{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 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 "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 272 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Ti mes" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 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 }{PSTYLE "Normal" -1 256 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 {PARA 3 "" 0 "" {TEXT -1 57 "The Duffing Equation - introducti on and free oscillations" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Ston e, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 2.4 .2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "load " }{TEXT 0 7 "desolve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " } {TEXT 272 7 "DEsol.m" }{TEXT -1 32 " is required by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives \+ its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\ \\Maple/procdrs/DEsol.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "load root-finding procedures etc. includi ng: " }{TEXT 0 7 "findmax" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m- file " }{TEXT 272 7 "roots.m" }{TEXT -1 38 " contains the code for the procedures " }{TEXT 0 7 "findmax" }{TEXT -1 25 " used in this workshe et. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple sess ion by a command similar to the one that follows, where the file path \+ gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \+ \"K:\\\\Maple/procdrs/roots.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "The double-well oscillator and th e Duffing equation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 113 "Acknowledgement: Adapted from work by A. Peter Young, Physics Department, University of California, Santa Cruz .\n " }{URLLINK 17 "http://bartok.ucsc.edu/peter/115/duffing/duffing.h tml" 4 "http://bartok.ucsc.edu/peter/115/duffing/duffing.html" "" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 261 16 "Duffing equation" }{TEXT -1 44 " is a differential equation used to model a " }{TEXT 261 22 "double well os cillator" }{TEXT -1 222 " such as the following magneto-elastic mechan ical system. This system consists of a steel ball attached to a flexib le rod positioned vertically between two magnets, with the top end fix ed, and the bottom end free to swing." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 288 273 273 {PLOTDATA 2 "62-%%TEXTG6$7$$\"$d\"!\"#$\"$X \"F)%\"mG-F$6$7$$\"#D!\"\"$\"$0&F)%3light~flexible~rodG-F$6$7$$\"$1$F) $\"$c\"F)%+steel~ballG-F$6$7$$\"#OF2$\"#eF)%'magnetG-F$6$7$$!#OF2FCFE- F$6$7$\"\"!$\"#)*F2%8external~periodic~forceG-%)POLYGONSG6%7&7$$\"\"&F NFN7$FW$!\"&F27$$FenFNFZ7$FgnFN7&7$$\"\"%FN$\"#&)F27$F[o$\"\")FN7$$!\" %FNF`o7$FcoF]o-%&COLORG6&%$RGBG$FaoF2FjoFjo-FS6%7&7$$\"#:F2$FXF27$$\"# BF2Fap7$FcpFN7$F_pFN7&7$$!#:F2Fap7$$!#BF2Fap7$F\\qFN7$FipFN-Fgo6&Fio$ \"\"'F2FbqFjo-%'CURVESG6%7S7$FN$\"1(*o%Q*******zFjp7$$\"1`XeH0#=j'!#>$ \"1Ce9'G]*eyFjp7$$\"1e&[=Kt#>B!#=$\"1$pQuRLit(Fjp7$$\"1!oi,q\\6Q&Fdr$ \"1DU!Q#pC)f(Fjp7$$\"1?LHhE\\X(*Fdr$\"1[\\sl,QfuFjp7$$\"1&z*[mrfO:!#<$ \"1Z:n[%>7K(Fjp7$$\"1vHc/RWr@Fds$\"1,iy_!yJ>(Fjp7$$\"1otOMG3WHFds$\"1R R!Q#3mgqFjp7$$\"13YsY!pk'QFds$\"16e@?))oBpFjp7$$\"1O_'y!Rh5\\Fds$\"1m. %fzXsy'Fjp7$$\"1(yl9.nW6'Fds$\"11JE%z-qk'Fjp7$$\"1%=?x\\(y$G(Fds$\"1!3 Z='\\dBlFjp7$$\"1!QG?v\"=A()Fds$\"1@5)zcXZQ'Fjp7$$\"1@Wk:$R'H5!#;$\"1! fq]&o[XiFjp7$$\"1f(R08[O>\"F]v$\"1=V4c+V6hFjp7$$\"1*R%oGb.`8F]v$\"1B7% R*f#)*)fFjp7$$\"1>;[W_\\b:F]v$\"1DB]rDSXeFjp7$$\"1abW!3%)zt\"F]v$\"1S= `'y:Ks&Fjp7$$\"1\"z9S&[\"H'>F]v$\"1v`X#Ry7e&Fjp7$$\"1VU@ZWNt@F]v$\"1\"*)*QqkT#F]v$\"1Qx*)oER=`Fjp7$$\"1>\"p9I])fEF]v$\"1 \"*3^61u(=&Fjp7$$\"18_MF77EHF]v$\"1jYT\"Ra;0&Fjp7$$\"1\\lEwBr\"=$F]v$ \"1/aF]v$\"1f$GjaW*4SFjp7$$\"1(=w\\[=%)z&F]v$ \"1R@7j5upQFjp7$$\"1=.yb)QU;'F]v$\"1o6;KP4WPFjp7$$\"1;>LiV\"pc'F]v$\"1 Cc\"3?'H5OFjp7$$\"1;oI@waUpF]v$\"1'z7\"*Q%Q*[$Fjp7$$\"1'3e,rv[O(F]v$\" 15^atQcdLFjp7$$\"1&*>L_`MtxF]v$\"1\\T8J=*QB$Fjp7$$\"1$>3z^v<@)F]v$\"1x (>$oJ+0JFjp7$$\"169_/Kr^')F]v$\"13$)e&p!QzHFjp7$$\"1#)Ga!o6U7*F]v$\"1A i&oM*G[GFjp7$$\"1XFQ=wx!f*F]v$\"1V#\\a'HXAFFjp7$$\"1u'RWI`z+\"Fjp$\"1x a*R5/Uf#Fjp7$$\"1l*\\M-\"ed5Fjp$\"1D8POTYnCFjp7$$\"1&)3PUf>/6Fjp$\"1D( )[NYS^BFjp7$$\"1(*G(R$)4)e6Fjp$\"1->R.3()=AFjp7$$\"1D%o%z\"G(37Fjp$\"1 %RVT)yx+@Fjp7$$\"1nL/u71j7Fjp$\"1Td-N6Mv>Fjp7$$\"1ylS'eLhJ\"Fjp$\"1$RT \"e1vb=Fjp7$$\"1sSxyuKu8Fjp$\"1F]5F)$\"$R\"F)7$$\"+\")e%o%>!\"*$\"+$HL,W\"Fcd l7$$\"+kKVP>Fcdl$\"+bfZ*[\"Fcdl7$$\"+%f5>#>Fcdl$\"+@)\\s`\"Fcdl7$$\"+s E_+>Fcdl$\"+q9q#e\"Fcdl7$$\"+)z1O(=Fcdl$\"+,T6D;Fcdl7$$\"+^ueT=Fcdl$\" +U)=Qm\"Fcdl7$$\"+'fp\\!=Fcdl$\"+(H0#)p\"Fcdl7$$\"+=2Lkv\"Fcdl7$$\"+)z1On\"Fcdl$\"+2EUq\"Fcdl$\"+p9q#e\"Fcdl7$$\"+1%*3y6FcdlF^el7$$\"+ Onci6FcdlFidl7$$\"+>T:`6FcdlFddl7$$\"$:\"F)$\"++++!R\"Fcdl7$$\"+?T:`6F cdl$\"+1n')R8Fcdl7$F^jl$\"+XS_!H\"Fcdl7$F[jl$\"+z,vU7Fcdl7$Ffil$\"+I&) H(>\"Fcdl7$Fcil$\"+**e)[:\"Fcdl7$F`il$\"+e6=;6Fcdl7$$\"+0/.&H\"Fcdl$\" +.Zz\"3\"Fcdl7$Fjhl$\"+I)oA0\"Fcdl7$Fghl$\"+z\"p!G5Fcdl7$$\"+.KRE9Fcdl $\"+$Rx&45Fcdl7$Fahl$\"+'*4&3(**!#57$F^hl$\"+'3$*y!**F_]m7$FiglFa]m7$F dgl$\"+(*4&3(**F_]m7$F_gl$\"+%Rx&45Fcdl7$FjflFe\\m7$FeflFb\\m7$F`flF_ \\m7$F[flFj[m7$Ffel$\"++f)[:\"Fcdl7$Fael$\"+J&)H(>\"Fcdl7$$\"+&f5>#>Fc dlFa[m7$$\"+lKVP>Fcdl$\"+YS_!H\"Fcdl7$Fadl$\"+2n')R8Fcdl7$F\\dlFfjlF`q -%+AXESLABELSG6$%!GFc_m-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$%(DEFAULTGF[`m " 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "C urve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" }}{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 263 "The bal l will be attracted to one of the two magnets, and, given some velocit y, will oscillate about that magnet until friction stops it. Each of t he magnets creates a fixed point where the ball may come to rest above that magnet and remain there in equilibrium. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 149 "When this system is shak en by an external periodic force, the ball may jump back and forth fro m one magnet to the other in a seemingly random manner." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "This system can be m odeled mathematically by the equation" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2x/(dt^2) + b" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*$%# dtGF&!\"\"F(%\"bGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt-x+x^3 = A*c os(omega*t);" "6#/,(*&%#dxG\"\"\"%#dtG!\"\"F'%\"xGF)*$F*\"\"$F'*&%\"AG F'-%$cosG6#*&%&omegaGF'%\"tGF'F'" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 20 "which is called the " }{TEXT 261 16 "Duffing equation" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "The D uffing equation for free oscillations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 233 "First we consider \+ a system without any driving force or friction. The Duffing equation d escribes the motion of a classical particle in a double well potential . We choose the units of length so that the minima of the potential ar e at " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "x = -1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 63 ", and the units of energy so that the depth of each well is at " }{XPPEDIT 18 0 "-1/4" "6#,$*&\"\"\"F%\"\"%!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "The potential is given by" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V(x)=-x^2/2+x^4/4" "6#/-%\"VG6#%\"xG,&*&F'\" \"#F*!\"\"F+*&F'\"\"%F-F+\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot(-x^2/2+ x^4/4,x=-2..2,-0.3..0.5);" }}{PARA 13 "" 1 "" {GLPLOT2D 288 223 223 {PLOTDATA 2 "6%-%'CURVESG6$7ao7$$!\"#\"\"!$\"\"#F*7$$!1LL$e%G?y>!#:$\" 1hrpc&4=(=F07$$!1nmm\"p0k&>F0$\"1PFf/>s[F0$\"1I5?,[hI ;F07$$!1LLL$Q6G\">F0$\"1EsWetO<:F07$$!1++v3-)[(=F0$\"1MzlAt_J8F07$$!1n m;M!\\p$=F0$\"1O9ey@Uf6F07$$!1++Dh9H%z\"F0$\"1&ocI=\\_\")*!#;7$$!1LLL) )Qj^)FQ7$$!1LLL`Np3O,+UP&FQ7$$!1++Drp,B;F0$\"1#>q:4%RwTFQ7$$!1nm;C2G!e\"F0$\"12.X_ @n/JFQ7$$!1++]_(e1a\"F0$\"1I_/kU9#FQ7$$!1++](p7U7\"F0$ !1Ut2CFQ7$$!1nm;arvU5F0$!1B\"[oAG4 [#FQ7$$!1,+++()>'***FQ$!1i5cb&)***\\#FQ7$$!1++++0\"*H\"*FQ$!1OpL0#Q2V# FQ7$$!1++++83&H)FQ$!1!HQLhqnD#FQ7$$!1LLL3k(p`(FQ$!1=H'*z)pN.#FQ7$$!1nm mmj^NmFQ$!1]6]U>%or\"FQ7$$!1ommm9'=(eFQ$!1:mn\"oTnU\"FQ7$$!1,++v#\\N) \\FQ$!1-h>$Q%e(3\"FQ7$$!1pmmmCC(>%FQ$!1&zb@:RD.)Fjp7$$!1*****\\FRXL$FQ $!1utWDs[]_Fjp7$$!1+++D=/8DFQ$!17=bq&zz0$Fjp7$$!1mmm;a*el\"FQ$!17%Gwa) >_8Fjp7$$!1pmm;Wn(o)Fjp$!1Fdp$!1NMU:;/\\>!#@ 7$$\"1Mmm;f`@')Fjp$!1SSBy8t-PFdp7$$\"1)****\\nZ)H;FQ$!1e@rB0c58Fjp7$$ \"1lmm;$y*eCFQ$!1c`8FX)=$HFjp7$$\"1*******R^bJ$FQ$!1G8fc>L%>&Fjp7$$\"1 '*****\\5a`TFQ$!1p5I77)=)yFjp7$$\"1(****\\7RV'\\FQ$!1&3XWH$R!3\"FQ7$$ \"1'*****\\@fkeFQ$!1l2'o&[%RU\"FQ7$$\"1JLLL&4Nn'FQ$!1J(RUnG4t\"FQ7$$\" 1*******\\,s`(FQ$!1wUBdJkL?FQ7$$\"1lmm\"zM)>$)FQ$!1%f`#GD9jAFQ7$$\"1** *****pfa<*FQ$!1CHm\\N]PCFQ7$$\"1HLLeg`!)**FQ$!1t(*>*=i**\\#FQ7$$\"1mm; W/8S5F0$!1<`&)3FC$[#FQ7$$\"1++]#G2A3\"F0$!14!e0&)\\nU#FQ7$$\"1mm\"H3XL 7\"F0$!1vRQ2b^GBFQ7$$\"1LLL$)G[k6F0$!1L_j7P7$=#FQ7$$\"1++]7yh]7F0$!1F) f/MLYq\"FQ7$$\"1nmm')fdL8F0$!1zm6,kO^)*Fjp7$$\"1nmm,FT=9F0$\"1I/'eXmE) fFdp7$$\"1LL$e#pa-:F0$\"1#Q*p.;>a9FQ7$$\"1nm\"HB-7a\"F0$\"1a'=/&\\lGAF Q7$$\"1+++Sv&)z:F0$\"1secf[m%4$FQ7$$\"1nm;%)3;C;F0$\"1ausw$zn?%FQ7$$\" 1LLLGUYo;F0$\"181^HUhaaFQ7$$\"1++]n'*33F0$\"1#*= =^mx=:F07$$\"1+]i0j\"[$>F0$\"143JaurJ;F07$$\"1++v.Uac>F0$\"1]%zQA)[\\< F07$$\"1+](=5s#y>F0$\"1?S)*4(3A(=F07$F+F+-%'COLOURG6&%$RGBG$\"#5!\"\"F *F*-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(F+;$!\"$Fbal$\"\"&Fbal" 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 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The force is given by " } {XPPEDIT 18 0 "F(x) =-dV/dx " "6#/-%\"FG6#%\"xG,$*&%#dVG\"\"\"%#dxG!\" \"F-" }{XPPEDIT 18 0 "`` = x-x^3;" "6#/%!G,&%\"xG\"\"\"*$F&\"\"$!\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "Applying Newton's 2nd law of motion is " }{XPPEDIT 18 0 "F(x) = m;" "6#/-%\"FG6#%\"xG%\"mG " }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)" "6#*(%\"dG\"\"#%\"xG \"\"\"*&F$F'*$%\"tGF%F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 50 "From now on we set the mass equal to 1, so we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)=x-x^3" "6#/*( %\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\",&F'F(*$F'\"\"$F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "The 2nd order differential eq uation" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^ 2)=x-x^3" "6#/*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\",&F'F(*$F' \"\"$F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 65 "is equivalent to the system of first order differential equations" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([du/dt = x-x^3, ``],[dx/ dt = u, ``]);" "6#-%*PIECEWISEG6$7$/*&%#duG\"\"\"%#dtG!\"\",&%\"xGF**$ F.\"\"$F,%!G7$/*&%#dxGF*F+F,%\"uGF1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " } {XPPEDIT 18 0 "du/dx=(x-x^3)/u" "6#/*&%#duG\"\"\"%#dxG!\"\"*&,&%\"xGF& *$F+\"\"$F(F&%\"uGF(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 90 "T his 1st order differential equation has separable variables, and so ca n be easily solved." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "de1 := diff(u(x),x)=(x-x^3)/u(x);\nic1 := u (x0)=y0;\ndesolve(\{de1,ic1\},format=implicit);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de1G/-%%diffG6$-%\"uG6#%\"xGF,*&,&F,\"\"\"*$)F,\"\"$ F/!\"\"F/F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ic1G/-%\"uG6#%#x0G %#y0G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"\"#F'*$)-%\"uG6 #%\"xGF(F'F'F',,*&#F'\"\"%F'*$)F.F2F'F'!\"\"*&F&F'*$)F.F(F'F'F'*&F&F'* $)%#y0GF(F'F'F'*&#F'F2F'*$)%#x0GF2F'F'F'*&#F'F(F'*$)FAF(F'F'F5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*u^2-2*x^2+x^4=c" "6#/,(*&\"\"#\"\"\"*$%\"uGF&F 'F'*&F&F'*$%\"xGF&F'!\"\"*$F,\"\"%F'%\"cG" }{TEXT -1 1 "," }}{PARA 0 " " 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "c = 2*y[0]^2-2*x[0]^2+x[0]^ 4" "6#/%\"cG,(*&\"\"#\"\"\"*$&%\"yG6#\"\"!F'F(F(*&F'F(*$&%\"xG6#F-F'F( !\"\"*$&F16#F-\"\"%F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 36 " Here is a plot of some trajectories." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "with(plots):\ncontourplot (2*u^2-2*x^2+x^4,x=-3..3,u=-4..4,\ncontours=[-.8,-.6,-.4,-.2,-.1,0,.1, .2,.4,.8,1.6,3,4,6],grid=[50,50]);" }}{PARA 13 "" 1 "" {GLPLOT2D 355 248 248 {PLOTDATA 2 "61-%'CURVESG6cw7$7$$!1oMn$=fz*=!#:$!1!e&3,F`yK!#; 7$$!11*G>xBG!>F*$!1F*$!1<2!fcz9Q#F-7$7$$! 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\"\"" }{TEXT -1 85 ", and, since we have taken the mass to be 1 unit, \+ the kinetic energy at the start is " }{XPPEDIT 18 0 "u^2/2" "6#*&%\"uG \"\"#F%!\"\"" }{TEXT -1 119 ". Using the principle of conservation of \+ energy, the minimum initial velocity needed to get to the top of the h ump is " }{XPPEDIT 18 0 "1/sqrt(2)" "6#*&\"\"\"F$-%%sqrtG6#\"\"#!\"\" " }{TEXT -1 119 ". This means that an initial velocity of 1 is big eno ugh to kick the particle over the hump in the potential energy at " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 24 " to the vicinity of \+ the " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 25 " min imum and back again. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 293 "de := diff(x(t),t$2)=x(t)-x(t)^3;\nic := x(0)=1,D(x)(0)=1;\n(fn,dfn) := evalf[15](desolve(\{de,ic\},t=0..20,\n type=numeric,method=taylor,output=procedure_pair));\np lot('fn'(t),t=0..20,color=red,labels=[`t`,`x`]);\nplot('dfn'(t),t=0..2 0,color=COLOR(RGB,0,.7,0),labels=[`t`,`u=dx/dt`]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F,\"\"#,&F)\"\" \"*$)F)\"\"$F2!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG 6#\"\"!\"\"\"/--%\"DG6#F(F)F+" }}{PARA 13 "" 1 "" {GLPLOT2D 470 193 193 {PLOTDATA 2 "6&-%'CURVESG6#7ir7$$\"\"!F)$\"\"\"F)7$$\"+5xI&F/$\"+FJ\"HY\"F27$$ \"+j&*)fD'F/$\"++v36:F27$$\"+Y#H,t'F/$\"+1MyG:F27$$\"+G*oU?(F/$\"+7%)) =a\"F27$$\"+5'3%ywF/$\"+`TA]:F27$$\"+#H[D:)F/$\"+L`n`:F27$$\"+azw&o)F/ $\"+`Qm^:F27$$\"+;w)*=#*F/$\"+C7XV:F27$$\"+ys?_(*F/$\"+B8=H:F27$$\"+%p U&G5F2$\"+i**44:F27$$\"+Em=N6F2$\"+1z#HX\"F27$$\"+e0$=C\"F2$\"+ZWXy8F2 7$$\"+LA`c9F2$\"+x))=!>\"F27$$\"+3RBr;F2$\"+z=I#y*F/7$$\"+W^\"\\)=F2$ \"+0C_\"o(F/7$$\"+zjf)4#F2$\"+TDy;dF/7$$\"+Qiq'H#F2$\"+_.J[SF/7$$\"+'4 ;[\\#F2$\"+Jj&3^#F/7$$\"+!QZ**p#F2$\"++/q95F/7$$\"+j'y]!HF2$!+!z#>&R%! 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This is shown below for " }{TEXT 271 1 "t" } {TEXT -1 19 " between 0 and 10. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "plot(['fn(t)','dfn(t)',t=0.. 8.073],color=coral,thickness=2,\n labels=[`x`,`u=dx/dt`]);" }} {PARA 13 "" 1 "" {GLPLOT2D 365 240 240 {PLOTDATA 2 "6'-%'CURVESG6#7ho7 $$\"\"\"\"\"!F(7$$\"+*yiR/\"!\"*$\"+)H$zz**!#57$$\"+\\@u(3\"F.$\"+F^v: **F17$$\"+ZS8J6F.$\"+m]!G!)*F17$$\"+=C\"R<\"F.$\"+8M.O'*F17$$\"+ks9Y7F .$\"+\\08/#*F17$$\"+eML98F.$\"+Z&*\\v&)F17$$\"+I_B%Q\"F.$\"+#)pA=wF17$ $\"+@KuW9F.$\"+E/f%R'F17$$\"+F-\"R\\\"F.$\"+bO)H\"\\F17$$\"+9aHH:F.$\" +ft1BKF17$$\"+6&*H\\:F.$\"+2L$yR\"F17$$\"+7WA`:F.$!+r;`7\\!#67$$\"+l.L U:F.$!+[EQCAF17$$\"+_/)y^\"F.$!+^H&4(QF17$$\"+_\"z#z9F.$!+y!yAU&F17$$ \"+'\\a'G9F.$!+.`LonF17$$\"+h9hl8F.$!+05)=\"zF17$$\"+G(3RH\"F.$!+z4\\! z)F17$$\"+$Rcg@\"F.$!+lcX4%*F17$$\"+6A%R8\"F.$!+_3j$z*F17$$\"+PY+Z5F.$ !+`!Go(**F17$$\"+5qg#f*F1$!+]'fS)**F17$$\"+u(RS#))F1$!+7htw)*F17$$\"+V M*p1)F1$!+4#*G!p*F17$$\"+9jyGkF1$!+Z=$))4*F17$$\"+%*\\Q'*[F1$!+U?7K%)F 17$$\"+#[8g_$F1$!+Q>Q_yF17$$\"+u)QvN#F1$!+w\\NVuF17$$\"+o!3P.\"F1$!+)) p#e9(F17$$!+v&*Qvf!#7$!+x-KrqF17$$!+<&\\_L\"F1$!+9'G\\>(F17$$!+sJv 'F1$!+O^rI#*F17$$!+#HQEE)F1$!+X72X(*F17$$!+%[/g5*F1$!+@YzE**F17$$!+vh` g**F1$!+t[%)****F17$$!+=z_S5F.$!+5kd)F17$$!+))>J#Q\"F.$!+q!f+l(F17$$!+%3j:W\"F.$!+@$H=Z 'F17$$!+ejM\"\\\"F.$!+;\">x+&F17$$!+Qr_F:F.$!+Ao@LLF17$$!+z]D[:F.$!+dF F^:F17$$!+))3d`:F.$\"+@z0()HFco7$$!+\\%=Qa\"F.$\"+Zn#p2#F17$$!+!>;)>:F .$\"+#)QQqPF17$$!+1glx9F.$\"+'35XZ&F17$$!+JI5@9F.$\"+,/>IpF17$$!+Nt-g8 F.$\"+g[\"R*zF17$$!+K?rkF 1$\"+pNP;\"*F17$$!+@3&z(\\F1$\"+!=h\"o%)F17$$!+!Hq6i$F1$\"+5d1!*yF17$$ !+I)3DH#F1$\"+_a6CuF17$$!+Lo([2\"F1$\"+\\4$=:(F17$$\"+QlN*R\"Fco$\"+'> _C2(F17$$\"+VOi[8F1$\"+0xP(>(F17$$\"+JZ)>\\#F1$\"+\"yhW[(F17$$\"+M%*Gt QF1$\"+j(4B*zF17$$\"+`aN'>&F1$\"+sK%[c)F17$$\"+YYi7nF1$\"+XF`9#*F17$$ \"+ylKi#)F1$\"+#Q()\\u*F17$$\"+#pw++\"F.$\"+T********F1-%*THICKNESSG6# \"\"#-%+AXESLABELSG6$%\"xG%(u=dx/dtG-%'COLOURG6&%$RGBG$\"*++++\"!\")$ \")AR!)\\F_dl$F*F*-%%VIEWG6$%(DEFAULTGFfdl" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 147 "You can see that the phase tra jectory closes on itself. In fact this must be so since energy is cons erved, that is, energy remains constant, so if " }{TEXT 262 1 "x" } {TEXT -1 21 " is given then so is " }{TEXT 263 1 "u" }{TEXT -1 16 ", u p to a sign. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The trajectory with " }{XPPEDIT 18 0 "u(1) = 1" "6#/-%\"u G6#\"\"\"F'" }{TEXT -1 35 " has the analytical equation . . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "de1 := diff(u(x),x)=(x-x^3)/u(x);\ndesolve(\{de1,u(1)=1\},format=i mplicit);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de1G/-%%diffG6$-%\"uG6 #%\"xGF,*&,&F,\"\"\"*$)F,\"\"$F/!\"\"F/F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*$)-%\"uG6#%\"xG\"\"#\"\"\"#F,F+,(*$)F*F+F,F-*$)F*\" \"%F,#!\"\"F3#F,F3F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "We can find the " }{TEXT 264 1 "x" }{TEXT -1 53 " coor dinates of the points where the curve meets the " }{TEXT 265 1 "x" } {TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 26 "fsolve(x^2-x^4/2+1/2=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+uRx`:!\"*$\"+uRx`:F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "The numerical trajectory coinci des with the analytical curve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "g := x -> sqrt(x^2-x^4/2+1/ 2):\np1 := plot([g(x),-g(x)],x=-1.553773974..1.553773974,\n nump oints=200,color=red):\np2 := plot(['fn(t)','dfn(t)',t=0..8.73],color=g reen,thickness=2):\nplots[display]([p1,p2],labels=[`x`,`u=dx/dt`]);" } }{PARA 13 "" 1 "" {GLPLOT2D 390 246 246 {PLOTDATA 2 "6'-%'CURVESG6$7ax 7$$!31+++uRx`:!#<$\"3K(4]9aO*[6!#A7$$!31(fO:pJ7 $$!3I%>t!4%*o\\:F*$\"3w&Q<%pM\"fL\"!#=7$$!3K\"z4m7Zwa\"F*$\"3WiQLzkxL; F97$$!3K)QYT%[gX:F*$\"3i\"f_6#)zP)=F97$$!3N#e>#z-_T:F*$\"3kL#ySAP/I#F9 7$$!3PwFH9dVP:F*$\"3qMo!=Mu&[EF97$$!3t[s3)zF.`\"F*$\"3!f\"*\\=A1n:$F97 $$!33@<)=))>K_\"F*$\"3IIWvBM@&e$F97$$!3y!*za\"pKs]\"F*$\"3;K1^&ywSP%F9 7$$!3Ak!Rx_R6\\\"F*$\"3b(3b(z5D:]F97$$!3cF&G>&G7v9F*$\"3_\"f$QR(=Xb&F9 7$$!3sV\\<^(z!R'F97 $$!3VRK@ce**G9F*$\"3qZ2!e_D3w'F97$$!3(*Q1Ms^989F*$\"3coYT8KO#4(F97$$!3 Uj4o&zSoR\"F*$\"38S.mDFH+uF97$$!31kW%=*)zCQ\"F*$\"3#\\')RvX%HZwF97$$!3 e)Q&\\:HJm8F*$\"3j>cRJ#)Q,zF97$$!3oK\"))fbz+N\"F*$\"3XI=:4kpL\")F97$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "de1 := diff(u(x),x)=(x- x^3)/u(x);\ndesolve(\{de1,u(1)=1/sqrt(2)+10^(-3)\},format=implicit);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de1G/-%%diffG6$-%\"uG6#%\"xGF,*&, &F,\"\"\"*$)F,\"\"$F/!\"\"F/F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/, $*$)-%\"uG6#%\"xG\"\"#\"\"\"#F,F+,**$)F*F+F,F-*$)F*\"\"%F,#!\"\"F3#F, \"(+++#F,*$-%%sqrtG6#F+F,#F,\"%+?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 16 "We can find the " }{TEXT 267 1 "x" } {TEXT -1 53 " coordinates of the points where the curve meets the " } {TEXT 266 1 "x" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "fsolve(x^2-x^4/2+1/1000000+ sqrt(2)/1000=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+uMr99!\"*$\"+ uMr99F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "The numerical trajectory coincides with the analytical curve." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 253 "g := x -> sqrt(x^2-x^4/2+1/1000000+sqrt(2)/1000):\np1 := plot([g( x),-g(x)],x=-1.414713474..1.413713474,\n numpoints=200,color=red ):\np2 := plot(['hn'(t),'dhn'(t),t=0..20.04533],color=green,thickness= 2):\nplots[display]([p1,p2],labels=[`x`,`u=dx/dt`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 326 195 195 {PLOTDATA 2 "6'-%'CURVESG6$7ex7$$!3.+++uMr99! 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