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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "thetadot:=(theta,v)-
>(v^2-cos(theta))/v;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "vdo
t:=(theta,v)->-sin(theta)-R*v^2;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 97 "with(DEtools):\nglider_eqs:=\{D(theta)(t)=thetadot(th
eta(t),v(t)) , D(v)(t)=vdot(theta(t),v(t)) \};\n" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 7 "R:=0.3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 179 "DEplot( glider_eqs, [theta(t),v(t)], t=-5..5, [[theta(0)=0,v(
0)=1], [theta(0)=0,v(0)=2], [theta(0)=0,v(0)=1.2]], linecolor=[black,g
reen,blue],stepsize=.1,\ntheta=-Pi..Pi,v=.2..4 );" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 172 "DEplot( glider_eqs, [theta(t),v(t)], t=0..5
, \n[seq([theta(0)=-Pi,v(0)=.4+0.1*i],i=1..20)], \nlinecolor=[seq(COLO
R(HUE,0.05*i),i=1..20)],stepsize=.1,\ntheta=-Pi..Pi,v=.2..4 );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "R:=.05;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 173 "DEplot( glider_eqs, [theta(t),v(t)], t=0..9, \n[seq(
[theta(0)=-Pi,v(0)=.2+0.15*i],i=1..20)], \nlinecolor=[seq(COLOR(HUE,0.
05*i),i=1..20)],stepsize=.1,\ntheta=-Pi..Pi,v=.2..4 );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "R:=.1;   DEplot( glider_eqs, [thet
a(t),v(t)], t=0..9, \n[seq([theta(0)=-Pi,v(0)=.2+0.15*i],i=1..20)], \n
linecolor=[seq(COLOR(HUE,0.05*i),i=1..20)],stepsize=.1,\ntheta=-Pi..Pi
,v=.2..4 );\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "R:=.1;   \+
DEplot( glider_eqs, [theta(t),v(t)], t=0..15, \n[seq([theta(0)=-Pi,v(0
)=.2+0.15*i],i=1..20)], arrows=none, \nlinecolor=[seq(COLOR(HUE,0.05*i
),i=1..20)],stepsize=.1,\ntheta=-Pi..3*Pi,v=.2..4 );\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "R:=.1;   DEplot( glider_eqs, [thet
a(t),v(t)], t=0..15, \n[seq([theta(0)=-Pi,v(0)=.2+0.15*i],i=1..20)], s
cene=[t,theta],\nlinecolor=[seq(COLOR(HUE,0.05*i),i=1..20)],stepsize=.
1,\ntheta=-Pi..3*Pi,v=.2..4 );\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "glider_eqs;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 7 "R:=0.1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "g := dsolv
e(\{op(glider_eqs),theta(0)=0,v(0)=2\},\n \{theta(t),v(t)\}, type=nume
ric, output=listprocedure);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "g(1);g(2);g(2.0000234);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "g(1)[2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "rhs(g(1)[2])
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "points:=[seq( [rhs(g(0
.01*i)[2]), rhs(g(0.01*i)[3]) ] ,i=1..500)]:" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 14 "Here dt = 0.01" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "plot(points);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "poin
ts:=[ seq( [rhs(g(0.05*i)[2]), rhs(g(0.05*i)[3]) ] ,i=1..1000) ]:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Here dt=0.05" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 13 "plot(points);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 231 "xh_poi
nts:=[[0,100]];\nfor i from 1 to nops(points) do\n  x:=xh_points[i][1]
;\n  h:=xh_points[i][2];\n  dx:=points[i][2]*cos(points[i][1])*0.1;\n \+
 dh:=points[i][2]*sin(points[i][1])*0.1;\n  xh_points:=[op(xh_points),
[x+dx,h+dh]];  \nod:\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "
plot(xh_points);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x:='x';
 h:='h';R:='R';" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "xdot:=(t
heta,v)->v*cos(theta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "h
dot:=(theta,v)->v*sin(theta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 69 "new_eqs:=\{D(x)(t)=xdot(theta(t),v(t)), D(h)(t)=hdot(theta(t),v(
t)) \};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "aug_eqs:=\{ op(glider_eq
s), op(new_eqs)\};" }}{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 2 "x;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "h;" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 226 "R:=
.1;   DEplot( aug_eqs, [theta,v,x,h], t=0..15, \n[seq([theta(0)=-Pi,v(
0)=.2+0.15*i,x(0)=0,h(0)=100],i=1..20)],\nlinecolor=[seq(COLOR(HUE,0.0
5*i),i=1..20)], stepsize=.1,\ntheta=-Pi..Pi,v=.2..4,x=0..15,h=95..100 \+
, scene=[x,h]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "v;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "theta;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 178 "R:=0.1;   DEplot( aug_eqs, [theta,v,x,h], t=0
..50, \n[[theta(0)=0,v(0)=1.2,x(0)=0,h(0)=100]],\nlinecolor=[red], ste
psize=.1,\ntheta=-Pi..Pi,v=.2..4,x=0..50,h=95..102 , scene=[x,h]);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "R:=.1;   DEplot( aug_eqs, \+
[theta,v,x,h], t=0..500, \n[[theta(0)=0,v(0)=2,x(0)=0,h(0)=100]],\nlin
ecolor=[red], stepsize=.04,\ntheta=-Pi..Pi,v=.2..4,x=0..100,h=91..103 \+
, scene=[x,h]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "One more exper
iment of that same sort:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
194 "R:=0.04;\n\nDEplot( aug_eqs, [theta(t),v(t),x(t),h(t)], t=0..100,
 \n[[v(0)=2,theta(0)=0,x(0)=0,h(0)=100]], \nlinecolor=[blue] ,stepsize
=.04, scene=[t,v],\ntheta=0..6*Pi,v=0..3, h=98..102, x=0..100 );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "DEplot( aug_eqs, [theta(t),
v(t),x(t),h(t)], t=0..100, \n[[v(0)=2.3,theta(0)=0,x(0)=0,h(0)=100]], \+
\nlinecolor=[blue] ,stepsize=.04, scene=[x,h],\ntheta=0..6*Pi,v=.2..3,
 h=98..102, x=0..50 );\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{MARK "43" 0 }{VIEWOPTS 1 1 0 1 1 1803 }