(41\"F *7$$\"3I\\6nZWvG\")F*$\"3gp!pZDRhA*!#>7$$\"3[;:`[*[&z\")F*$\"3WU1x(R(G ;zF[r7$$\"3yA#)RD#*HC#)F*$\"3#H\"o\"zw(pelF[r7$$\"3w:'R*oe-e#)F*$\"3)G ;E*p%)=@`F[r7$$\"3%)*yQ:melG)F*$\"3'yv)[\"pys*RF[r7$$\"3Q;$ffe]wI)F*$ \"3UOezWdCNEF[r7$$\"3*o(*)\\K**)*=$)F*$\"3[skDhUTh9F[r7$$\"3gp$o;isRK) F*$\"3')H9!=<%Rp?!#?7$$\"3rNy.-lG@$)F*$!3WMy*p1rN2\"F[r7$$\"3>m$em**\\ 5J)F*$!3:d8rc855BF[r7$$\"3Q:Dsss2%H)F*$!35woR$yu.\\$F[r7$$\"3Y@=USW?n# )F*$!3%R+-9es7y%F[r7$$\"3!*o'4n1ifB)F*$!36mMVJ%\\F#fF[r7$$\"3#\\4lY&=J &>)F*$!3W/!*f&*eM@rF[r7$$\"3?&p$)R0\"3_\")F*$!3u[ZfxRs)=)F[r7$$\"3Uf'y qW&)z4)F*$!32nkX(*z[M$*F[r7$$\"3kJlXs!*oS!)F*$!3\"\\)f\"H3n\"R5F*7$$\" 3-?_.L2IuzF*$!3Ih#)fHcVZ6F*7$$\"3-(>.W+2J!zF*$!3GoHK*=45D\"F*7$$\"3.lZ $QHi?#yF*$!35J'fx$o&pN\"F*7$$\"3(4+jV0dyt(F*$!3c6(\\\"f+`c9F*7$$\"3?n \"HfKfck(F*$!3yh`5w9zb:F*7$$\"3/UuLKXI[vF*$!3kMROX$3;l\"F*7$$\"3;biRo` y`uF*$!3wH*eJ7.tt\"F*7$$\"3[a)zFoj'RtF*$!3rm'38>&pK=F*7$$\"3@xc7PG]KsF *$!3:)>gh29a\">F*7$$\"3cZ)o8U>J6(F*$!3ZZD2o%e3+#F*7$$\"3)ol7bWbS*pF*$! 3#\\YZP3S*z?F*7$$\"3ugf)*)3Ul&oF*$!37NsEc]CF*7$$\"3#yRT/&F*$!3_L;@*[+)HGF*7$ $\"3`)))QRG&3A]F*$!3'eG.Qya<(GF*7$$\"3'>;;A0=8%[F*$!3WuLYfpl7HF*7$$\"3 %p!)3\"=+cvYF*$!3ouZ!H*z!o%HF*7$$\"3w6L=)*[@!\\%F*$!3[93;DsP\")HF*7$$ \"3'4?8lta2J%F*$!3yN(e'*zq8,$F*7$$\"3+KB2KB.ITF*$!3w^:d'eh#QIF*7$$\"3e \\]r'z]'RRF*$!3!=$orj68jIF*7$$\"3mQ)4;>:Iw$F*$!3;.x]AB9$3$F*7$$\"3ix)) =.x+\"e$F*$!3\"3*[yHfx+JF*7$$\"3aUc(GT\\*yLF*$!3#eC@BZ0p6$F*7$$\"3ur5z e%*=&>$F*$!3!>fMVn3&GJF*7$$\"3D*yA$*yTm+$F*$!376M#Gm]u8$F*7$$\"3w)=FUP LW\"GF*$!3%Qj'4h%RN9$F*7$$\"3aV@33X]PEF*$!3'*\\Wg!)**[YJF*7$$\"3)GXy'G D!)[CF*$!3?\"3gT1uo9$F*7$$\"3ekA&pT(fhAF*$!3-.8(H%4ZWJF*7$$\"3wb$f8,*f h?F*$!3rqi7H(p)QJF*7$$\"3GI^H&p![&)=F*$!3mLRQ@-OJJF*7$$\"35\"=f)zA*Go \"F*$!3')H9k6/v>JF*7$$\"3(=nV@*p$e F[r$!3J$fQB%[5,IF*7$$\"3KxR&>L')p2%F[r$!3WB(zqy.K(HF*7$$\"3[4t!*)*yZ)= #F[r$!3%HO?viK/%HF*7$$\"3MOi#)Q?()=dF_t$!3wY%G@j0+\"HF*7$$!3awg)eTvLE \"F[r$!3C9nW]'[F(GF*7$$!3CqcXNVQmHF[r$!34Bm@P6`NGF*7$$!3f=*HEY_/j%F[r$ !3C(p%49fj'z#F*7$$!3MGABW([AF'F[r$!3?B1kcytbFF*7$$!3&)GZcA-F@zF[r$!3\\ ;jafK07FF*7$$!3XtH7y^# F*7$$!3i^&GjC')*z:F*$!3+e%*=UZ'QY#F*7$$!3r%eF5D2^r\"F*$!3G'R&4\\#>ST#F *7$$!3?1OS5Ear=F*$!3-\"yo')R0LN#F*7$$!3)[qGX_R$**>F*$!3.:*)[$Rt6I#F*7$ $!3;#)*H@?N^9#F*$!3<5.(piO(QAF*7$$!3I\\8=a#*\\!G#F*$!3w&=)*)H'\\x<#F*7 $$!3q'o&*Hzs\")R#F*$!3_d(yYG.A7#F*7$$!3KJ#okJUo_#F*$!3=+1u>Dde?F*7$$!3 m-T/sEP_EF*$!39[s23&\\L*>F*7$$!3#fQq\\-gKx#F*$!3$GVNcNYt#>F*7$$!3wr0`! 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(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Cu rve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 " That one never got suf ficiently damped out, so let's do it for a longer time." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "plot(subs(betalis t,b=1,[xcrit,vcrit,t=0..40]),x=0..2,labels=[x,v],scaling=constrained,t itle=`critically damped,\\n beta=omega0/2, b positive`);" }}{PARA 13 " " 1 "" {GLPLOT2D 682 682 682 {PLOTDATA 2 "6'-%'CURVESG6$7^o7$$\"\"\"\" \"!$\"3++++++++v!#=7$$\"37rDYyx$F-7$$\"3;;.&pXiDq\"F1$\"3$yrpD6o)=FF-7$$\"3O(z+O A&))\\7$$\"3g)=CjV8!*)=F1$\"3u/'*zyH>\\5Fip7$$\"3 7aK!f$R#H)=F1$!3=qzbl+T7PFip7$$\"3qw-N%3b'e=F1$!3A)H+ZIv%*\\(Fip7$$\"3 *fE*)=wq*>=F1$!3Y3p/_E4[5F-7$$\"3)>G@v@**Rx\"F1$!3e(\\H@U6ME\"F-7$$\"3 utfxk?^?Ao)QLY3&F-$!3)p['GKw&=<*Fip7$$\"3KTvWBtFC WF-$!3zF-$!3!HhrCjq,\"RFip7$$\"3)G3^uOyao \"F-$!3#Qq%GG/XwLFip7$$\"3u$4Z*yLj?9F-$!3s*RLcuaW(GFip7$$\"3Wvgd.ZW(= \"F-$!3jUyI@4YDCFip7$$\"3kFVIzp\\95F-$!39kd+Z?%z3#Fip7$$\"3%4KV-RUya)F 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