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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 31 "Megszoritott h\341romtes
t probl\351ma" }}{PARA 257 "" 0 "" {TEXT 256 25 "esettanulm\341ny: Tad
pole (~" }{TEXT 257 6 "ebihal" }{TEXT 258 17 ") \351s Horseshoe (~" }
{TEXT 259 7 "l\363patk\363" }{TEXT 260 8 ") p\341ly\341k" }}{PARA 257 
"" 0 "" {TEXT -1 34 "Bartha Ferenc, SZTE, 2002. m\341rcius" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart;with(plots):" }}}{EXCHG 
{PARA 0 "" 0 "" {MPLTEXT 0 21 11 "Definici\363k:" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {XPPEDIT 18 0 "s1=sqrt(r1^2+r^2+2*r*r1*cos(phi)):\ns2=
sqrt(r2^2+r^2-2*r*r2*cos(phi)):\nv=-omega^2*(r1+r2)^2*(r2/s1+r1/s2):\n
vr = diff(v,r); vp := diff(v,p)" "6#C'/%#s1G-%%sqrtG6#,(*$%#r1G\"\"#\"
\"\"*$%\"rGF,F-**F,F-F/F-F+F--%$cosG6#%$phiGF-F-/%#s2G-F'6#,(*$%#r2GF,
F-*$F/F,F-**F,F-F/F-F;F--F26#F4F-!\"\"/%\"vG,$*(%&omegaGF,,&F+F-F;F-F,
,&*&F;F-F%F@F-*&F+F-F6F@F-F-F@/%#vrG-%%diffG6$FBF/>%#vpG-FM6$FB%\"pG" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "s1:=(r,p) ->sqrt(r1^2+r^
2+2*r*r1*cos(p)):\ns2:=(r,p) ->sqrt(r2^2+r^2-2*r*r2*cos(p)):\nv :=(r,p
) ->-omega^2*(r1+r2)^2*(r2/s1(r,p)+r1/s2(r,p)):" }{TEXT -1 1 "\n" }
{MPLTEXT 1 0 83 "subf:=\{r=r(t),p=p(t)\}:\nvr:=subs(subf,diff(v(r,p),r
)):vp:=subs(subf,diff(v(r,p),p)):" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 
0 21 17 "Egyenletrendszer:" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }{XPPEDIT 19 1 "eqs:=\ndiff(r(t),t)=Pr(t),\ndiff(p(t
),t)=Pp(t)/r(t)^2-omega,\ndiff(Pr(t),t)=Pp(t)^2/r(t)^3-vr,\ndiff(Pp(t)
,t)=-vp:" "6#>%$eqsG6&/-%%diffG6$-%\"rG6#%\"tGF--%#PrG6#F-/-F(6$-%\"pG
6#F-F-,&*&-%#PpG6#F-\"\"\"*$-F+6#F-\"\"#!\"\"F<%&omegaGFA/-F(6$-F/6#F-
F-,&*&-F:6#F-F@*$-F+6#F-\"\"$FAF<%#vrGFA/-F(6$-F:6#F-F-,$%#vpGFA" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "fcns := \{r(t), p(t),Pr(t),Pp(t)\}:
" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 7 "Adatok:" }}{PARA 0 "> " 
0 "" {XPPEDIT 19 1 "d:=1.0:omega:=1:a:=0.000953875: " "6#C%>%\"dG-%&Fl
oatG6$\"#5!\"\">%&omegaG\"\"\">%\"aG-F'6$\"'vQ&*!\"*" }{TEXT -1 25 " ~
 Nap-Jupiter t\366megar\341ny" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 
21 23 "A k\351t test koordin\341t\341i:" }{MPLTEXT 1 0 145 "\nr2:=d/(1
+a):r1:=d-r2:\nbodies:=[-r1,0],[r2,0]:\nppb:=plot([bodies],style=point
,\n          symbol=[circle,circle],symbolsize=[30,10],colour=brown):
" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 19 "Lagrange helyzetek:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 376 "l3:=d^2*(r2/(r-r1)^2+r1/(r+r2)^2)-
r:L3:=fsolve(l3,r=r1..infinity):\nl2:=d^2*(r2/(r+r1)^2+r1/(r-r2)^2)-r:
L2:=fsolve(l2,r=r2..infinity):\nl1:=d^2*(r2/(r+r1)^2-r1/(r-r2)^2)-r:L1
:=fsolve(l1,r=r1..r2):\nr0:=sqrt(d^2+r2^2-2*r2*d*cos(Pi/3)):\np0:=eval
f(arcsin(d/r0*sin(Pi/3))):\nx0:=r0*cos(p0):y0:=r0*sin(p0):\nL[i](r,phi
),[L[1]=(L1,0),L[2]=(L2,0),L[3]=(L3,Pi),L[4]=(r0,p0),L[5]=(r0,-p0)];" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "pnts:=[[L1,0]],[[L2,0]],[[-L3,0]
],[[x0,y0]],[[x0,-y0]]:\nppl:=plot([pnts],style=point,symbol=diamond,s
ymbolsize=15,colour=blue):" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 
62 "Tadpole ebihalak a stabil pont k\366rny\351k\351n\nKezdeti felt
\351tel I.:\n" }{MPLTEXT 1 0 207 "ini0:=r(0)=r0*1.1,p(0)=p0,Pr(0)=0,Pp
(0)=r0^2*omega:\nran:=0..20:\ndsol0:=dsolve([eqs,ini0],numeric,fcns,ra
nge=ran):\npl0:=odeplot(dsol0,[r(t)*cos(p(t)),r(t)*sin(p(t))],ran,labe
ls=[x,y]):\ndisplay([pl0,ppl,ppb]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 49 "\311s a Horseshoe pat
k\363k.....\nKezdeti felt\351tel II.:\n" }{MPLTEXT 1 0 23 "X0:=1.0275:
Y0:=0.04033:" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }{MPLTEXT 1 0 212 "
ini0:=r(0)=X0,p(0)=Pi,Pr(0)=0,Pp(0)=X0^2*(omega-Y0/X0):\nran:=0..200:
\ndsol0:=dsolve([eqs,ini0],numeric,fcns,range=ran):\npl0:=odeplot(dsol
0,[r(t)*cos(p(t)),r(t)*sin(p(t))],ran,labels=[x,y]):\ndisplay([pl0,ppl
,ppb]);" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 23 "Kezdeti felt\351t
el III.:\n" }{MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }{MPLTEXT 1 0 241 "X0:=0
.976683:Y0:=-0.0611916:\nini0:=r(0)=X0,p(0)=Pi,Pr(0)=0,Pp(0)=X0^2*(ome
ga-Y0/X0):\nran:=0..185:\ndsol0:=dsolve([eqs,ini0],numeric,fcns,range=
ran):\npl0:=odeplot(dsol0,[r(t)*cos(p(t)),r(t)*sin(p(t))],ran,labels=[
x,y]):\ndisplay([pl0,ppl,ppb]);" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 
21 21 "Kezdeti felt\351tel IV.:" }{MPLTEXT 1 0 243 "\nX0:=0.766650:Y0:
=-0.51230159:\nini0:=r(0)=X0,p(0)=Pi,Pr(0)=0,Pp(0)=X0^2*(omega-Y0/X0):
\nran:=0..177:\ndsol0:=dsolve([eqs,ini0],numeric,fcns,range=ran):\npl0
:=odeplot(dsol0,[r(t)*cos(p(t)),r(t)*sin(p(t))],ran,labels=[x,y]):\ndi
splay([pl0,ppl,ppb]);" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 21 "Kez
deti felt\351tel V.:\n" }{MPLTEXT 1 0 238 "X0:=1.09137:Y0:=0.1430196:
\nini0:=r(0)=X0,p(0)=Pi,Pr(0)=0,Pp(0)=X0^2*(omega-Y0/X0):\nran:=0..83:
\ndsol0:=dsolve([eqs,ini0],numeric,fcns,range=ran):\npl0:=odeplot(dsol
0,[r(t)*cos(p(t)),r(t)*sin(p(t))],ran,labels=[x,y]):\ndisplay([pl0,ppl
,ppb]);" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 22 "Kezdeti felt\351t
el VI.:\n" }{MPLTEXT 1 0 238 "X0:=1.10137:Y0:=0.1535425:\nini0:=r(0)=X
0,p(0)=Pi,Pr(0)=0,Pp(0)=X0^2*(omega-Y0/X0):\nran:=0..65:\ndsol0:=dsolv
e([eqs,ini0],numeric,fcns,range=ran):\npl0:=odeplot(dsol0,[r(t)*cos(p(
t)),r(t)*sin(p(t))],ran,labels=[x,y]):\ndisplay([pl0,ppl,ppb]);" }}}
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