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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 38 "Maple workbook:  Anharm
onic Oscillator" }}{PARA 0 "" 0 "" {TEXT -1 24 "RL Dewar,  29 March 19
99" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 293 92 "Problem 2.
8.2:  Variational calculation of harmonic generation and nonlinear fre
quency shift" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Statement of prob
lem" }}{PARA 0 "" 0 "" {TEXT 32 1197 "Consider the following potential
 V, corresponding to a particle of \nmass m oscillating along the x-ax
is under the influence of a \nnonideal spring (i.e. one with a nonline
ar restoring force),\n\\begin\{displaymath\}\n\011V(x) = (1/2)(m omega
_0^2)\n\011(x^2 + sigma (x^4/l_0^2))\n\\end\{displaymath\}\nwhere the \+
constant $\\omega_0$ is the angular frequency of oscillations \nhaving
 amplitude small compared with the characteristic length $l_0$, \nand \+
$\\sigma = \\pm 1$ depends on whether the spring is ``soft'' \n($\\sig
ma = -1$) or ``hard'' ($\\sigma = +1$).\n\nBy using the trial function
\\index\{trial function\}\n\\begin\{displaymath\}\n\011x = l_0(A \\cos
\\omega t + B\\cos(3\\omega t)\n\011\011+ C\\sin (3\\omega t))\n\\end
\{displaymath\}\nin the time-averaged Hamilton's Principle, find relat
ions giving \napproximate expressions for the nonlinearly shifted freq
uency $\\omega$ \nand the nondimensionalized third harmonic amplitude \+
$B$ as functions \nof the nondimensionalized fundamental amplitude $A$
.  The trial \nfunction is strictly appropriate only to the case $A \\
ll 1$, but plot \n$\\omega/\\omega_0$ and $B$ \\emph\{vs.\} $A$ from 0
 to 1 in the case of \nboth a hard and a soft spring.  (You are encour
aged to use Maple and \nMatLab in this problem.)" }}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 52 "Initializations & Calculation of Averaged Lagrangi
an" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 108 "Define pote
ntial with quadratic term (note that we have changed notation slightly
 from that in the notes -- " }{XPPEDIT 18 0 "omega[0];" "6#&%&omegaG6#
\"\"!" }{TEXT -1 12 " has become " }{XPPEDIT 18 0 "Omega;" "6#%&OmegaG
" }{TEXT -1 6 "  and " }{XPPEDIT 18 0 "l[0];" "6#&%\"lG6#\"\"!" }
{TEXT -1 12 " has become " }{TEXT 258 1 "l" }{TEXT -1 1 ":" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "V := x-> m*(Omega^2)*(x^2 + sigma*x
^4/l^2)/2;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VGR6#
%\"xG6\"6$%)operatorG%&arrowGF(,$*(%\"mG\"\"\")%&OmegaG\"\"#\"\"\",&*$
)9$F2F3F/*&*&%&sigmaGF/)F7\"\"%F3F3*$)%\"lG\"\"#F3!\"\"F/F/#F/F2F(F(F(
" }}}{PARA 0 "" 0 "" {TEXT -1 53 "Define trial function (2nd harmonic \+
vanishes because " }{TEXT 266 1 "V" }{TEXT -1 1 "(" }{TEXT 267 1 "x" }
{TEXT -1 99 ") is even. So does sin component of 3rd harmonic, but lea
ve this in as a check -- should find that " }{TEXT 268 1 "C" }{TEXT 
-1 8 " = 0.) :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "\011xtr :=
 t->l*(A*cos(omega*t) + B*cos(3*omega*t)\011+ C*sin (3*omega*t));" }
{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xtrGR6#%\"tG6\"6$%)
operatorG%&arrowGF(*&%\"lG\"\"\",(*&%\"AGF.-%$cosG6#*&%&omegaGF.9$F.F.
F.*&%\"BGF.-F36#,$F5\"\"$F.F.*&%\"CGF.-%$sinGF;F.F.F.F(F(F(" }}}{PARA 
0 "" 0 "" {TEXT -1 89 "Average the potential and kinetic energy, then \+
define averaged Lagrangian as function of " }{TEXT 257 8 "A, B, C," }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 1 ":
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Vbar := simplify(simplif
y(\nint(V(xtr(t)),t=0..(2*Pi/omega))\n)*omega/(2*Pi));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%VbarG,$**)%\"lG\"\"#\"\"\"%\"mG\"\"\")%&OmegaGF
)F*,6*$)%\"AGF)F*\"\"%*$)%\"BGF)F*F3*$)%\"CGF)F*F3*&%&sigmaGF,)F2F3F*
\"\"$*&F;F*)F6F3F*F=*&F;F*)F9F3F*F=*(F;F*)F2F=F*F6F,F3*(F;F*F1F*F8F*\"
#7*(F;F*F1F*F5F*FE*(F;F*F5F*F8F*\"\"'F,#F,\"#;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 89 "Tbar := simplify(simplify(\nint((m/2)*diff(xtr
(t),t)^2,t=0..(2*Pi/omega))\n)*omega/(2*Pi));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%TbarG,$**%\"mG\"\"\")%\"lG\"\"#\"\"\")%&omegaGF+F,,(
*$)%\"AGF+F,F(*$)%\"BGF+F,\"\"**$)%\"CGF+F,F6F(#F(\"\"%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Lbar:=unapply(Tbar-Vbar,A,B,C,omega
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LbarGR6&%\"AG%\"BG%\"CG%&omeg
aG6\"6$%)operatorG%&arrowGF+,&**%\"mG\"\"\")%\"lG\"\"#\"\"\")9'F5F6,(*
$)9$F5F6F2*$)9%F5F6\"\"**$)9&F5F6F@F2#F2\"\"%**F3F6F1F6)%&OmegaGF5F6,6
F:FEF=FEFAFE*&%&sigmaGF2)F<FEF6\"\"$*&FKF6)F?FEF6FM*&FKF6)FCFEF6FM*(FK
F6)F<FMF6F?F2FE*(FKF6F;F6FBF6\"#7*(FKF6F;F6F>F6FU*(FKF6F>F6FBF6\"\"'F2
#!\"\"\"#;F+F+F+" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VGR6#%\"xG6\"
6$%)operatorG%&arrowGF(,$*(%\"mG\"\"\")%&OmegaG\"\"#\"\"\",&*$)9$F2F3F
/*&*&%&sigmaGF/)F7\"\"%F3F3*$)%\"lG\"\"#F3!\"\"F/F/#F/F2F(F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xtrGR6#%\"tG6\"6$%)operatorG%&arrow
GF(*&%\"lG\"\"\",(*&%\"AGF.-%$cosG6#*&%&omegaGF.9$F.F.F.*&%\"BGF.-F36#
,$F5\"\"$F.F.*&%\"CGF.-%$sinGF;F.F.F.F(F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%VbarG,$**)%\"lG\"\"#\"\"\"%\"mG\"\"\")%&OmegaGF)F*,6
*(%&sigmaGF,)%\"AGF)F*)%\"BGF)F*\"#7*(F1F*F2F*)%\"CGF)F*F6*(F1F*)F3\"
\"$F*F5F,\"\"%*(F1F*F4F*F8F*\"\"'*&F1F*)F9F=F*F<*&F1F*)F3F=F*F<*$F2F*F
=*$F4F*F=*$F8F*F=*&F1F*)F5F=F*F<F,#F,\"#;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%TbarG,$**%\"mG\"\"\")%\"lG\"\"#\"\"\")%&omegaGF+F,,(
*$)%\"AGF+F,F(*$)%\"BGF+F,\"\"**$)%\"CGF+F,F6F(#F(\"\"%" }}{PARA 0 "" 
0 "" {XPPMATH 20 "6#>%%LbarGR6&%\"AG%\"BG%\"CG%&omegaG6\"6$%)operatorG
%&arrowGF+,&**%\"mG\"\"\")%\"lG\"\"#\"\"\")9'F5F6,(*$)9$F5F6F2*$)9%F5F
6\"\"**$)9&F5F6F@F2#F2\"\"%**F3F6F1F6)%&OmegaGF5F6,6*(%&sigmaGF2F;F6F>
F6\"#7*(FKF6F;F6FBF6FL*(FKF6)F<\"\"$F6F?F2FE*(FKF6F>F6FBF6\"\"'*&FKF6)
FCFEF6FP*&FKF6)F<FEF6FPF:FEF=FEFAFE*&FKF6)F?FEF6FPF2#!\"\"\"#;F+F+F+" 
}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "Euler-Lagrange equations" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Ma
ke action stationary with respect to variation of " }{TEXT 259 1 "A" }
{TEXT -1 2 ", " }{TEXT 260 1 "B" }{TEXT -1 6 ", and " }{TEXT 261 1 "C
" }{TEXT -1 90 " by differentiating averaged Lagrangian and setting to
 zero (also simplify by dividing by " }{TEXT 264 1 "A" }{TEXT -1 57 ",
 since we are not interested in the zero-amplitude case " }{TEXT 265 
1 "A" }{TEXT -1 7 " = 0 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "ELA:=
unapply(simplify(diff(Lbar(A, B, C, omega),A)/A),A,B,C,omega);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "ELB:=unapply(simplify(diff(Lbar(A, \+
B, C, omega),B)/A),A,B,C,omega);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 
"ELC:=unapply(simplify(diff(Lbar(A, B, C, omega),C)/A),A,B,C,omega);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ELAGR6&%\"AG%\"BG%\"CG%&omegaG6\"
6$%)operatorG%&arrowGF+,$*(%\"mG\"\"\")%\"lG\"\"#\"\"\",.*$)9'F5F6!\"#
*$)%&OmegaGF5F6F5*(F=F6%&sigmaGF2)9$F5F6\"\"$**F=F6F@F6FBF29%F2FC*(F=F
6F@F6)9&F5F6\"\"'*(F=F6F@F6)FEF5F6FIF2#!\"\"\"\"%F+F+F+" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$ELBGR6&%\"AG%\"BG%\"CG%&omegaG6\"6$%)operator
G%&arrowGF+,$*&*(%\"mG\"\"\")%\"lG\"\"#\"\"\",.*&)9'F6F79%F3!#=*&)%&Om
egaGF6F7F<F7F6*(F?F7%&sigmaGF3)F<\"\"$F7FD*(F?F7FBF7)9$FDF7F3**F?F7FBF
7)FGF6F7F<F7\"\"'**F?F7FBF7F<F7)9&F6F7FDF3F7FG!\"\"#!\"\"\"\"%F+F+F+" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ELCGR6&%\"AG%\"BG%\"CG%&omegaG6\"
6$%)operatorG%&arrowGF+,$*&**%\"mG\"\"\")%\"lG\"\"#\"\"\"9&F3,,*$)9'F6
F7!#=*$)%&OmegaGF6F7F6*(F?F7%&sigmaGF3)F8F6F7\"\"$*(F?F7FBF7)9$F6F7\"
\"'*(F?F7FBF7)9%F6F7FDF3F7FG!\"\"#!\"\"\"\"%F+F+F+" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 34 "Show that Euler-Lagrange equation " }{TEXT 263 3 
"ELC" }{TEXT -1 23 " = 0 is satisfied when " }{TEXT 262 1 "C" }{TEXT 
-1 5 " = 0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "ELC(A,B,0,omega);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 85 "Create set of remaining Euler-Lagrange equations for Mapl
e to solve (definining w2 = " }{XPPEDIT 18 0 "omega^2;" "6#*$%&omegaG
\"\"#" }{TEXT -1 2 "/ " }{XPPEDIT 18 0 "Omega^2;" "6#*$%&OmegaG\"\"#" 
}{TEXT -1 51 " to reduce order of equations) and then solve them:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "eqns := \{ELA(A,B,0,Omega*sqrt(w2))
=0, ELB(A,B,0,Omega*sqrt(w2))=0\};" }{TEXT -1 0 "" }{MPLTEXT 1 0 30 "E
Lsolns := solve(eqns,\{B,w2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%
eqnsG<$/,$*(%\"mG\"\"\")%\"lG\"\"#\"\"\",,*&)%&OmegaGF-F.%#w2GF*!\"#*$
F1F.F-*(F1F.%&sigmaGF*)%\"AGF-F.\"\"$**F1F.F7F.F9F*%\"BGF*F:*(F1F.F7F.
)F<F-F.\"\"'F*#!\"\"\"\"%\"\"!/,$*&*(F)F.F+F.,,*(F1F.F3F.F<F.!#=*&F1F.
F<F.F-*(F1F.F7F.)F<F:F.F:*(F1F.F7F.)F9F:F.F***F1F.F7F.F8F.F<F.F?F*F.F9
!\"\"F@FC" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(ELsolnsG<$/%#w2G,*\"\"
\"F)*&%&sigmaGF))%\"AG\"\"#\"\"\"#\"\"$F.*(F+F/F-F/-%'RootOfG6#,**&F+F
/)%#_ZGF1F/\"#^*(F+F/F-F))F9F.F/\"#F*&,&F*\"#@\"#;F)F)F9F)F)*&F+F/)F-F
1F/!\"\"F)F0*&F+F/)F3F.F/F1/%\"BGF3" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 24 "Case of 'hard' potential" }}{PARA 0 "" 0 "" {TEXT -1 23 "
Do a specific value of " }{TEXT 272 1 "A" }{TEXT -1 92 " to see which \+
of the three roots of the cubic equation is real (the only physical so
lution):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "allvalues(eval(E
Lsolns,\{sigma=1,A=0.1\}));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%<$/%#w2
G,&$\"*(>&*ph!#5\"\"\"%\"IG$\"*[TK2&!#6/%\"BG,&$!+3L9]EF.F*F+$!+zYbJcF
)<$/F%,&F'F*F+$!*[TK2&F./F0,&F2F*F+$\"+zYbJcF)<$/F%$\"+k#4],\"!\"*/F0$
\"++!o*oh!#9" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Now define functions " }
{TEXT 269 3 "Bfn" }{TEXT -1 6 ", and " }{TEXT 273 3 "wfn" }{TEXT -1 8 
" giving " }{TEXT 270 1 "B" }{TEXT -1 5 " and " }{TEXT 274 2 "w2" }
{TEXT -1 33 ", respectively, as a function of " }{TEXT 271 1 "A" }
{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "Bfn := Ava
l -> eval(B,allvalues(eval(ELsolns,\{A=Aval,sigma=1\}))[3]);\nwfn := A
val -> eval(w2,allvalues(eval(ELsolns,\{A=Aval,sigma=1\}))[3]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$BfnGR6#%%AvalG6\"6$%)operatorG%&arr
owGF(-%%evalG6$%\"BG&-%*allvaluesG6#-F-6$%(ELsolnsG<$/%&sigmaG\"\"\"/%
\"AG9$6#\"\"$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$wfnGR6#%%Ava
lG6\"6$%)operatorG%&arrowGF(-%%evalG6$%#w2G&-%*allvaluesG6#-F-6$%(ELso
lnsG<$/%&sigmaG\"\"\"/%\"AG9$6#\"\"$F(F(F(" }}}{PARA 0 "" 0 "" {TEXT 
-1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Bfn(0.1);wfn
(0.1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++!o*oh!#9" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"+k#4],\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 34 
"Plot amplitude of third harmonic, " }{TEXT 275 1 "B" }{TEXT -1 6 ", v
s. " }{TEXT 276 1 "A" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 77 "plot(Bfn,0..1,labels=[A,B],axesfont=[SYMBOL,12],label
font=[TIMES,ITALIC,14]);" }}{PARA 13 "" 1 "" {GLPLOT2D 266 180 180 
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+$>]hM#F*7$$\"+!oK0e*FG$\"+e'[oW#F*7$$\"+<5s#y*FG$\"+gY#Ra#F*7$%%FAILG
Fiz-%'COLOURG6&%$RGBG$\"#5!\"\"\"\"!Fa[l-%+AXESLABELSG6%%\"AG%\"BG-%%F
ONTG6%%&TIMESG%'ITALICG\"#9-%*AXESTICKSG6%%(DEFAULTGF`\\l-Fh[l6$%'SYMB
OLG\"#7-%%VIEWG6$;Fa[l$\"\"\"Fa[lF`\\l" 1 2 0 1 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 }}}}{PARA 0 "" 0 "" {TEXT -1 114 "Thus the harmo
nic content is very small, even for quite large amplitudes.\n\nNow plo
t non-dimensionalized frequency " }{TEXT 287 1 "w" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 1 "/" }{XPPEDIT 18 0 "Om
ega;" "6#%&OmegaG" }{TEXT -1 18 " as a function of " }{TEXT 277 1 "A" 
}{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot(sqrt(
wfn),0..1,labels=[A,w],axesfont=[SYMBOL,12],labelfont=[TIMES,ITALIC,14
]);" }}{PARA 13 "" 1 "" {GLPLOT2D 272 165 165 {PLOTDATA 2 "6&-%'CURVES
G6$7gn7$\"\"!$\"+^iNDC!#57$$\"+v1h6o!#8$\"+[.++5!\"*7$$\"+N@Ki8!#7$\"+
#R,++\"F27$$\"+.K[V?F6$\"+KJ++5F27$$\"+qUkCFF6$\"+ob++5F27$$\"+0k'p3%F
6$\"+FD,+5F27$$\"+S&)G\\aF6$\"+rA-+5F27$$\"+5G$R<)F6$\"+4,0+5F27$$\"+3
x&)*3\"!#6$\"+\"3*3+5F27$$\"+ilyM;FU$\"+B/?+5F27$$\"+;arz@FU$\"+%Gc.+
\"F27$$\"+\"y%*z7$FU$\"+-Ot+5F27$$\"+XTFwSFU$\"+abC,5F27$$\"+\"z_\"4iF
U$\"+U!))G+\"F27$$\"+S&phN)FU$\"+JbA05F27$$\"+*=)H\\5F+$\"+Z%H#35F27$$
\"+[!3uC\"F+$\"+.Rh65F27$$\"+J$RDX\"F+$\"+83s:5F27$$\"+)R'ok;F+$\"+#R1
1-\"F27$$\"+1J:w=F+$\"+)Q9h-\"F27$$\"+3En$4#F+$\"+]bVK5F27$$\"+/RE&G#F
+$\"+$pX&Q5F27$$\"+D.&4]#F+$\"+bA-Y5F27$$\"+vB_<FF+$\"+`a:a5F27$$\"+v'
Hi#HF+$\"+x]di5F27$$\"+(*ev:JF+$\"+8cqq5F27$$\"+347TLF+$\"+%)4'43\"F27
$$\"+LY.KNF+$\"+(pO,4\"F27$$\"+\"o7Tv$F+$\"+4&f85\"F27$$\"+$Q*o]RF+$\"
+<Ux66F27$$\"+\"=lj;%F+$\"+TgqB6F27$$\"+V&R<P%F+$\"++faN6F27$$\"+Xh-'e
%F+$\"+'4#Q[6F27$$\"+R\"3Gy%F+$\"+$4\"fg6F27$$\"+.T1&*\\F+$\"+<')>u6F2
7$$\"+(RQb@&F+$\"+,'*z)=\"F27$$\"+=>Y2aF+$\"+CW)=?\"F27$$\"+yXu9cF+$\"
+b_R;7F27$$\"+\\y))GeF+$\"+8fyJ7F27$$\"+i_QQgF+$\"+:?AZ7F27$$\"+!y%3Ti
F+$\"+%)=]i7F27$$\"+O![hY'F+$\"+p.&)z7F27$$\"+#Qx$omF+$\"+4*pdH\"F27$$
\"+u.I%)oF+$\"+*\\+JJ\"F27$$\"+(pe*zqF+$\"+^44H8F27$$\"+C\\'QH(F+$\"+@
F(oM\"F27$$\"+8S8&\\(F+$\"+c'zQO\"F27$$\"+0#=bq(F+$\"+l<$>Q\"F27$$\"+2
s?6zF+$\"+eD%)*R\"F27$$\"+IXaE\")F+$\"+Z*f)=9F27$$\"+l*RRL)F+$\"+JPUP9
F27$$\"+`<.Y&)F+$\"+a&\\mX\"F27$$\"+8tOc()F+$\"+.#[fZ\"F27$$\"+\\Qk\\*
)F+$\"+fw(Q\\\"F27$$\"+p0;r\"*F+$\"+I)[Y^\"F27$$\"+lxGp$*F+$\"+=,UL:F2
7$$\"+!oK0e*F+$\"+$\\FOb\"F27$$\"+<5s#y*F+$\"+M$[Jd\"F27$%%FAILGFh]l-%
'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6%%\"AG%\"wG-%%FONTG6%%&TI
MESG%'ITALICG\"#9-%*AXESTICKSG6%%(DEFAULTGF^_l-Ff^l6$%'SYMBOLG\"#7-%%V
IEWG6$;F($\"\"\"F(F^_l" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 }}}}{PARA 0 "" 0 "" {TEXT -1 78 "Thus the frequency increa
ses with amplitude in the case of the hard potential." }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 24 "Case of 'soft' potential" }}{PARA 0 "" 0 
"" {TEXT -1 23 "Do a specific value of " }{TEXT 278 1 "A" }{TEXT -1 
92 " to see which of the three roots of the cubic equation is real (th
e only physical solution):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
44 "allvalues(eval(ELsolns,\{sigma=-1,A=0.001\}));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6%<$/%\"BG$!+WVw.c!#5/%#w2G$\"*/#QxeF(<$/F%$!+.#3+D'!#?
/F*$\"++])*****F(<$/F%$\"+F-Z)f&F(/F*$\"*Jms)eF(" }}}{PARA 0 "" 0 "" 
{TEXT -1 142 "In this case we have three real solutions. However, we s
ee from the above example that only the second root is close to the li
near value when " }{TEXT 279 1 "A" }{TEXT -1 52 " is small. Therefore \+
the other two must be spurious." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 21 "Now define functions " }{TEXT 280 3 "Bfn
" }{TEXT -1 6 ", and " }{TEXT 283 3 "wfn" }{TEXT -1 8 " giving " }
{TEXT 281 1 "B" }{TEXT -1 5 " and " }{TEXT 284 2 "w2" }{TEXT -1 33 ", \+
respectively, as a function of " }{TEXT 282 1 "A" }{TEXT -1 1 ":" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "Bfn := Aval -> eval(B,allva
lues(eval(ELsolns,\{A=Aval,sigma=-1\}))[2]);\nwfn := Aval -> eval(w2,a
llvalues(eval(ELsolns,\{A=Aval,sigma=-1\}))[2]);" }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$BfnGR6#%%AvalG6\"6$%)operatorG%&arr
owGF(-%%evalG6$%\"BG&-%*allvaluesG6#-F-6$%(ELsolnsG<$/%\"AG9$/%&sigmaG
!\"\"6#\"\"#F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$wfnGR6#%%Aval
G6\"6$%)operatorG%&arrowGF(-%%evalG6$%#w2G&-%*allvaluesG6#-F-6$%(ELsol
nsG<$/%\"AG9$/%&sigmaG!\"\"6#\"\"#F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Plot amplitude of third harmonic, \+
" }{TEXT 285 1 "B" }{TEXT -1 6 ", vs. " }{TEXT 286 1 "A" }{TEXT -1 1 "
:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plot(Bfn,0.001..0.87,la
bels=[A,B],axesfont=[SYMBOL,12],labelfont=[TIMES,ITALIC,14]);" }}
{PARA 13 "" 1 "" {GLPLOT2D 266 180 180 {PLOTDATA 2 "6&-%'CURVESG6$7Y7$
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$F17$$\"+g9FBlFN$!+DV*[j$F17$$\"+A`41nFN$!+3%=`<%F17$$\"+l!R[)oFN$!+;!
4hy%F17$$\"+*>n>2(FN$!+<2')GbF17$$\"+KQ>_sFN$!+QLyijF17$$\"+'f,lV(FN$!
+icPftF17$$\"+)>$G>wFN$!+](Q/_)F17$$\"+20C(y(FN$!+&zL6x*F17$$\"+Q&Q(zz
FN$!+jJNZ6FN7$$\"+0[#e1)FN$!+T^%[B\"FN7$$\"+r5\">:)FN$!+]=\"3L\"FN7$$
\"+()ppV#)FN$!+'f1TW\"FN7$$\"+-H[N$)FN$!+by8r:FN7$$\"+NPLB%)FN$!+*yo*3
<FN7$$\"+nX=6&)FN$!+`D)z'=FN7$$\"+E%)Qe&)FN$!+H!3_'>FN7$$\"+%G#f0')FN$
!+98kt?FN7$$\"+Uhz_')FN$!+zJR(>#FN7$$\"#()!\"#$!+(=KUM#FN-%'COLOURG6&%
$RGBG$\"#5!\"\"\"\"!Fe]l-%+AXESLABELSG6%%\"AG%\"BG-%%FONTG6%%&TIMESG%'
ITALICG\"#9-%*AXESTICKSG6%%(DEFAULTGFd^l-F\\^l6$%'SYMBOLG\"#7-%%VIEWG6
$;F(Fi\\lFd^l" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}
{PARA 0 "" 0 "" {TEXT -1 97 "Unlike the case of the hard potential, th
e harmonic content increases rapidly once the amplitude " }{TEXT 292 
1 "A" }{TEXT -1 64 "  gets above about 0.6.\n\nNow plot non-dimensiona
lized frequency " }{TEXT 288 1 "w" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "o
mega;" "6#%&omegaG" }{TEXT -1 1 "/" }{XPPEDIT 18 0 "Omega;" "6#%&Omega
G" }{TEXT -1 18 " as a function of " }{TEXT 289 1 "A" }{TEXT -1 3 "  :
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "plot(sqrt(wfn),0..0.87,l
abels=[A,w],axesfont=[SYMBOL,12],labelfont=[TIMES,ITALIC,14]);" }}
{PARA 13 "" 1 "" {GLPLOT2D 295 167 167 {PLOTDATA 2 "6&-%'CURVESG6$7in7
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w2mbNF5$\"+z^!*****F+7$$\"+L5)3u%F5$\"+H9$)****F+7$$\"+]:K6rF5$\"+72i*
***F+7$$\"+l?w\"[*F5$\"+.dK****F+7$$\"+5VEA9!#6$\"+xF[)***F+7$$\"+8CN'
*=FY$\"+#e-t***F+7$$\"+haN@FFY$\"+$QWW***F+7$$\"+2&eja$FY$\"+'zj0***F+
7$$\"+HH'>S&FY$\"+5T4y**F+7$$\"+,v')psFY$\"+EhHg**F+7$$\"+\\U*)G\"*FY$
\"+'*RLP**F+7$$\"+-]C&3\"F+$\"+p6M6**F+7$$\"+=#4PE\"F+$\"+u_iz)*F+7$$
\"+mrF[9F+$\"+eYkT)*F+7$$\"+-KDK;F+$\"+eX\\)z*F+7$$\"+p^\\@=F+$\"+iq`[
(*F+7$$\"+(fz\"))>F+$\"+;zz*p*F+7$$\"+$yEe<#F+$\"+q*\\&R'*F+7$$\"+mWCk
BF+$\"+EyBt&*F+7$$\"+<)>ea#F+$\"+Mxs.&*F+7$$\"+Jwq5FF+$\"+^pvN%*F+7$$
\"+!>vn!HF+$\"+BG\")[$*F+7$$\"+J,(G2$F+$\"+Va&)p#*F+7$$\"+L!ygE$F+$\"+
#)=vr\"*F+7$$\"+k(*4PMF+$\"+Z<<z!*F+7$$\"+3xtCOF+$\"+cxBr*)F+7$$\"+.MT
.QF+$\"+sV6i))F+7$$\"+ZF%)*)RF+$\"+)4r9u)F+7$$\"+\"3V5;%F+$\"+kPSC')F+
7$$\"+rdqXVF+$\"+uP=\"\\)F+7$$\"+1%=v`%F+$\"+o;#\\M)F+7$$\"+p=\\/ZF+$
\"+Vo$3@)F+7$$\"+%yF[)[F+$\"+\"p+(e!)F+7$$\"+IC8r]F+$\"+^FD$*yF+7$$\"+
y^R`_F+$\"+d\\$Hs(F+7$$\"+gPuHaF+$\"+Zh&)\\vF+7$$\"+$z[bi&F+$\"+4%>xM(
F+7$$\"+C$)[,eF+$\"+@,vcrF+7$$\"+E8M*)fF+$\"+8ViUpF+7$$\"+nScfhF+$\"+!
HP!RnF+7$$\"+&[icM'F+$\"+/\\_0lF+7$$\"+$fm2_'F+$\"+DF![F'F+7$$\"+R3!Qq
'F+$\"+6j^@gF+7$$\"+r-v#)oF+$\"+?:2hdF+7$$\"+UP4qqF+$\"+8CotaF+7$$\"+r
x_]sF+$\"+!zc6=&F+7$$\"+Ew/NuF+$\"+??'Q'[F+7$$\"+k&R!=wF+$\"+?v#y_%F+7
$$\"+],>'y(F+$\"+oF\"f>%F+7$$\"+(p4*yzF+$\"+\\'H%zPF+7$$\"+d.G^\")F+$
\"+UP6gLF+7$$\"+LM1N$)F+$\"+d&HI$GF+7$$\"+g`,B%)F+$\"+4\"[9`#F+7$$\"+'
Gn4^)F+$\"+G#eM<#F+7$$\"+VO[0')F+$\"+&3d*o;F+7$$\"#()!\"#$\"+%plhb(FY-
%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6%%\"AG%\"wG-%%FONTG6%%&T
IMESG%'ITALICG\"#9-%*AXESTICKSG6%%(DEFAULTGF[`l-Fc_l6$%'SYMBOLG\"#7-%%
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0 }}}}{PARA 0 "" 0 "" {TEXT -1 27 "In this case the frequency " }
{TEXT 291 9 "decreases" }{TEXT -1 40 " with amplitude, going to zero a
t about " }{TEXT 290 1 "A" }{TEXT -1 8 " = 0.87." }}}}{SECT 0 {PARA 4 
"" 0 "" {TEXT -1 0 "" }{TEXT 294 56 "Problem 3.5.2:  Phase-space plot \+
in soft potential case " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "Statem
ent of problem (see Chapter 5.3 for discussion)" }}{PARA 0 "" 0 "" 
{TEXT 32 410 "Find the Hamiltonian corresponding to the Lagrangian of \+
the anharmonic \noscillator in Problem~\\ref\{sec:AHosc\}.  Sketch con
tours of $H(x,p)$ in \nthe soft potential case for (a) $H < m\\omega_0
^2 l_0^2/8$ and (b) $H > \nm\\omega_0^2 l_0^2/8$.  Indicate representa
tive trajectories of the \nsystem in phase-space, showing the directio
n of travel.  Try to \ninclude representatives of each qualitatively d
istinct orbit." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "Hamiltonian Pl
ots" }}{PARA 0 "" 0 "" {TEXT -1 24 "Work in units such that " }
{XPPEDIT 18 0 "omega[0] = 1,l[0] = 1.;" "6$/&%&omegaG6#\"\"!\"\"\"/&%
\"lG6#F'$\"\"\"F'" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "H := (
x,p) ->  p^2/2 + x^2/2 - x^4/2;\nxdot:=unapply(diff(H(x,p),p),x,p);\np
dot:=unapply(-diff(H(x,p),x),x,p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"HGR6$%\"xG%\"pG6\"6$%)operatorG%&arrowGF),(*$)9%\"\"#\"\"\"#\"\"\"
F1*$)9$F1F2F3*$)F7\"\"%F2#!\"\"F1F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%xdotGR6$%\"xG%\"pG6\"6$%)operatorG%&arrowGF)9%F)F)F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%pdotGR6$%\"xG%\"pG6\"6$%)operatorG%
&arrowGF),&9$!\"\"*$)F.\"\"$\"\"\"\"\"#F)F)F)" }}}{PARA 0 "" 0 "" 
{TEXT -1 69 "First plot the potential, showing that there is confineme
nt only for " }{TEXT 296 1 "x" }{TEXT -1 35 " in the range of about -0
.8 to 0.8:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "plot(H(x,0),x=
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