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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(plots):with(DEt
ools):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords \+
has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Equilibri
a points." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "
Recall that the differential equations give \"moving directions\" for \+
a particle, by telling the particle how fast its x and y coordinates h
ave to be changing at any given point:\n" }{XPPEDIT 18 0 "diff(x,t) = \+
f(x,y);" "6#/-%%diffG6$%\"xG%\"tG-%\"fG6$F'%\"yG" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{XPPEDIT 18 0 "diff(y,t) = g(x,y);" "6#/-%%diffG6$%\"yG
%\"tG-%\"gG6$%\"xGF'" }{TEXT -1 78 "\n\nThus we can find out where a p
article could be stopped by finding out where " }{XPPEDIT 18 0 "diff(x
,t);" "6#-%%diffG6$%\"xG%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dif
f(y,t);" "6#-%%diffG6$%\"yG%\"tG" }{TEXT -1 362 " are both zero. Put d
ifferently, assume that a particle could be stopped at a point (x0,y0)
. Then plugging x(t)=x0 and y(t) =y0 into out system of diffential equ
ations we get a solution exactly when both f(x0,y0)=0 and g(x0,y0)=0. \+
Thus we can find the equilibria points (or rather the equilibria solut
ions) by solving ther system of equations f(x,y)=0, g(x,y)=0." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "eq:=\{diff(x(t),t)=x(t)^2-y(t)^2, d
iff(y(t),t)=2*x(t)*y(t)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG<$
/-%%diffG6$-%\"xG6#%\"tGF-,&*$)F*\"\"#\"\"\"F2*$)-%\"yGF,F1F2!\"\"/-F(
6$F5F-,$*&F*F2F5F2F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "sol
ve(\{x^2-y^2=0,2*x*y=0\},\{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$
<$/%\"yG\"\"!/%\"xGF&F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "We can
 see that the origin is an equilibrium solution in a field plot." }
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "fieldplot([x^2-
y^2,2*x*y],x=-1..1,y=-1..1,scaling=constrained);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 48 "A solution to this might look like the following" 
}{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "DEplot(eq,[x(t
),y(t)],t=-5..5,[[x(0)=.5,y(0)=.5]],stepsize=.1,scaling=constrained);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 394 "There
 are different types of equilibrium points. The type of equilibium poi
nt determines how a particle will move near that point. We will not gi
ve a mathematical calssification of different types of equilibrium poi
nts but we will give examples of the common critical points in the pla
ne so that, armed with maple, you can classify a critical point that b
elongs to one of the common categories." }{MPLTEXT 1 0 1 "\n" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "A CENTER." }{MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "eqcenter:=\{diff(x(t
),t)=-2*y(t),diff(y(t),t)=3*x(t)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%)eqcenterG<$/-%%diffG6$-%\"xG6#%\"tGF-,$-%\"yGF,!\"#/-F(6$F/F-,$F*
\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "FPcenter:=fieldplo
t([-2*y,3*x],x=-1..1,y=-1..1):display(FPcenter);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 177 "PLcenter:=DEplot(eqcenter,[x(t),y(t)],t=-10.
.10,[[x(0)=0,y(0)=.2],[x(0)=0,y(0)=.4],[x(0)=0,y(0)=.6],[x(0)=0,y(0)=.
8],[x(0)=0,y(0)=1]],stepsize=.1,arrows=none):display(PLcenter);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "display(\{FPcenter,PLcenter
\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 14 "A stable NODE." }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "eqnode:=\{diff(x(t),t)=-y(t)-2*x(t),diff(y(t),t)=x(t)
\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'eqnodeG<$/-%%diffG6$-%\"yG6#
%\"tGF--%\"xGF,/-F(6$F.F-,&F*!\"\"*&\"\"#\"\"\"F.F7F4" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "FPnode:=fieldplot([-y-2*x,x],x=-1..
1,y=-1..1):display(FPnode);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
297 "PLnode:=DEplot(eqnode,[x(t),y(t)],t=-3..10,[[x(0)=0,y(0)=.2],[x(0
)=0,y(0)=.4],[x(0)=0,y(0)=.6],[x(0)=0,y(0)=.8],[x(0)=0,y(0)=1],[x(0)=0
,y(0)=-.2],[x(0)=0,y(0)=-.4],[x(0)=0,y(0)=-.6],[x(0)=0,y(0)=-.8],[x(0)
=0,y(0)=-1]],stepsize=.1,arrows=none,view=[-1..1,-1..1]):display(PLnod
e,view=[-1..1,-1..1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "d
isplay(\{FPnode,PLnode\},view=[-1..1,-1..1]);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "An unstable NODE." }{MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "A stable FOCUS (or SPIRA
L)." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "eqspiral:=\{diff(x(t),t)=
-y(t)-.4*x(t),diff(y(t),t)=x(t)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%)eqspiralG<$/-%%diffG6$-%\"yG6#%\"tGF--%\"xGF,/-F(6$F.F-,&F*!\"\"*&$
\"\"%F4\"\"\"F.F8F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "FPsp
iral:=fieldplot([-y-.4*x,x],x=-1..1,y=-1..1):display(FPspiral);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "PLspiral:=DEplot(eqspiral,[
x(t),y(t)],t=-3..10,[[x(0)=0,y(0)=.2],[x(0)=0,y(0)=.4],[x(0)=0,y(0)=.6
],[x(0)=0,y(0)=.8],[x(0)=0,y(0)=1]],stepsize=.1,arrows=none,view=[-1..
1,-1..1]):display(PLspiral);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 48 "display(\{FPspiral,PLspiral\},view=[-1..1,-1..1]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "An unstable F
OCUS (or SPIRAL)." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "A SADDLE" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "eqsad
dle:=\{diff(x(t),t)=-x(t)+.5*y(t),diff(y(t),t)=y(t)+.5*x(t)\};" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)eqsaddleG<$/-%%diffG6$-%\"yG6#%\"tG
F-,&F*\"\"\"*&$\"\"&!\"\"F/-%\"xGF,F/F//-F(6$F4F-,&F4F3*&F1F/F*F/F/" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "FPsaddle:=fieldplot([-x+.5
*y,y+.5*x],x=-1..1,y=-1..1):display(FPsaddle);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 490 "PLsaddle:=DEplot(eqsaddle,[x(t),y(t)],t=-5..5
,[[x(0)=0,y(0)=.2],[x(0)=0,y(0)=.4],[x(0)=0,y(0)=.6],[x(0)=0,y(0)=.8],
[x(0)=0,y(0)=1],[x(0)=0,y(0)=-.2],[x(0)=0,y(0)=-.4],[x(0)=0,y(0)=-.6],
[x(0)=0,y(0)=-.8],[x(0)=0,y(0)=-1],[x(0)=.2,y(0)=.5],[x(0)=.4,y(0)=.5]
,[x(0)=.6,y(0)=.5],[x(0)=.8,y(0)=.5],[x(0)=1,y(0)=.5],[x(0)=-.2,y(0)=-
.5],[x(0)=-.4,y(0)=-.5],[x(0)=-.6,y(0)=-.5],[x(0)=-.8,y(0)=-.5],[x(0)=
-1,y(0)=-.5]],stepsize=.1,arrows=none,view=[-1..1,-1..1]):display(PLsa
ddle,view=[-1..1,-1..1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
48 "display(\{FPsaddle,PLsaddle\},view=[-1..1,-1..1]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Do rigid body
 example." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "PHASE SPACE" }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "The motion of a block \+
hanging from a spring." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{MARK "52 0 0" 0 }{VIEWOPTS 1 1 0 2 1 1805 1 
1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }