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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 79 "Rigid Body Dynamics of a \nSpinn
ing Truncated Cone \nImmersed in a Pressure Field" }}{PARA 19 "" 0 "" 
{TEXT -1 15 "Marc A. Murison" }}{PARA 259 "" 0 "" {TEXT -1 74 "Astrono
mical Applications Department\nU.S. Naval Observatory\nWashington, DC
" }}{PARA 260 "" 0 "" {TEXT -1 62 "murison@riemann.usno.navy.mil\nhttp
://aa.usno.navy.mil/murison/" }}{PARA 257 "" 0 "" {TEXT -1 14 "17 Marc
h, 1998" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 42 "Euler Equations of Mot
ion for a Rigid Body" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "restart;
\nalias( I=I, psi=psi(t), theta=theta(t), phi=phi(t),\n       Omega[x]
=Omega[x](t), Omega[y]=Omega[y](t), Omega[z]=Omega[z](t),\n       Omeg
a[phi]=Omega[phi](t), Omega[psi]=Omega[psi](t),\n       Omega[theta]=O
mega[theta](t) ):" "6#C$%(restartG-%&aliasG6,/%\"IGF)/%$psiG-F+6#%\"tG
/%&thetaG-F06#F./%$phiG-F46#F./&%&OmegaG6#%\"xG-&F96#F;6#F./&F96#%\"yG
-&F96#FC6#F./&F96#%\"zG-&F96#FK6#F./&F96#F4-&F96#F46#F./&F96#F+-&F96#F
+6#F./&F96#F0-&F96#F06#F." }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new
 definition for norm" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "In the f
rame of reference attached to the rotating body (the body frame), the \+
equations of motion are" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Eule
rEqs := mat(I[x]*diff(Omega[x],t)+(I[z]-I[y])*Omega[y]*Omega[z]-K[x],I
[y]*diff(Omega[y],t)+(I[x]-I[z])*Omega[x]*Omega[z]-K[y],I[z]*diff(Omeg
a[z],t)+(I[y]-I[x])*Omega[x]*Omega[y]-K[z]);" "6#>%)EulerEqsG-%$matG6%
,(*&&%\"IG6#%\"xG\"\"\"-%%diffG6$&%&OmegaG6#F-%\"tGF.F.*(,&&F+6#%\"zGF
.&F+6#%\"yG!\"\"F.&F36#F=F.&F36#F:F.F.&%\"KG6#F-F>,(*&&F+6#F=F.-F06$&F
36#F=F5F.F.*(,&&F+6#F-F.&F+6#F:F>F.&F36#F-F.&F36#F:F.F.&FD6#F=F>,(*&&F
+6#F:F.-F06$&F36#F:F5F.F.*(,&&F+6#F=F.&F+6#F-F>F.&F36#F-F.&F36#F=F.F.&
FD6#F:F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)EulerEqsG-%'matrixG6#7%
7#,(*&&%\"IG6#%\"xG\"\"\"-%%diffG6$&%&OmegaGF.%\"tGF0F0*(,&&F-6#%\"zGF
0&F-6#%\"yG!\"\"F0&F5F=F0&F5F:F0F0&%\"KGF.F?7#,(*&F<F0-F26$F@F6F0F0*(,
&F,F0F9F?F0F4F0FA\"\"\"F0&FCF=F?7#,(*&F9F0-F26$FAF6F0F0*(,&F<F0F,F?F0F
4FKF@FKF0&FCF:F?" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "latex(Euler
Eqs,\"d:/dynamics/precession/EulerEqs.tex\");" "6#-%&latexG6$%)EulerEq
sGQDd:/dynamics/precession/EulerEqs.tex6\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 6 "where " }{XPPEDIT 18 0 "I[x], I[y], I[z]" "6%&%\"IG6#%\"xG
&F$6#%\"yG&F$6#%\"zG" }{TEXT -1 51 " are the principal moments of iner
tia of the body, " }{XPPEDIT 18 0 "Omega[x], Omega[y], Omega[z]" "6%&%
&OmegaG6#%\"xG&F$6#%\"yG&F$6#%\"zG" }{TEXT -1 70 " are the angular vel
ocities of the body about the principal axes, and " }{XPPEDIT 18 0 "K[
x], K[y], K[z]" "6%&%\"KG6#%\"xG&F$6#%\"yG&F$6#%\"zG" }{TEXT -1 53 " a
re the components of the torque acting on the body." }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 29 "Eulerian Angle Transformation" }}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 15 "Rotation Matrix" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 165 "Construct a rotation matrix that transforms coordinate
s from the fixed frame (X,Y,Z) to the body frame (x,y,z).  First, rota
te the coordinates ccw around the Z axis." }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "r1 := matrix([[cos(phi), sin(phi), 0], [-sin(phi), cos(
phi), 0], [0, 0, 1]]):" "6#>%#r1G-%'matrixG6#7%7%-%$cosG6#%$phiG-%$sin
G6#F-\"\"!7%,$-F/6#F-!\"\"-F+6#F-F17%F1F1\"\"\"" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 36 "Next, rotate ccw around the X' axis." }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "r2 := matrix([[1, 0, 0], [0, cos(psi), \+
sin(psi)], [0, -sin(psi), cos(psi)]]):" "6#>%#r2G-%'matrixG6#7%7%\"\"
\"\"\"!F+7%F+-%$cosG6#%$psiG-%$sinG6#F07%F+,$-F26#F0!\"\"-F.6#F0" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Next, rotate ccw around the Z'' ax
is." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "r3 := matrix([[cos(theta
),sin(theta),0],[-sin(theta),cos(theta),0],[0,0,1]]):" "6#>%#r3G-%'mat
rixG6#7%7%-%$cosG6#%&thetaG-%$sinG6#F-\"\"!7%,$-F/6#F-!\"\"-F+6#F-F17%
F1F1\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Now combine the rot
ations into a single rotation matrix." }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "R := (p,q,r)->evalm( subs( theta=r, eval(r3) ) &* \n   \+
                  subs( psi=q,   eval(r2) ) &* \n                     \+
subs( phi=p,   eval(r1) ) ):" "6#>%\"RGR6%%\"pG%\"qG%\"rG7\"6$%)operat
orG%&arrowG6\"-%&evalmG6#-%#&*G6$-F36$-%%subsG6$/%&thetaGF)-%%evalG6#%
#r3G-F86$/%$psiGF(-F=6#%#r2G-F86$/%$phiGF'-F=6#%#r1GF.F.F." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 44 "Hence, we have the coordinate transformat
ion" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "mat(x,y,z) = R(phi,psi,t
heta) &* mat(X,Y,Z);" "6#/-%$matG6%%\"xG%\"yG%\"zG-%#&*G6$-%\"RG6%%$ph
iG%$psiG%&thetaG-F%6%%\"XG%\"YG%\"ZG" }}{PARA 12 "" 1 "" {XPPMATH 20 "
6#/-%'matrixG6#7%7#%\"xG7#%\"yG7#%\"zG-%#&*G6$-F%6#7%7%,&*&-%$cosG6#%&
thetaG\"\"\"-F86#%$phiGF;F;*(-%$sinGF9F;-F86#%$psiGF;-FAF=F;!\"\",&*&F
7\"\"\"FEFIF;*(F@FIFBFIF<FIF;*&F@FI-FAFCF;7%,&*&F@FIF<FIFF*(F7FIFBFIFE
FIFF,&*&F@FIFEFIFF*(F7FIFBFIF<FIF;*&F7FIFLFI7%*&FLFIFEFI,$*&FLFIF<FIFF
FB-F%6#7%7#%\"XG7#%\"YG7#%\"ZG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 
1 "latex(%,\"d:/dynamics/precession/FixedToBody.tex\");" "6#-%&latexG6
$%\"%GQGd:/dynamics/precession/FixedToBody.tex6\"" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 52 "The diagram below illustrates the three rotations.
  " }}{PARA 0 "" 0 "" {OLE 1 8300 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:
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C:?j::W;j>>:<J<jPv:jJP:OmDLj_AJ<<J:f?Cj<j>vY:=:<jD>:qQ]yyFJ:^QB:b>c\\_
;:>ACB:f?KB:C:^:CB:>;:O:<J<J;F:JQ:F;;Z:^:f?KJ?jJPJ><J]`pmPQsPPJ@Mvvk>^
:f?Cj;>Z:>:JBAj:<J:DJ:DZJVdsgg\\;jPF;CB:>XKj>>:_c<;X:QW:=Z:Ng:nVM:_;or
:=B:FIKR:MJ:N@Vs>F:FHIp>JSjpaj:<J:FH:_;SX==B:FI:_;aY>=B:NGf<VY;fy<Z:JB
AJ:B:D:cTTP@:JT^:f?KJ<JrG;MJ:N@fp<F:;JP>FMB:N@Ff>F:^?><<jw;<:[V:b:DZJV
DDpq:f?MJ<Jr?;MJ:N`DJ@Uj:>:<J::::::::::::::::::5:" }}}}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 41 "Angular Velocity Vector in the Body Frame" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 288 "The angular velocity vector may b
e decomposed into components along each of the rotation axes used to c
onstruct the transformation matrix.  If we transform those components \+
to the body frame, then we can express the angular velocity vector in \+
the body frame in terms of the Euler angles (" }{XPPEDIT 18 0 "phi,psi
,theta" "6%%$phiG%$psiG%&thetaG" }{TEXT -1 21 ").  The component of " 
}{XPPEDIT 18 0 "Omega" "6#%&OmegaG" }{TEXT -1 63 " along the first rot
ation axis, as viewed in the body frame, is" }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "omega[phi] := evalm(`&*`(R(0,psi,theta),mat(0,0,1)))*
diff(phi,t);" "6#>&%&omegaG6#%$phiG*&-%&evalmG6#-%#&*G6$-%\"RG6%\"\"!%
$psiG%&thetaG-%$matG6%F2F2\"\"\"\"\"\"-%%diffG6$F'%\"tGF9" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>&%&omegaG6#%$phiG*&-%'matrixG6#7%7#*&-%$sinG6#
%&thetaG\"\"\"-F06#%$psiGF37#*&-%$cosGF1F3F4\"\"\"7#-F:F5F3-%%diffG6$F
'%\"tGF3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "The component along t
he Y' axis is, in the body frame," }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
19 1 "omega[psi] := evalm(`&*`(R(0,0,theta),mat(1,0,0)))*diff(psi,t);
" "6#>&%&omegaG6#%$psiG*&-%&evalmG6#-%#&*G6$-%\"RG6%\"\"!F2%&thetaG-%$
matG6%\"\"\"F2F2\"\"\"-%%diffG6$F'%\"tGF8" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%&omegaG6#%$psiG*&-%'matrixG6#7%7#-%$cosG6#%&thetaG7#
,$-%$sinGF0!\"\"7#\"\"!\"\"\"-%%diffG6$F'%\"tGF9" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 51 "Finally, the component along the Z'' axis is simpl
y" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "omega[theta] := mat( 0, 0,
 1 ) * diff(theta,t);" "6#>&%&omegaG6#%&thetaG*&-%$matG6%\"\"!F,\"\"\"
\"\"\"-%%diffG6$F'%\"tGF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&omega
G6#%&thetaG*&-%'matrixG6#7%7#\"\"!F-7#\"\"\"F0-%%diffG6$F'%\"tGF0" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Hence, we have the angular velocit
y vector in the body frame," }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "
Omega = mat( seq( evalm(omega[phi])[i,1] + \n                  evalm(o
mega[psi])[i,1] + \n                  evalm(omega[theta])[i,1], i=1..3
 ) );" "6#/%&OmegaG-%$matG6#-%$seqG6$,(&-%&evalmG6#&%&omegaG6#%$phiG6$
%\"iG\"\"\"\"\"\"&-F.6#&F16#%$psiG6$F5\"\"\"F7&-F.6#&F16#%&thetaG6$F5
\"\"\"F7/F5;\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&OmegaG-%
'matrixG6#7%7#,&*(-%%diffG6$%$phiG%\"tG\"\"\"-%$sinG6#%&thetaGF1-F36#%
$psiGF1F1*&-F-6$F8F0F1-%$cosGF4F1F17#,&*(F,\"\"\"F<FAF6FAF1*&F:FAF2FA!
\"\"7#,&*&F,FA-F=F7F1F1-F-6$F5F0F1" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "latex(%,\"d:/dynamics/precession/OmegaBodyFrame.tex\");
" "6#-%&latexG6$%\"%GQJd:/dynamics/precession/OmegaBodyFrame.tex6\"" }
}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Obody := convert(rhs(%),vector
):" "6#>%&ObodyG-%(convertG6$-%$rhsG6#%\"%G%'vectorG" }}}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 35 "Rigid Body Equations \227 General Case" }
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Equations of Motion" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 51 "Substituting back into the Euler equation
s, we find" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "xyz := [x, y, z]:
" "6#>%$xyzG7%%\"xG%\"yG%\"zG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "MI := \+
[I[x], I[y], I[z]]:" "6#>%#MIG7%&%\"IG6#%\"xG&F'6#%\"yG&F'6#%\"zG" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(seq(Omega[xyz[k]](t) = Obod
y[k],k = 1 .. 3),evalm(EulerEqs));" "6#-%%subsG6$-%$seqG6$/-&%&OmegaG6
#&%$xyzG6#%\"kG6#%\"tG&%&ObodyG6#F1/F1;\"\"\"\"\"$-%&evalmG6#%)EulerEq
sG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7#,(*&&%\"IG6#%\"x
G\"\"\"-%%diffG6$,&*(-F06$%$phiG%\"tGF.-%$sinG6#%&thetaGF.-F96#%$psiGF
.F.*&-F06$F>F7F.-%$cosGF:F.F.F7F.F.*(,&&F+6#%\"zGF.&F+6#%\"yG!\"\"F.,&
*(F4\"\"\"FBFOF<FOF.*&F@FOF8FOFLF.,&*&F4FO-FCF=F.F.-F06$F;F7F.F.F.&%\"
KGF,FL7#,(*&FIF.-F06$FMF7F.F.*(,&F*F.FFFLF.F2F.FQFOF.&FWFJFL7#,(*&FFF.
-F06$FQF7F.F.*(,&FIF.F*FLF.F2FOFMFOF.&FWFGFL" }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "[ seq( collect( %[i,1]/MI[i],\n                [diff(
theta,t,t), diff(psi,t,t), diff(phi,t,t), \n                 sin(psi),
 diff(psi,t), sin(theta), cos(theta), diff],\n                simplify
 ),\n       i = 1 .. 3) ];" "6#7#-%$seqG6$-%(collectG6%*&&%\"%G6$%\"iG
\"\"\"\"\"\"&%#MIG6#F.!\"\"7*-%%diffG6%%&thetaG%\"tGF:-F76%%$psiGF:F:-
F76%%$phiGF:F:-%$sinG6#F=-F76$F=F:-FB6#F9-%$cosG6#F9F7%)simplifyG/F.;
\"\"\"\"\"$" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%,,*&-%%diffG6$%$psiG-
%\"$G6$%\"tG\"\"#\"\"\"-%$cosG6#%&thetaGF/F/*(-F'6$%$phiGF*F/-%$sinGF2
F/-F96#F)F/F/*(,&*&*(,&&%\"IG6#%\"zGF/&FB6#%\"yG!\"\"F/-F1F;F/)-F'6$F7
F-F.\"\"\"FM&FB6#%\"xG!\"\"F/*&*(,(FNF/FAF/FEFHF/-F'6$F3F-F/FKF/FMFNFQ
F/F/F0FMF:FMF/*(,&*&*(FIFM,(FNF/FAFHFEF/F/FKFMFMFNFQF/*&*&FTFMFUFMFMFN
FQFHF/F8FM-F'6$F)F-F/F/*&&%\"KGFOFMFNFQFH,,*&F&FMF8FMFH*(F5FMF0FMF:FMF
/*(,&*&*(,&FNF/FAFHF/FIFMFJFMFMFEFQF/*&*(,(FEFHFNF/FAFHF/FUFMFKFMFMFEF
QF/F/F8FMF:FMF/*(,&*&*(FIFMFenFMFKFMFMFEFQF/*&*&FgoFMFUFMFMFEFQF/F/F0F
MFhnFMF/*&&F\\oFFFMFEFQFH,.-F'6$F3F*F/*&F5FMFIFMF/*&*,,&FEFHFNF/F/FJFM
F0FMF8FM)F:F.FMFMFAFQFH*(,(*&*(FfpFMFKFM)F8F.FMFMFAFQF/*&*(FfpFMFKFM)F
0F.FMFMFAFQFHFKFHF/FhnFMF:FMF/*&**FfpFMF0FMF8FM)FhnF.FMFMFAFQF/*&&F\\o
FCFMFAFQFH" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "tmp := %:" "6#>%$
tmpG%\"%G" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 33 "Clean up the third \+
equation a bit" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "algsubs(cos(th
eta)*sin(theta) = A,tmp[3]);" "6#-%(algsubsG6$/*&-%$cosG6#%&thetaG\"\"
\"-%$sinG6#F+F,%\"AG&%$tmpG6#\"\"$" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "a
lgsubs(sin(psi)*diff(psi,t) = B/diff(phi,t),%):" "6#-%(algsubsG6$/*&-%
$sinG6#%$psiG\"\"\"-%%diffG6$F+%\"tGF,*&%\"BGF,-F.6$%$phiGF0!\"\"%\"%G
" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "algsubs(I[x]-I[y] = C*I[z],%);" "6#
-%(algsubsG6$/,&&%\"IG6#%\"xG\"\"\"&F)6#%\"yG!\"\"*&%\"CGF,&F)6#%\"zGF
,%\"%G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,*(,(*&*(,&&%\"IG6#%\"yG!\"
\"&F*6#%\"xG\"\"\"F1-%%diffG6$%$phiG%\"tGF1)-%$sinG6#%&thetaG\"\"#\"\"
\"F=&F*6#%\"zG!\"\"F1*&*(F(F=F2F=)-%$cosGF:F<F=F=F>FAF-F2F-F1-F36$%$ps
iGF6F1-F96#FIF1F1*&*(F(F=%\"AGF1,&*$)FGF<F=F-*&)F2F<F=)FJF<F=F1F1F=F>F
AF--F36$F;-%\"$G6$F6F<F1*&-F36$F5FWF1-FFFKF1F1*&&%\"KGF?F=F>FAF-" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#,,*(%\"AG\"\"\",&*$)-%%diffG6$%$psiG%
\"tG\"\"#\"\"\"!\"\"*&)-F+6$%$phiGF.F/F0)-%$sinG6#F-F/F0F&F&%\"CGF&F1*
&*&%\"BGF&,(*()-%$cosG6#%&thetaGF/F0F;F0&%\"IG6#%\"zGF&F1*()-F9FDF/F0F
;F0FFF0F&FFF1F&F0FF!\"\"F&-F+6$FE-%\"$G6$F.F/F&*&-F+6$F6FPF&-FCF:F&F&*
&&%\"KGFHF0FFFMF1" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Collect( %
, \n         [ diff(theta,t,t), diff(phi,t,t),\n           A, B, C, di
ff(phi,t)^2, cos(theta) ], \n         simplify, alg ):" "6#-%(CollectG
6&%\"%G7)-%%diffG6%%&thetaG%\"tGF,-F)6%%$phiGF,F,%\"AG%\"BG%\"CG*$-F)6
$F/F,\"\"#-%$cosG6#F+%)simplifyG%$algG" }}{PARA 0 "" 0 "" {XPPEDIT 19 
1 "subs(A = cos(theta)*sin(theta), B = sin(psi)*diff(psi,t)*diff(phi,t
), C = (I[x]-I[y])/I[z], %);" "6#-%%subsG6&/%\"AG*&-%$cosG6#%&thetaG\"
\"\"-%$sinG6#F,F-/%\"BG*(-F/6#%$psiGF--%%diffG6$F6%\"tGF--F86$%$phiGF:
F-/%\"CG*&,&&%\"IG6#%\"xGF-&FC6#%\"yG!\"\"F-&FC6#%\"zGFI%\"%G" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#,,-%%diffG6$%&thetaG-%\"$G6$%\"tG\"\"#
\"\"\"*&-F%6$%$phiGF(F--%$cosG6#%$psiGF-F-*&**,&*&,&!\"\"F-*$)F2F,\"\"
\"F-F-)-F%6$F1F+F,F>F-*$)-F%6$F5F+F,F>F-F-,&&%\"IG6#%\"yGF;&FH6#%\"xGF
-F--F36#F'F--%$sinGFOF-F>&FH6#%\"zG!\"\"F-**,&*&*&,&*$)FNF,F>!\"#F-F-F
-FFF>F>FRFUF-F;F-F-F@F--FQF4F-FDF-F-*&&%\"KGFSF>FRFUF;" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "tmp[3] := %:" "6#>&%$tmpG6#\"\"$%\"%G" 
}}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "BodyEqs := eval(tmp):" "6#>%
(BodyEqsG-%%evalG6#%$tmpG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "la
tex(BodyEqs,\"d:/dynamics/precession/RigidBodyEqs.tex\");" "6#-%&latex
G6$%(BodyEqsGQHd:/dynamics/precession/RigidBodyEqs.tex6\"" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "mat(BodyEqs);" "6#-%$matG6#%(BodyEqsG" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7#,,*&-%%diffG6$%$psiG
-%\"$G6$%\"tG\"\"#\"\"\"-%$cosG6#%&thetaGF3F3*(-F+6$%$phiGF.F3-%$sinGF
6F3-F=6#F-F3F3*(,&*&*(,&&%\"IG6#%\"zGF3&FF6#%\"yG!\"\"F3-F5F?F3)-F+6$F
;F1F2\"\"\"FQ&FF6#%\"xG!\"\"F3*&*(,(FRF3FEF3FIFLF3-F+6$F7F1F3FOF3FQFRF
UF3F3F4FQF>FQF3*(,&*&*(FMFQ,(FRF3FEFLFIF3F3FOFQFQFRFUF3*&*&FXFQFYFQFQF
RFUFLF3F<FQ-F+6$F-F1F3F3*&&%\"KGFSFQFRFUFL7#,,*&F*FQF<FQFL*(F9FQF4FQF>
FQF3*(,&*&*(,&FRF3FEFLF3FMFQFNFQFQFIFUF3*&*(,(FIFLFRF3FEFLF3FYFQFOFQFQ
FIFUF3F3F<FQF>FQF3*(,&*&*(FMFQFinFQFOFQFQFIFUF3*&*&F\\pFQFYFQFQFIFUF3F
3F4FQF\\oFQF3*&&F`oFJFQFIFUFL7#,,-F+6$F7F.F3*&F9FQFMFQF3*&**,&*&,&FLF3
*$)FMF2FQF3F3FNFQF3*$)F\\oF2FQF3F3,&FIFLFRF3F3F4FQF<FQFQFEFUF3**,&*&*&
,&*$)F4F2FQ!\"#F3F3F3FcqFQFQFEFUF3FLF3F3FOFQF>FQF\\oFQF3*&&F`oFGFQFEFU
FL" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "Rigid Body Equations \+
\227 Symmetric Top" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "General Mot
ion of a Symmetric Top" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 19 "Equatio
ns of Motion" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "SymTop := subs(I
[x] = I[y],I[y] = I[xy],BodyEqs);" "6#>%'SymTopG-%%subsG6%/&%\"IG6#%\"
xG&F*6#%\"yG/&F*6#F/&F*6#%#xyG%(BodyEqsG" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%'SymTopG7%,,*&-%%diffG6$%$psiG-%\"$G6$%\"tG\"\"#\"\"\"-%$cosG6
#%&thetaGF1F1*(-F)6$%$phiGF,F1-%$sinGF4F1-F;6#F+F1F1*(,&*&*(,&&%\"IG6#
%\"zGF1&FD6#%#xyG!\"\"F1-F3F=F1)-F)6$F9F/F0\"\"\"FOFG!\"\"F1*&*(FCF1-F
)6$F5F/F1FMF1FOFGFPF1F1F2FOF<FOF1*(,&*&*(FKFO,&FGF0FCFJF1FMFOFOFGFPF1*
&*&FCFOFSFOFOFGFPFJF1F:FO-F)6$F+F/F1F1*&&%\"KG6#%\"xGFOFGFPFJ,,*&F(FOF
:FOFJ*(F7FOF2FOF<FOF1*(,&*&*(,&FGF1FCFJF1FKFOFLFOFOFGFPF1FQFJF1F:FOF<F
OF1*(FVFOF2FOFfnFOF1*&&Fjn6#%\"yGFOFGFPFJ,*-F)6$F5F,F1*&F7FOFKFOF1*(FM
FOF<FOFfnFOFJ*&&FjnFEFOFCFPFJ" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 
1 "SymTop := [seq(collect(simplify(SymTop[k],\{1-I[z]/I[xy] = beta\},[
I[z]]),[diff(psi,t,t), diff(phi,t,t), sin(theta), cos(theta), sin(psi)
, diff(psi,t), diff],factor),k = 1 .. 3)];" "6#>%'SymTopG7#-%$seqG6$-%
(collectG6%-%)simplifyG6%&F$6#%\"kG<#/,&\"\"\"\"\"\"*&&%\"IG6#%\"zGF6&
F96#%#xyG!\"\"F?%%betaG7#&F96#F;7)-%%diffG6%%$psiG%\"tGFI-FF6%%$phiGFI
FI-%$sinG6#%&thetaG-%$cosG6#FP-FN6#FH-FF6$FHFIFF%'factorG/F1;\"\"\"\"
\"$" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'SymTopG7%,,*&-%%diffG6$%$psi
G-%\"$G6$%\"tG\"\"#\"\"\"-%$cosG6#%&thetaGF1F1*(-F)6$%$phiGF,F1-%$sinG
F4F1-F;6#F+F1F1*(,&*(-F3F=F1,&F1F1%%betaGF1F1-F)6$F9F/F1F1*&,&!\"\"F1F
CF1F1-F)6$F5F/F1F1F1-F)6$F+F/F1F:\"\"\"F1*(,&*(FAFMFCF1)FDF0FMFH*(,&F1
F1FCFHF1FIFMFDFMF1F1F<FMF2FMF1*&&%\"KG6#%\"xGFM&%\"IG6#%#xyG!\"\"FH,,*
&F(FMF:FMFH*(F7FMF2FMF<FMF1*(,&FPF1*(FGFMFIFMFDFMF1F1F<FMF:FMF1*(F?FMF
KFMF2FMF1*&&FV6#%\"yGFMFYFgnFH,**&F7FMFAFMF1*(FDFMF<FMFKFMFH-F)6$F5F,F
1*&&FV6#%\"zGFM*&FY\"\"\"FG\"\"\"FgnF1" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "latex( SymTop[1], \"d:/dynamics/precession/SymTop1.tex
\" );\nlatex( SymTop[2], \"d:/dynamics/precession/SymTop2.tex\" );\nla
tex( SymTop[3], \"d:/dynamics/precession/SymTop3.tex\" );" "6#C%-%&lat
exG6$&%'SymTopG6#\"\"\"QCd:/dynamics/precession/SymTop1.tex6\"-F%6$&F(
6#\"\"#QCd:/dynamics/precession/SymTop2.texF,-F%6$&F(6#\"\"$QCd:/dynam
ics/precession/SymTop3.texF," }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 42 
"Conversion to a System of First-Order ODEs" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 87 "We may write the equations of motion as a system of first
-order differential equations." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 
1 "subslist := [diff(phi,t) = Omega[phi], diff(psi,t) = Omega[psi], di
ff(theta,t) = Omega[theta]]:" "6#>%)subslistG7%/-%%diffG6$%$phiG%\"tG&
%&OmegaG6#F*/-F(6$%$psiGF+&F-6#F2/-F(6$%&thetaGF+&F-6#F8" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "subs( subslist, SymTop ):" "6#-%%subsG6
$%)subslistG%'SymTopG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "solve(
 \{op(%)\}, \n       \{diff(Omega[phi],t), diff(Omega[psi],t), diff(Om
ega[theta],t)\} ):" "6#-%&solveG6$<#-%#opG6#%\"%G<%-%%diffG6$&%&OmegaG
6#%$phiG%\"tG-F-6$&F06#%$psiGF3-F-6$&F06#%&thetaGF3" }}}{EXCHG {PARA 
0 "" 0 "" {XPPEDIT 19 1 "collect( %, [I[xy],Omega[psi],sin(psi),Omega[
theta]], factor );" "6#-%(collectG6%%\"%G7&&%\"IG6#%#xyG&%&OmegaG6#%$p
siG-%$sinG6#F/&F-6#%&thetaG%'factorG" }}{PARA 12 "" 1 "" {XPPMATH 20 "
6#<%/-%%diffG6$&%&OmegaG6#%$phiG%\"tG,&*&*&,&*&,&\"\"\"F3%%betaG!\"\"F
3&F)6#%&thetaGF3F3*(-%$cosG6#%$psiGF3,&F3F3F4F3F3F(F3F5F3&F)6#F=F3\"\"
\"-%$sinGF<!\"\"F3*&,&*&-F;6#F8F3&%\"KG6#%\"yGF3F3*&&FK6#%\"xGF3-FCFIF
3F3FA*&FB\"\"\"&%\"IG6#%#xyG\"\"\"FDF3/-F&6$F6F,,&*&,&*(F(FAF4F3FBF3F5
*&,&*(,&F5F3F4F3F3F:FAF6FAF3*&F>FAF(FAF3FAFBFDF3F3F?FAF3*&,&*&&FK6#%\"
zGFAF^oFDF5*&*&FFF3F:FAFAFBFDF5FAFUFDF3/-F&6$F?F,,&*&,&*(F:FAF4FA)F(\"
\"#FAF3*(F^oFAF6FAF(FAF3F3FBFAF3*&,&*&FJFAFRFAF5*&FHFAFOFAF3FAFUFDF3" 
}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "foo := %:" "6#>%$fooG%\"%G" }
}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "select( has, foo, diff(Omega[p
hi],t) ):" "6#-%'selectG6%%$hasG%$fooG-%%diffG6$&%&OmegaG6#%$phiG%\"tG
" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "collect( op(%)*sin(psi), [I
[xy],Omega[psi],Omega[theta]], factor ):" "6#-%(collectG6%*&-%#opG6#%
\"%G\"\"\"-%$sinG6#%$psiGF+7%&%\"IG6#%#xyG&%&OmegaG6#F/&F66#%&thetaG%'
factorG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "foo := foo minus %% \+
union \{%\}:" "6#>%$fooG-%&unionG6$-%&minusG6$F$%#%%G<#%\"%G" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "select( has, foo, diff(Omega[the
ta],t) ):" "6#-%'selectG6%%$hasG%$fooG-%%diffG6$&%&OmegaG6#%&thetaG%\"
tG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "collect( sinfix(op(%)*sin
(psi),psi), \n         [I[xy],Omega[psi],Omega[phi],K[z]], factor ):" 
"6#-%(collectG6%-%'sinfixG6$*&-%#opG6#%\"%G\"\"\"-%$sinG6#%$psiGF.F27&
&%\"IG6#%#xyG&%&OmegaG6#F2&F96#%$phiG&%\"KG6#%\"zG%'factorG" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "foo := foo minus %% union \{%\};
" "6#>%$fooG-%&unionG6$-%&minusG6$F$%#%%G<#%\"%G" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$fooG<%/*&-%$sinG6#%$psiG\"\"\"-%%diffG6$&%&OmegaG6#%
$phiG%\"tGF,,&*&,&*&,&F,F,%%betaG!\"\"F,&F16#%&thetaGF,F,*(-%$cosGF*F,
,&F,F,F:F,F,F0F,F;F,&F16#F+F,F,*&,&*&-FA6#F>F,&%\"KG6#%\"yGF,F,*&&FK6#
%\"xGF,-F)FIF,F,\"\"\"&%\"IG6#%#xyG!\"\"F,/-F.6$FCF4,&*&,&*(F@FSF:F,)F
0\"\"#FSF,*(,&F;F,F:F,F,F<FSF0FSF,F,F(FSF,*&,&*&FJFSFRFSF;*&FHFSFOFSF,
FSFTFXF,/*&F(FS-F.6$F<F4F,,&*&,&*&,&*&)F@F[oFSF:FSF,F,F,F,F0FSF,*(F]oF
SF@FSF<FSF,F,FCFSF,*&,&*&*&F(FS&FK6#%\"zGF,FSF]oFXF;*&FFF,F@FSF;FSFTFX
F," }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "FirstOrderODEsK := [diff(
phi,t) = Omega[phi], diff(psi,t) = Omega[psi], diff(theta,t) = Omega[t
heta], op(select(has,foo,diff(Omega[phi],t))), op(select(has,foo,diff(
Omega[psi],t))), op(select(has,foo,diff(Omega[theta],t)))]:" "6#>%0Fir
stOrderODEsKG7(/-%%diffG6$%$phiG%\"tG&%&OmegaG6#F*/-F(6$%$psiGF+&F-6#F
2/-F(6$%&thetaGF+&F-6#F8-%#opG6#-%'selectG6%%$hasG%$fooG-F(6$&F-6#F*F+
-F<6#-F?6%FAFB-F(6$&F-6#F2F+-F<6#-F?6%FAFB-F(6$&F-6#F8F+" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "mat(FirstOrderODEsK);" "6#-%$matG6#%0Fi
rstOrderODEsKG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7#/-%%
diffG6$%$phiG%\"tG&%&OmegaG6#F,7#/-F*6$%$psiGF-&F/6#F57#/-F*6$%&thetaG
F-&F/6#F<7#/*&-%$sinG6#F5\"\"\"-F*6$F.F-FE,&*&,&*&,&FEFE%%betaG!\"\"FE
F=FEFE*(-%$cosGFDFE,&FEFEFMFEFEF.FEFNFEF6FEFE*&,&*&-FQ6#F<FE&%\"KG6#%
\"yGFEFE*&&FY6#%\"xGFE-FCFWFEFE\"\"\"&%\"IG6#%#xyG!\"\"FE7#/-F*6$F6F-,
&*&,&*(FPF[oFMFE)F.\"\"#F[oFE*(,&FNFEFMFEFEF=F[oF.F[oFEFEFBF[oFE*&,&*&
FXF[oFjnF[oFN*&FVF[oFgnF[oFEF[oF\\oF`oFE7#/*&FBF[o-F*6$F=F-FE,&*&,&*&,
&*&)FPFjoF[oFMF[oFEFEFEFEF.F[oFE*(F\\pF[oFPF[oF=F[oFEFEF6F[oFE*&,&*&*&
FBF[o&FY6#%\"zGFEF[oF\\pF`oFN*&FTFEFPF[oFNF[oF\\oF`oFE" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "latex(%,\"d:/dynamics/precession/FirstO
rderODEsK.tex\");" "6#-%&latexG6$%\"%GQKd:/dynamics/precession/FirstOr
derODEsK.tex6\"" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "Force-Free \+
Motion of a Symmetric Top" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "FFS
ymTop := subs( K[x]=0, K[y]=0, K[z]=0, SymTop );" "6#>%)FFSymTopG-%%su
bsG6&/&%\"KG6#%\"xG\"\"!/&F*6#%\"yGF-/&F*6#%\"zGF-%'SymTopG" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%)FFSymTopG7%,**&-%%diffG6$%$psiG-%\"$G6$%
\"tG\"\"#\"\"\"-%$cosG6#%&thetaGF1F1*(-F)6$%$phiGF,F1-%$sinGF4F1-F;6#F
+F1F1*(,&*(-F3F=F1,&F1F1%%betaGF1F1-F)6$F9F/F1F1*&,&!\"\"F1FCF1F1-F)6$
F5F/F1F1F1-F)6$F+F/F1F:\"\"\"F1*(,&*(FAFMFCF1)FDF0FMFH*(,&F1F1FCFHF1FI
FMFDFMF1F1F<FMF2FMF1,**&F(FMF:FMFH*(F7FMF2FMF<FMF1*(,&FPF1*(FGFMFIFMFD
FMF1F1F<FMF:FMF1*(F?FMFKFMF2FMF1,(*&F7FMFAFMF1*(FDFMF<FMFKFMFH-F)6$F5F
,F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "We notice that the third e
quation," }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "FFSymTop[3];" "6#&%
)FFSymTopG6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&-%%diffG6$%$p
hiG-%\"$G6$%\"tG\"\"#\"\"\"-%$cosG6#%$psiGF.F.*(-F&6$F(F,F.-%$sinGF1F.
-F&6$F2F,F.!\"\"-F&6$%&thetaGF)F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
17 "can be written as" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "FFSymT
op[3] = ('diff(diff(phi,t)*cos(psi)+diff(theta,t),t)');" "6#/&%)FFSymT
opG6#\"\"$.-%%diffG6$,&*&-F*6$%$phiG%\"tG\"\"\"-%$cosG6#%$psiGF2F2-F*6
$%&thetaGF1F2F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&-%%diffG6$%$ph
iG-%\"$G6$%\"tG\"\"#\"\"\"-%$cosG6#%$psiGF/F/*(-F'6$F)F-F/-%$sinGF2F/-
F'6$F3F-F/!\"\"-F'6$%&thetaGF*F/-F'6$,&*&F5\"\"\"F0FCF/-F'6$F>F-F/F-" 
}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "rhs(%)-lhs(%);" "6#,&-%$rhsG6
#%\"%G\"\"\"-%$lhsG6#F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Hence, " }{XPPEDIT 18 0 "diff(
phi,t)*cos(psi)+diff(theta,t) = const" "6#/,&*&-%%diffG6$%$phiG%\"tG\"
\"\"-%$cosG6#%$psiGF+F+-F'6$%&thetaGF*F+%&constG" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 97 "This is the projection of the angular velocity ont
o the symmetry axis.  Recall the components of " }{XPPEDIT 18 0 "Omega
" "6#%&OmegaG" }{TEXT -1 19 " in the body frame:" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "mat(Omega[x],Omega[y],Omega[z]) = convert(Obody,
matrix);" "6#/-%$matG6%&%&OmegaG6#%\"xG&F(6#%\"yG&F(6#%\"zG-%(convertG
6$%&ObodyG%'matrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%
7#&%&OmegaG6#%\"xG7#&F*6#%\"yG7#&F*6#%\"zG-F%6#7%7#,&*(-%%diffG6$%$phi
G%\"tG\"\"\"-%$sinG6#%&thetaGF@-FB6#%$psiGF@F@*&-F<6$FGF?F@-%$cosGFCF@
F@7#,&*(F;\"\"\"FKFPFEFPF@*&FIFPFAFP!\"\"7#,&*&F;FP-FLFFF@F@-F<6$FDF?F
@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "We see that the angular velo
city about the symmetry axis, " }{XPPEDIT 18 0 "Omega[z]" "6#&%&OmegaG
6#%\"zG" }{TEXT -1 68 ", is a constant of the motion.  We also have co
nservation of energy," }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "E = (I
[x]*Omega[x]^2+I[y]*Omega[y]^2+I[z]*Omega[z]^2)/2:" "6#/%\"EG*&,(*&&%
\"IG6#%\"xG\"\"\"*$&%&OmegaG6#F+\"\"#F,F,*&&F)6#%\"yGF,*$&F/6#F5\"\"#F
,F,*&&F)6#%\"zGF,*$&F/6#F=\"\"#F,F,F,\"\"#!\"\"" }}{PARA 0 "" 0 "" 
{XPPEDIT 19 1 "KE := %:" "6#>%#KEG%\"%G" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 31 "Substituting the components of " }{XPPEDIT 18 0 "Omega" "
6#%&OmegaG" }{TEXT -1 38 " in terms of the Euler angles, we have" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "collect(subs(seq(Omega[xyz[k]](t
) = Obody[k],k = 1 .. 3),KE),[diff, sin(psi)],factor);" "6#-%(collectG
6%-%%subsG6$-%$seqG6$/-&%&OmegaG6#&%$xyzG6#%\"kG6#%\"tG&%&ObodyG6#F4/F
4;\"\"\"\"\"$%#KEG7$%%diffG-%$sinG6#%$psiG%'factorG" }}{PARA 0 "" 0 "
" {XPPEDIT 19 1 "KE := %:" "6#>%#KEG%\"%G" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/%\"EG,**&,&*&,&*&&%\"IG6#%\"yG\"\"\")-%$cosG6#%&thetaG
\"\"#\"\"\"#F/F5*&&F,6#%\"xGF/)-%$sinGF3F5F6F7F/)-F>6#%$psiGF5F6F/*&&F
,6#%\"zGF/)-F2FAF5F6F7F/)-%%diffG6$%$phiG%\"tGF5F6F/*&,&*,F=F/F1F/,&F+
!\"\"F9F/F/F@F/-FK6$FBFNF/F/*(FDF6-FK6$F4FNF/FHF/F/F/FJF/F/*&,&*&F9F6F
0F6F7*&F+F6F<F6F7F/)FTF5F6F/*&FDF6)FWF5F6F7" }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "subslist := [diff(psi,t) = Omega[psi], diff(phi,t) = \+
Omega[phi], diff(theta,t) = Omega[theta]]:" "6#>%)subslistG7%/-%%diffG
6$%$psiG%\"tG&%&OmegaG6#F*/-F(6$%$phiGF+&F-6#F2/-F(6$%&thetaGF+&F-6#F8
" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(subslist,I[x]=I[y],I[y]=I[xy],
KE);" "6#-%%subsG6&%)subslistG/&%\"IG6#%\"xG&F)6#%\"yG/&F)6#F.&F)6#%#x
yG%#KEG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%\"EG,**&,&*&,&*&&%\"IG6#%
#xyG\"\"\")-%$cosG6#%&thetaG\"\"#\"\"\"#F/F5*&F+F6)-%$sinGF3F5F6F7F/)-
F;6#%$psiGF5F6F/*&&F,6#%\"zGF/)-F2F>F5F6F7F/)&%&OmegaG6#%$phiGF5F6F/**
FAF6&FH6#F4F/FEF/FGF/F/*&F)F6)&FH6#F?F5F6F/*&FAF6)FLF5F6F7" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "collect( simplify(%/I[xy]), [Omega[phi]
,Omega[psi],Omega[theta],I[xy]], simplify );" "6#-%(collectG6%-%)simpl
ifyG6#*&%\"%G\"\"\"&%\"IG6#%#xyG!\"\"7&&%&OmegaG6#%$phiG&F36#%$psiG&F3
6#%&thetaG&F-6#F/F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"EG\"\"\"&
%\"IG6#%#xyG!\"\",**&,(#\"\"\"\"\"#F0*$)-%$cosG6#%$psiGF1F&#!\"\"F1*&*
&&F(6#%\"zGF0F3F&F&F'F+F/F0)&%&OmegaG6#%$phiGF1F&F0*&**F<F&&FA6#%&thet
aGF0F4F0F@F0F&F'F+F0*$)&FA6#F7F1F&F/*&*&F<F&)FFF1F&F&F'F+F/" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "collect(algsubs(I[z] = I[xy]*(1-
beta),%),[Omega],factor);" "6#-%(collectG6%-%(algsubsG6$/&%\"IG6#%\"zG
*&&F+6#%#xyG\"\"\",&\"\"\"F2%%betaG!\"\"F2%\"%G7#%&OmegaG%'factorG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"EG\"\"\"&%\"IG6#%#xyG!\"\",**&)&
%&OmegaG6#%$phiG\"\"#F&,&#\"\"\"F3F6*&)-%$cosG6#%$psiGF3F&%%betaGF6#!
\"\"F3F6F6**,&F?F6F=F6F6F9F6&F06#%&thetaGF6F/F6F?*$)&F06#F<F3F&F5*&,&F
5F6F=F>F6)FBF3F&F6" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "KE := %:
" "6#>%#KEG%\"%G" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 26 "Steady Prece
ssion Solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Suppose we look \+
for a solution such that " }{XPPEDIT 18 0 "psi" "6#%$psiG" }{TEXT -1 
27 " is constant.  Then we have" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
19 1 "eval( subs( diff(psi,t)=0, FFSymTop ) );" "6#-%%evalG6#-%%subsG6
$/-%%diffG6$%$psiG%\"tG\"\"!%)FFSymTopG" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#7%,&*(-%%diffG6$%$phiG-%\"$G6$%\"tG\"\"#\"\"\"-%$sinG6#%&thetaGF
/-F16#%$psiGF/F/*(,&*(-%$cosGF5F/%%betaGF/)-F'6$F)F-F.\"\"\"!\"\"*(,&F
/F/F<FAF/-F'6$F3F-F/F>F/F/F/F4F@-F;F2F/F/,&*(F&F@FFF@F4F@F/*(,&F9F/*(,
&FAF/F<F/F/FDF@F>F@F/F/F4F@F0F@F/,&*&F&F@F:F@F/-F'6$F3F*F/" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 48 "The first two equations can be combined t
o yield" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "collect( (%[1]*sin(t
heta) + %[2]*cos(theta))/sin(psi), \n         [diff], simplify ) = 0;
" "6#/-%(collectG6%*&,&*&&%\"%G6#\"\"\"\"\"\"-%$sinG6#%&thetaGF.F.*&&F
+6#\"\"#F.-%$cosG6#F2F.F.F.-F06#%$psiG!\"\"7#%%diffG%)simplifyG\"\"!" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$%$phiG-%\"$G6$%\"tG\"\"#
\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Hence, solutions with " 
}{XPPEDIT 18 0 "psi=const" "6#/%$psiG%&constG" }{TEXT -1 30 " force a \+
constant precession, " }{XPPEDIT 18 0 "diff(phi,t) = const" "6#/-%%dif
fG6$%$phiG%\"tG%&constG" }{TEXT -1 45 ".  From the third equation, we \+
then see that " }{XPPEDIT 18 0 "diff(theta,t) = const" "6#/-%%diffG6$%
&thetaG%\"tG%&constG" }{TEXT -1 11 " as well.  " }}}{EXCHG {PARA 0 "" 
0 "" {XPPEDIT 19 1 "subs( diff(phi,t,t)=0, %%[1] ):" "6#-%%subsG6$/-%%
diffG6%%$phiG%\"tGF+\"\"!&%#%%G6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "solve(%,diff(phi,t)):" "6#-%&solveG6$%\"%G-%%diffG6$%$p
hiG%\"tG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "diff(phi,t) = %[2];
" "6#/-%%diffG6$%$phiG%\"tG&%\"%G6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%%diffG6$%$phiG%\"tG,$*&*&-F%6$%&thetaGF(\"\"\",&!\"\"F/%%beta
GF/F/\"\"\"*&-%$cosG6#%$psiG\"\"\"F2\"\"\"!\"\"F1" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "normal( subs( beta = (I[xy]-I[z])/I[xy], % ) );
" "6#-%'normalG6#-%%subsG6$/%%betaG*&,&&%\"IG6#%#xyG\"\"\"&F.6#%\"zG!
\"\"F1&F.6#F0F5%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$%$p
hiG%\"tG,$*&*&-F%6$%&thetaGF(\"\"\"&%\"IG6#%\"zGF/\"\"\"*&-%$cosG6#%$p
siG\"\"\",&&F16#%#xyG!\"\"F0F/\"\"\"!\"\"F?" }}}}}}{SECT 1 {PARA 3 "" 
0 "" {TEXT -1 54 "A Spinning Truncated Cone Embedded in a Pressure Fie
ld" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Consider a truncated cone wi
th small and large radii " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 5 " \+
and " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 24 " and cone opening ang
le " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 67 ".  Let the cent
er of mass reside on the (local) z axis, a distance " }{XPPEDIT 18 0 "
h" "6#%\"hG" }{TEXT -1 119 " below the smaller flat surface (the top s
urface of the truncated cone).  Then the equation for the conical surf
ace is " }{XPPEDIT 18 0 "tan*alpha = (r-a)/(h-z):" "6#/*&%$tanG\"\"\"%
&alphaGF&*&,&%\"rGF&%\"aG!\"\"F&,&%\"hGF&%\"zGF,F," }{TEXT -1 5 ", or \+
" }{XPPEDIT 19 1 "x^2+y^2-(a+(h-z)*tan(alpha))^2 = 0:" "6#/,(*$%\"xG\"
\"#\"\"\"*$%\"yG\"\"#F(*$,&%\"aGF(*&,&%\"hGF(%\"zG!\"\"F(-%$tanG6#%&al
phaGF(F(\"\"#F3\"\"!" }{TEXT -1 2 ". " }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "latex(%,\"d:/dynamics/precession/ConeEq
uation.tex\");" "6#-%&latexG6$%\"%GQHd:/dynamics/precession/ConeEquati
on.tex6\"" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Conical Coordinates
" }}{EXCHG {PARA 0 "" 0 "" {OLE 1 16952 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;
:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::fyyyyyqyyyYJ<JyK=j=J>j>J?j?J@j@>:W:YJ:><<j
BJCjCJDjDJEjEJFjFJGjGJHjHJIjIJyky;::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::_lqvGcMJ<j\\nGcMsFI>:::::::JEf:yyyxIF:>Z=B:
<:;`:Z@[::JjtO@Lj_]rXAj;J:H:<:=ja^GE=;:::::::::N;?R:yyyyyy?Z:vYxY:B:::
::::Yi:B:N:Y<>D_mlVH[KRJ:<:::::::>=?jyyyyyI:;Z::::::j<>Z:^unZ:::::::::
::::yay=J:B::::::::F:;J;>:A:C:E:GJ:v:>;F;N;;j?Jyky;:::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::F:Dj:<Z:F:>::
e=Ar:B:>R<:Tn:f;VbIocCWb;oikn_PWd>syyYew@;:::D=l<HJ:H^icU``]SRl`=[icVL
`ccU@`nSSbETamCSDM;r:A:;Z:M:J]<Z^Z>OaV_fsog[ggcWfm_ejOh_Wf_WhmGgiogVWb
IoC:ry:;:::\\w:bPr:;BKd]VCS`ah[gSTH`mCQ<alSSH`_sVHacsUt`VsJZcsR@`f;x`l
CS``hSRLa_[h:L`lG:PZF[;^\\<J:<Z:M:J]HZ:BB>Z:GsAHhQxbwH:KdN?eDITi=JDHlr
Xs]Jr>Z:Z;>:;B::Z]D?x[ZycJA^dQac?\\\\@uBrD[CFuV@HL\\dMh<:;::jPaPJF@>:A
hBAia[Jl`Qq:;BKmdM=TMMtJaLKF@>:ihBOdV[JrDql:>ZJOd?Wb?WdxG@_KS>Zj?:CECc
B^;k[xPq`pk@PpHnqxPsLpntpjLqlHQQTqsL`=wC:J:FJ^=V:;B:s:];e;g\\:f<V:V`AF
tK>\\:>Z:^b;_d;CKo[[Z<>:L[:B:Z<BQsTKeDMEdNsDK=DO^fskpJh:ZVsUW=rUZn;ryx
yyAKGD:<Z:M:J]<:;ryvyZfnTo]RxbwHZZpqmF_RxbwD:c:J:>::::::::::::::::::::
:::::::::::::::::::::::::jysyA:CJ:F:vYxY;J:J[@=gsG^pnj:F;HJgd;QJ?@:;B:
EJ:V[<Z:f:^[<fj@Ft?f:F[<vyyyy:f:FZ<fsGiSCZ:NZ;N:EB:F:jP<:?T:A:K;[B:<J>
j:<j:>:;::::jykyyY:<jdP?yA:::^:f_;j><Z:Ny<j<B:;:?K<jPF:W:Cs:EjAvh@Vr?v
h@VRYjAv;vHMJ:NY>Z::N>jPN:IZ:^y<F:<J:J<jPV:C:[Y:;J<:=j<j>D:f:FZ<>Z:B:C
:U;M:oAAB:>:N>jP>:J;F:K:Es:?jNLjNMj_PjwMJ<>:UK;^:>X;j<j>:j:vyyAZ:^:N:?
j<j::UC:NB>:K;[J:<J>j:<j:>:;::::jywY[CF>`jj;HJ:V:>@<JRj;j;J<jP@J:F[:Jw
[:>Z:>:N>jPF:?]:^>Yk;DjmDj<DjiDJ?DJeDjCDjaDJJDJ^DJRDjZDJ[DjWDjeDJUDjqD
jRDJ>EJQDJKEJPDjXEjODJfEjODjsEJPDJAHJQDjMHjRDJYHJUDjcHjWDJmHjZDJuHJ^DJ
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:^:f?I:F^?jv^\\?vd?^:f?G:F;yAj:Z:^:>;:O:<J<J;F:JQ:F;;Z:^:f?I:F^?JvN[?N
e:^Z:jPn:j>vY:=:<J<J>:N;B:C:?j::W;j>>:<J<jPv:jJP:GE:MV:yc:C:UK=:Mjy;F:
B:C:K:J?Z:^:N:=:n?:MJ:B:C:Uk=:=S;Vd:^l=^e;^Z:jPn:j>vY:=:<J<J>:N;B:C:?j
::W;j>>:<JB>Z:VYGxyAC:;J:^Q;Z<>:DkcfGYEU^:f?Kj=jJP:mPqDJqIJ<<:UK=:Mjy;
F:;Z:^:>;f:N;B:C:?R:=:n?:MJ:B:C:UK>:=S;j<HJMLJ<<:UK=:Mjy;F:B:C:K:J?Z:^
:N:=:n?:MJ:B:C:UK>:=S;jxIJOIJ<jPn:j>vY:=:<J<jPV:M:oAA:;:?;f?MJ<<:?j:f:
^<AS;Qh:jCn>Q@^<eS;Qh:jC>a?Vs<:cj]Pj_;aJ_Pj_;cjfPj_E:V<sT;Qh:JDVg?VSjC
ng?Vs<:cjxPj_;aJ:Qj_E:^<YV;Q@V<_V;Q`:JDFn?Vs<:aJLQj_E:^<ao_;aJUQj_;cj
\\Qj_;aJ^Qj__:f??J<JrG[:fZ:j>:j:B:;B:eJ:VYF=>:s?<:DKd``_WhZnc_whZNdigg
[og:>[;^Z:jPF;C:n:C:KJ::O:<J<J;J<JQ:F;;Z:^:f?MJ?>Z:F^?>;GTTIUW_US;cFqm
CMJ<<:UK=f:F;yAj:>:<J<jPV:Jwk;jyKK:UK?^:N:=J>jLjeJAIj?LJ<<:UK;^:>XO:N:
?j<j:B:;B:C:UK<<J<:=Jk>:kS;?B:=C:UjfFa=Ve=>a=fe=v`=ne=f`=ve=^`=>f=N`=F
f=>`=Ff=n?=u:Ss:;u:Os:yt:Ms:ulM^e=n^=NEEs:kt:Es:gt:Es:ct:Es:_t:Gs:]t:I
s:Yt:Ks:Wt:Ms:UlOHj`HjPHj`HjQHj`HjRHJaHjSHjaHjTHJbHJUHjbv@at:ks:et:ms:
it:ms:mt:mkCIjILjyHJ>LJ;IJ>Lj<IJ>Lj=IJ>LJ?IJ>Lj@IJ>LJBIj=LJCIJ=LjDIJ<L
jEIj;LjFIj:LjCIjI^Z:jPN:C:V:E:M::=Z:B:C:K:J?Z:^:N:=J<JQ:F;;Z:^:f?I:F^?
F:_mdMJAH:f?Gj<j>vY:=:<J<jPV:M:oAA:;:?;f?O:N:=J>jLjPEjKgo<Vb=^Z:jPN:C:
[AEB:F;:F:<J:<J<jPV:Jwk;J:JK:UK?^:N::E;Vl=fOsC;CB:f??J<Jr;N:?j<j:B:;B:
C:UK:^:j:n=<JLH:Ug:]D;yf:gC;?g:iC;Cg:kC;Ig:kC;OoVLjPEjVLjQEjVLJSEJVLjT
EJVLJVEjULJWo`>fo<FDC:UK;^:j;f:F;:F:<Z:^:f?F;NY;j;J:JK:UK?:?j:f:^<Ef:U
g:E:aJ>EjPE:^<if:Ug:jCNm<fo<^Z:jPN:C:[AJ;N:E:=Z:>Z:^:f?;:j:n=^>jrDJWEJ
@EJVEj?EjTEJ?EJSEj>EjQEj>EjPEj>EJOEj>EjMGk<^n<Nk<Nn<Vk<vm<^k<fo<FHC:UK
;^:j;VZ:J:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::3:
" }}{PARA 0 "" 0 "" {TEXT -1 60 "Transformation matrix from conical to
 Cartesian coordinates:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "matr
ix([[sin(alpha)*cos(eta), -sin(eta), cos(alpha)*cos(eta)], [sin(alpha)
*sin(eta), cos(eta), cos(alpha)*sin(eta)], [-cos(alpha), 0, sin(alpha)
]]):" "6#-%'matrixG6#7%7%*&-%$sinG6#%&alphaG\"\"\"-%$cosG6#%$etaGF-,$-
F*6#F1!\"\"*&-F/6#F,F--F/6#F1F-7%*&-F*6#F,F--F*6#F1F--F/6#F1*&-F/6#F,F
--F*6#F1F-7%,$-F/6#F,F5\"\"!-F*6#F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 64 "For later convenience, make this and its inverse into functions
." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "M := %:" "6#>%\"MG%\"%G" }
}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "ConToCart := (a,b)->evalm( sub
s(alpha=a,eta=b,eval(M)) ):" "6#>%*ConToCartGR6$%\"aG%\"bG7\"6$%)opera
torG%&arrowG6\"-%&evalmG6#-%%subsG6%/%&alphaGF'/%$etaGF(-%%evalG6#%\"M
GF-F-F-" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "CartToCon := (a,b)->map(simp
lify,inverse(ConToCart(a,b))):" "6#>%*CartToConGR6$%\"aG%\"bG7\"6$%)op
eratorG%&arrowG6\"-%$mapG6$%)simplifyG-%(inverseG6#-%*ConToCartG6$F'F(
F-F-F-" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "latex(ConToCart(alpha
,eta),\"d:/dynamics/precession/ConicalToCartesian.tex\");" "6#-%&latex
G6$-%*ConToCartG6$%&alphaG%$etaGQNd:/dynamics/precession/ConicalToCart
esian.tex6\"" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "latex(CartToCon(alpha,e
ta),\"d:/dynamics/precession/CartesianToConical.tex\");" "6#-%&latexG6
$-%*CartToConG6$%&alphaG%$etaGQNd:/dynamics/precession/CartesianToConi
cal.tex6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Hence," }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "mat(xhat,yhat,zhat) = `&*`(ConToCart(al
pha,eta),mat(rhohat,etahat,alphahat));" "6#/-%$matG6%%%xhatG%%yhatG%%z
hatG-%#&*G6$-%*ConToCartG6$%&alphaG%$etaG-F%6%%'rhohatG%'etahatG%)alph
ahatG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#%%xhatG7#%%y
hatG7#%%zhatG-%#&*G6$-F%6#7%7%*&-%$sinG6#%&alphaG\"\"\"-%$cosG6#%$etaG
F:,$-F7F=!\"\"*&-F<F8F:F;\"\"\"7%*&F6FDF@F:F;*&FCFDF@FD7%,$FCFA\"\"!F6
-F%6#7%7#%'rhohatG7#%'etahatG7#%)alphahatG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 3 "and" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "mat(rhohat,e
tahat,alphahat) = `&*`(CartToCon(alpha,eta),mat(xhat,yhat,zhat));" "6#
/-%$matG6%%'rhohatG%'etahatG%)alphahatG-%#&*G6$-%*CartToConG6$%&alphaG
%$etaG-F%6%%%xhatG%%yhatG%%zhatG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-
%'matrixG6#7%7#%'rhohatG7#%'etahatG7#%)alphahatG-%#&*G6$-F%6#7%7%*&-%$
sinG6#%&alphaG\"\"\"-%$cosG6#%$etaGF:*&F6\"\"\"-F7F=F:,$-F<F8!\"\"7%,$
FAFDF;\"\"!7%*&FCF:F;F@*&FCF@FAF@F6-F%6#7%7#%%xhatG7#%%yhatG7#%%zhatG
" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Radiation Pressure" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "[P[X], P[Y], \+
P[Z]];" "6#7%&%\"PG6#%\"XG&F%6#%\"YG&F%6#%\"ZG" }{TEXT -1 92 " be the \+
pressure vector in the fixed frame.  Then the conical coordinates repr
esentation of " }{XPPEDIT 18 0 "P;" "6#%\"PG" }{TEXT -1 4 " is " }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "mat(P[rho],P[eta],P[alpha]) = `&
*`(`&*`(CartToCon(alpha,eta),R(phi,psi,theta)),mat(P[X],P[Y],P[Z]));" 
"6#/-%$matG6%&%\"PG6#%$rhoG&F(6#%$etaG&F(6#%&alphaG-%#&*G6$-F26$-%*Car
tToConG6$F0F--%\"RG6%%$phiG%$psiG%&thetaG-F%6%&F(6#%\"XG&F(6#%\"YG&F(6
#%\"ZG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"PG6#%$r
hoG7#&F*6#%$etaG7#&F*6#%&alphaG-%#&*G6$-F66$-F%6#7%7%*&-%$sinGF3\"\"\"
-%$cosGF/FA*&F?\"\"\"-F@F/FA,$-FCF3!\"\"7%,$FFFIFB\"\"!7%*&FHFAFBFE*&F
HFEFFFEF?-F%6#7%7%,&*&-FC6#%&thetaGFA-FC6#%$phiGFAFA*(-F@FWFA-FC6#%$ps
iGFA-F@FZFAFI,&*&FVFEF[oFEFA*(FgnFEFhnFEFYFEFA*&FgnFE-F@FinFA7%,&*&Fgn
FEFYFEFI*(FVFEFhnFEF[oFEFI,&*&FgnFEF[oFEFI*(FVFEFhnFEFYFEFA*&FVFEF`oFE
7%*&F`oFEF[oFE,$*&F`oFEFYFEFIFhn-F%6#7%7#&F*6#%\"XG7#&F*6#%\"YG7#&F*6#
%\"ZG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "latex(%,\"d:/dynamics/
precession/PressureBodyFrame.tex\");" "6#-%&latexG6$%\"%GQMd:/dynamics
/precession/PressureBodyFrame.tex6\"" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "Pbody := convert( evalm(rhs(%)), vector ):" "6#>%&Pbody
G-%(convertG6$-%&evalmG6#-%$rhsG6#%\"%G%'vectorG" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "simplify(mag(%));" "6#-%)simplifyG6#-%$magG6#%\"
%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#,(*$)&%\"PG6#%\"ZG\"
\"#\"\"\"\"\"\"*$)&F+6#%\"YGF.F/F0*$)&F+6#%\"XGF.F/F0F/" }}}{EXCHG 
{PARA 0 "" 0 "" {OLE 1 6200 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:
nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::fyyyyyqyyyYJ<j<J=j=J>j>J?Jyky;::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::::::_lqvGcMJ<j\\nGcMs
FI>:::::::JEf:yyyxIF:>Z=B:<:;`:Z@[::JZXlHLj_]rXAj;ZIB:<:=ja^GE=;::::::
:::N;?R:yyyyyy?Z:vYxY:B:::::::uh:B:N:Y<>D_mlVH[KRJ:<:::::::>=?jyyiy=J:
B::::::^;^\\FZ:::::::::::::yay=J:B::::::::F:;J;>:A:CJ:f:;B:G:I:K:M:OJ:
V;nYf;n;v;><<jBJCjCJD<:e:gJ:v<>=;jFJGjGJHjHJIjIJJB:=C:N>V>^>f>n>v>>?F?
N?V?^?f?n?v?>@<jRJSjSJTjTJUjUJVjVJWjWJXJyky;::::::::::::::::::::::::::
:::::::::::::::::j:b:=Z:F:>::YM:VZ=Z:JZC:bK=j@>:xNcHn\\HNy@MQTocL^yyyg
pU>:::BEbCr:;bKx`lsRP`=sUp`jsUt`_cULamcETamCSTL;r:A:;Z:M:J]<J:L<LNWlOl
`Qq@pp`pkHQh<QsPpkPpsHqnxPqlO]xNc:Zy=>:::Bm=ZN[=>Z>cHl_^SXt`[SU``]sVl_
jcVP`]SSHamSTx@l_@sUDa][=[gCVx`hSSt`nsVt<BSlp:R;l>@JDD:;B:<F;:Gt:;B:<:
<T?XV[Ys]pu:>[]o`RVcwH:HM[Ys]JkJ:R:;J:<::Bb;B?x[ZycJA^dQac?\\\\@uBrD[C
FuV@HL\\dMh<:;::jPaPJF@>:AhBAia[Jl`Qq:;BKmdM=TMMtJaLKF@>:ihBOdV[JrDql:
>ZJOd?Wb?WdxG`:v?_[\\KlLLh<CGtU_uREuJYUU[uUWUSWEWaeEguVCM]x>E<W;<Zx[VZ
=B:;BARFRGbO>ZOZ=ZmvZKI\\J>Z:>Z>GbNGBCdHF>NZ:JD>:;:N:sDQETOSTK_DQCTJc
\\^GIZV;:slqDA::ryxyyAKGD:<Z:M:J]<J:>Zymy=naYJsyt_pu:<psdP;<u_pu:<J>:>
:;:::::jysyAJ:^:F:vYxY;J:Jah<gxB<Jpi:=j>r:gY;<j?f?>:EJ:<j?D:<j<J@DjWLj
HPj<j>Djyyyy=j<j:Dj`qv`_:NZ;N:EB:F:jPJ[@j;B:K;[B:<J>j:<j:>:;::::jykyyY
:<jdP?yA:::^:f_;j><Z:Ny<j<B:;:?K<jPF:W:Cs:EjAf_>v[?f_>v;YjAv;f?M:oA;B:
:?;f??j=>Z:^y<F:<J:J<jPV:C:[Y:;J<:=j<j>D:f:FZ<vyyAZ:^:N:f:F:jP<:?<;JNJ
B>Z:>;FZ:F:;J:::::vYyQR<=[CFVZ=>:A:[[:>@V:V:^:f_;>:MB:NYVZ:>Z:>:N>jPF:
W:C;IS:Ec:ik<LjSLj<LjwMjLDjM@jL^:f??J<:AJ<Jr?:C:F:E:Mb:vYE:=b:;B:;B:C:
U;M:oAE:;:?;f?;:N:=J><:Es:?jAEjDHjAEjJ_:f??J<Jr?:J;N:E:=Z:>Z:v:^Y;B:<:
^:f?;j><:oq<J:vYN>jPF:w:Ck@<jAEj:=JLEJP?n<NOsf:MOFEJNWl<vNYf:IO?EjMoj<
>O=f:MoxDJOGi<^OYf:=N<jPN:C:V:E:Mb:j<j:B:<J<jPj>Jw?:E:yI:<JK:UK<^:N:=J
>jL:eJ[IjDHJ<jPN:C:[Q<:?J;f:F:<J:<j=JxkyK:^:f?C:j:n=^>Vv=f\\=>q=N_=Fq=
n^=Vq=N^=^q=n]=fq=F]=fq=f\\=fq=>\\=^q=^[=Vq=>[=Fq=VZ=>q=vy<Vv=f<C:UK;^
:j;f:F;:F:<J:<J<jPj>Jwk<jyKK:Uk<^:N:=j<JDDjDHjAg:V\\<N]=vk<:cjNHjAE:V<
Ws:Yf:JDfa=vk<:aJ[HjAE:^<]t:Yf:jCnd=vk<:cjlHjA;aJoHjA;cjvHjA;aJyHjA;cj
@IjA;aJCIjA_:f??J<Jrg:J;N:E:=Z:>Z:^:f?C:j:n=^>vKew:me:Ov:ue:Sv:=f:Uv:G
f:Wv:Of:Yv:Yf:Yv:af:Yv:kf:Wv:sf:Uv:;g:Sv:Cg:Ov:YnT_:f??J<:Aj<j>:j:B:<J
<jPj>Jwk;<:;:?;f?E:N:=JD>:EK>F\\<Fd<N^<vb<V`<va<^b<Na<nd<>a<vf<v`<NImc
:Uf:sc:C:UK;^:>Xf:F;:F:<J:<J<J>@J:<j<J?DJ:B:C:?:C:WS:kj<j>>:<JB>Z:Vy<f
pya?>:[V:b:;B:Dkc`qpDpql`:^:f?Ej=jJPj:VFAf:Wd:[B:<jw?<I:;JXE:<j:NZ\\Vd
sWhngfgcUC:UK=f:F;yAj:>:<J<J>j<J?Z:^:NZ;F:JQ:F;;Z:><;B:qqT_=<J:>LB:bBS
TXCUT=eU^:f?Ij=jJP:Wl?wo<^:f?G:F;yAj:Z:^:>;f:N;B:C:?j::W;j>>:<J<jPv:jJ
P:[D:AI:=T:C:UK=Z:F;yAj:Z:^:>;:O:<J<J;F:JQ:F;;Z:^:f?I:F^?N:CeK]i:_s:C:
UK=:Mjy;F:B:C:K:J?Z:^:N:=:n?:MJ:B:[J:<jwosyq=<J:^Qb:;ZLVDU=^:f?Kj=jJPZ
:NytY[=Vf=^:f?G:F;yAj:>:<J<J>j<J?Z:^:NZ;F:JQ:F;;Z:^:f?I:F^?Jlfk=Fq<^:f
?G:F;yAj:Z:^:>;:O:<J<J;F:JQ:F;;Z:^:f?K:F^?F:wLJIJqEJ<jPn:j>vY:=:<J<jPv
:C:n:C:K:J?Z:^:N:=:n?:MJ:B:C:Uk=v:F^?N:Ho`IJd_:;jPn:E:Mjy;F:B:C:K:J?Z:
^:N:=:n?:MJ:B:C:UK>:=S;=JQGq<fb=^:f?G:F;yAj:Z:^:>;:O:<J<J;F:JQ:F;;Z:^:
f?K:F^?:ct:Qh:C:UK=:Mjy;F:B:C:K:J?Z:^:N:=:n?:MJ:B:C:UK>:=S;JOWa;^:f?G:
F;yAj:Z:^:>;:O:<J<J;F:JQ:F;;Z:^:f?K:F^?jfoCgg:C:UK=:Mjy;F:B:C:K:J?Z:^:
N:=:n?:MJ:B:e:qQ]yyF>:;JX[:ZL^dcSSaEBWTSiEB_tUUURWmJ<J<jPF;OJ:<jJPJ>nb
cwfoOh_?^m>p:vu=^:f?G:F;yAj:>:<J<jPV:Jwk<J:vY?;f?OJ<J;F:K:EK;<jC@jbgi<
Ny<^:f??J<JrO;J;N:E:=Z:>Z:v:^YB:yQ:^:f?O:j:n]:B:C;f<Ee:ui:Oe:si:We:qi:
_e:mi:ge:ii:omtEJyDJsEj:EJrEJ<EJpEj=EjnEj>EjlEjAEjD^:f??J<:Aj;B:;:::::
::::::::::::::::::::::::::::::::::::::::::::3:" }}{PARA 0 "" 0 "" 
{TEXT -1 29 "The force on an area element " }{XPPEDIT 18 0 "dS;" "6#%#
dSG" }{TEXT -1 4 " is " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "mat(d
F[perp],dF[parallel]) = abs(P)*dS*('abs(cos(gamma))')\n    * mat( (1+A
[C])*cos(gamma), (1-A[C])*sin(gamma) );" "6#/-%$matG6$&%#dFG6#%%perpG&
F(6#%)parallelG**-%$absG6#%\"PG\"\"\"%#dSGF3.-F06#-%$cosG6#%&gammaGF3-
F%6$*&,&\"\"\"F3&%\"AG6#%\"CGF3F3-F96#F;F3*&,&\"\"\"F3&FB6#FD!\"\"F3-%
$sinG6#F;F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7$7#&%#d
FG6#%%perpG7#&F*6#%)parallelG**-%$absG6#%\"PG\"\"\"%#dSGF6-F36#-%$cosG
6#%&gammaGF6-F%6#7$7#*&,&F6F6&%\"AG6#%\"CGF6F6F:F67#*&,&F6F6FD!\"\"F6-
%$sinGF<F6F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 
18 0 "F[perp];" "6#&%\"FG6#%%perpG" }{TEXT -1 5 " and " }{XPPEDIT 18 
0 "F[parallel];" "6#&%\"FG6#%)parallelG" }{TEXT -1 49 " are the compon
ent perpendicular and parallel to " }{XPPEDIT 18 0 "dS;" "6#%#dSG" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "gamma;" "6#%&gammaG" }{TEXT -1 22 " is
 the angle between " }{XPPEDIT 18 0 "P;" "6#%\"PG" }{TEXT -1 31 " and \+
the (unit) surface normal " }{XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 6 
", and " }{XPPEDIT 18 0 "A[C]" "6#&%\"AG6#%\"CG" }{TEXT -1 65 " is the
 albedo of the cone surface.  From Figure 3, we see that  " }{XPPEDIT 
18 0 "dot(P,N) < 0;" "6#2-%$dotG6$%\"PG%\"NG\"\"!" }{TEXT -1 26 " for \+
our problem.  Hence, " }{XPPEDIT 18 0 "cos*gamma = -cos*chi;" "6#/*&%$
cosG\"\"\"%&gammaGF&,$*&F%F&%$chiGF&!\"\"" }{TEXT -1 11 " (that is, " 
}{XPPEDIT 18 0 "cos*chi = -dot(P,N)/abs(P);" "6#/*&%$cosG\"\"\"%$chiGF
&,$*&-%$dotG6$%\"PG%\"NGF&-%$absG6#F-!\"\"F2" }{TEXT -1 13 ") and we h
ave" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "mat(dF[perp],dF[parallel
]) = abs(P)*dS*cos(chi)*mat(-(1+A[C])*cos(chi),(1-A[C])*sin(chi));" "6
#/-%$matG6$&%#dFG6#%%perpG&F(6#%)parallelG**-%$absG6#%\"PG\"\"\"%#dSGF
3-%$cosG6#%$chiGF3-F%6$,$*&,&\"\"\"F3&%\"AG6#%\"CGF3F3-F66#F8F3!\"\"*&
,&\"\"\"F3&F@6#FBFEF3-%$sinG6#F8F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/-%'matrixG6#7$7#&%#dFG6#%%perpG7#&F*6#%)parallelG**-%$absG6#%\"PG\"
\"\"%#dSGF6-%$cosG6#%$chiG\"\"\"-F%6#7$7#,$*&,&F6F6&%\"AG6#%\"CGF6F6F8
F6!\"\"7#*&,&F6F6FDFHF6-%$sinGF:F6F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 17 "The magnitude of " }{XPPEDIT 18 0 "dF;" "6#%#dFG" }{TEXT -1 3 "
 is" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "dFpp := convert(evalm(rh
s(%)),vector):" "6#>%%dFppG-%(convertG6$-%&evalmG6#-%$rhsG6#%\"%G%'vec
torG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "simplify(sqrt(dot(%,%)),assume \+
= real):" "6#-%)simplifyG6$-%%sqrtG6#-%$dotG6$%\"%GF,/%'assumeG%%realG
" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "abs(dF) = rootfunc(%,collect,cos,fa
ctor);" "6#/-%$absG6#%#dFG-%)rootfuncG6&%\"%G%(collectG%$cosG%'factorG
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$absG6#%#dFG*0-%'signumG6#%\"PG
\"\"\"F,F--F*6#%#dSGF-F0F--F*6#-%$cosG6#%$chiGF-F3F--%%sqrtG6#,&*&)F3
\"\"#\"\"\"&%\"AG6#%\"CGF-\"\"%*$),&F?F-!\"\"F-F=F>F-F>" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "for p in select(has,indets(%),signum) d
o subs(p = 1,%) od:" "6#?&%\"pG-%'selectG6%%$hasG-%'indetsG6#%\"%G%'si
gnumG%%trueG-%%subsG6$/F$\"\"\"F," }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "su
bs(P = abs(P),%);" "6#-%%subsG6$/%\"PG-%$absG6#F'%\"%G" }}{PARA 0 "" 
0 "" {XPPEDIT 19 1 "mag_dF := rhs(%):" "6#>%'mag_dFG-%$rhsG6#%\"%G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$absG6#%#dFG**-F%6#%\"PG\"\"\"%#dSG
F,-%$cosG6#%$chiGF,-%%sqrtG6#,&*&)F.\"\"#\"\"\"&%\"AG6#%\"CGF,\"\"%*$)
,&F:F,!\"\"F,F8F9F,F9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Now, " }
{XPPEDIT 18 0 "cos*gamma = P[alpha]/abs(P);" "6#/*&%$cosG\"\"\"%&gamma
GF&*&&%\"PG6#%&alphaGF&-%$absG6#F*!\"\"" }{TEXT -1 10 ".  Hence, " }
{XPPEDIT 18 0 "cos*chi = -P[alpha]/abs(P);" "6#/*&%$cosG\"\"\"%$chiGF&
,$*&&%\"PG6#%&alphaGF&-%$absG6#F+!\"\"F1" }{TEXT -1 4 " or " }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "cos*chi = subs(P[X] = pi[X],P[Y]
 = pi[Y],P[Z] = pi[Z],-Pbody[3]);" "6#/*&%$cosG\"\"\"%$chiGF&-%%subsG6
&/&%\"PG6#%\"XG&%#piG6#F//&F-6#%\"YG&F16#F6/&F-6#%\"ZG&F16#F<,$&%&Pbod
yG6#\"\"$!\"\"" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "cos_chi := rhs(%):" "
6#>%(cos_chiG-%$rhsG6#%\"%G" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "latex(%,
\"d:/dynamics/precession/cos_chi.tex\");" "6#-%&latexG6$%\"%GQCd:/dyna
mics/precession/cos_chi.tex6\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*&%
$cosG\"\"\"%$chiGF&,(*&,(*(-F%6#%&alphaGF&-F%6#%$etaGF&,&*&-F%6#%&thet
aGF&-F%6#%$phiGF&F&*(-%$sinGF5F&-F%6#%$psiGF&-F<F8F&!\"\"F&F&*(F,\"\"
\"-F<F0F&,&*&F;FCF7FCFA*(F4FCF=FCF@FCFAF&F&*(-F<F-F&-F<F>F&F@FCF&F&&%#
piG6#%\"XGF&FA*&,(*(F,FCF/FC,&*&F4FCF@FCF&*(F;FCF=FCF7FCF&F&F&*(F,FCFD
FC,&*&F;FCF@FCFA*(F4FCF=FCF7FCF&F&F&*(FIFCFJFCF7FCFAF&&FL6#%\"YGF&FA*&
,(**F,FCF/FCF;FCFJFCF&**F,FCFDFCF4FCFJFCF&*&FIFCF=FCF&F&&FL6#%\"ZGF&FA
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "pi[X] \+
= P[X]/abs(P),pi[Y] = P[Y]/abs(P),pi[Z] = P[Z]/abs(P);" "6%/&%#piG6#%
\"XG*&&%\"PG6#F'\"\"\"-%$absG6#F*!\"\"/&F%6#%\"YG*&&F*6#F4F,-F.6#F*F0/
&F%6#%\"ZG*&&F*6#F=F,-F.6#F*F0" }{TEXT -1 3 ".  " }{XPPEDIT 18 0 "dF[p
arallel];" "6#&%#dFG6#%)parallelG" }{TEXT -1 16 " can be written " }
{XPPEDIT 18 0 "dF[parallel] = abs(dF[parallel])*(P-dot(P,N)*N)/abs(P-d
ot(P,N)*N);" "6#/&%#dFG6#%)parallelG*(-%$absG6#&F%6#F'\"\"\",&%\"PGF.*
&-%$dotG6$F0%\"NGF.F5F.!\"\"F.-F*6#,&F0F.*&-F36$F0F5F.F5F.F6F6" }
{TEXT -1 7 ".  But " }{XPPEDIT 18 0 "abs(P-dot(P,N)*N) = abs(cross(P,N
));" "6#/-%$absG6#,&%\"PG\"\"\"*&-%$dotG6$F(%\"NGF)F.F)!\"\"-F%6#-%&cr
ossG6$F(F." }{TEXT -1 4 " or " }{XPPEDIT 18 0 "abs(P-dot(P,N)*N) = abs
(P)*sin(chi);" "6#/-%$absG6#,&%\"PG\"\"\"*&-%$dotG6$F(%\"NGF)F.F)!\"\"
*&-F%6#F(F)-%$sinG6#%$chiGF)" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "dot
(P,N)/abs(P) = -cos(chi);" "6#/*&-%$dotG6$%\"PG%\"NG\"\"\"-%$absG6#F(!
\"\",$-%$cosG6#%$chiGF." }{TEXT -1 19 ", so we can write  " }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "dF[parallel]/abs(dF[parallel]) = (P+abs
(P)*cos(chi)*N)/(abs(P)*sin(chi));" "6#/*&&%#dFG6#%)parallelG\"\"\"-%$
absG6#&F&6#F(!\"\"*&,&%\"PGF)*(-F+6#F2F)-%$cosG6#%$chiGF)%\"NGF)F)F)*&
-F+6#F2F)-%$sinG6#F9F)F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#dFG6
#%)parallelG\"\"\"-%$absG6#F%!\"\"*&,&%\"PG\"\"\"*(-F+6#F0F1-%$cosG6#%
$chiGF1%\"NGF1F1F)*&F3\"\"\"-%$sinGF7\"\"\"F-" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 18 "In component form," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }{XPPEDIT 19 1 "subs(abs(P) = Q,P = mat(P[rho],P[eta],P[alpha]
),Q = abs(P),N = mat(0,0,1),%);" "6#-%%subsG6'/-%$absG6#%\"PG%\"QG/F*-
%$matG6%&F*6#%$rhoG&F*6#%$etaG&F*6#%&alphaG/F+-F(6#F*/%\"NG-F.6%\"\"!F
@\"\"\"%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#dFG6#%)parallelG
\"\"\"-%$absG6#F%!\"\"*&,&-%'matrixG6#7%7#&%\"PG6#%$rhoG7#&F66#%$etaG7
#&F66#%&alphaG\"\"\"*(-F+6#F6FA-%$cosG6#%$chiGFA-F16#7%7#\"\"!FL7#FAFA
FAF)*&FC\"\"\"-%$sinGFG\"\"\"F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 
"For " }{XPPEDIT 18 0 "dF[perp];" "6#&%#dFG6#%%perpG" }{TEXT -1 10 ", \+
we have " }{XPPEDIT 19 1 "dF[perp]/abs(dF[perp]) = matrix([[0], [0], [
1]]):" "6#/*&&%#dFG6#%%perpG\"\"\"-%$absG6#&F&6#F(!\"\"-%'matrixG6#7%7
#\"\"!7#F57#\"\"\"" }{TEXT -1 14 ".  Therefore, " }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "mat(dF[rho],dF[eta],dF[alpha]) = evalm(dFpp[2]*r
hs(%%)+dFpp[1]*rhs(%));" "6#/-%$matG6%&%#dFG6#%$rhoG&F(6#%$etaG&F(6#%&
alphaG-%&evalmG6#,&*&&%%dFppG6#\"\"#\"\"\"-%$rhsG6#%#%%GF:F:*&&F76#\"
\"\"F:-F<6#%\"%GF:F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7
%7#&%#dFG6#%$rhoG7#&F*6#%$etaG7#&F*6#%&alphaG-F%6#7%7#**%#dSG\"\"\"-%$
cosG6#%$chiGF;,&F;F;&%\"AG6#%\"CG!\"\"F;&%\"PGF+F;7#**F:\"\"\"F<FJF@FJ
&FGF/F;7#,&**F:FJF<FJF@FJ,&&FGF3F;*&-%$absG6#FGF;F<FJF;F;F;**FRFJF:FJ)
F<\"\"#FJ,&F;F;FAF;F;FE" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "The co
mponent perpendicular to the surface simplifies:" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "factormat(subs(P[rho] = pi[rho]*abs(P),P[eta] = \+
pi[eta]*abs(P),P[alpha] = pi[alpha]*abs(P),%%), collect, [dS, cos], fa
ctor);" "6#-%*factormatG6&-%%subsG6&/&%\"PG6#%$rhoG*&&%#piG6#F-\"\"\"-
%$absG6#F+F2/&F+6#%$etaG*&&F06#F9F2-F46#F+F2/&F+6#%&alphaG*&&F06#FBF2-
F46#F+F2%#%%G%(collectG7$%#dSG%$cosG%'factorG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&&%#dFG6#%%perpG\"\"\"-%$absG6#F%!\"\"-%'matrixG6#7%7
#\"\"!F27#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Thus, the for
ce integrated over the cone surface is" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "mat(F[rho],F[eta],F[alpha]) = Int(Int(subs(dS = rho*sin
(alpha),rhs(%)),rho = f .. f+S),eta = 0 .. 2*Pi);" "6#/-%$matG6%&%\"FG
6#%$rhoG&F(6#%$etaG&F(6#%&alphaG-%$IntG6$-F26$-%%subsG6$/%#dSG*&F*\"\"
\"-%$sinG6#F0F<-%$rhsG6#%\"%G/F*;%\"fG,&FFF<%\"SGF</F-;\"\"!*&\"\"#F<%
#PiGF<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"FG6#%$r
hoG7#&F*6#%$etaG7#&F*6#%&alphaG-%$IntG6$-F66$-F%6#7%7#\"\"!F=7#\"\"\"/
F,;%\"fG,&FCF@%\"SGF@/F0;F>,$%#PiG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "Fintegral := %:" "6#>%*FintegralG%\"%G" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "latex(%,\"d:/dynamics/precession/ConeFo
rceIntegral.tex\");" "6#-%&latexG6$%\"%GQMd:/dynamics/precession/ConeF
orceIntegral.tex6\"" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "Torque D
ue to Radiation Pressure on the Cone Surface" }}{SECT 1 {PARA 5 "" 0 "
" {TEXT -1 29 "Radius Vector to Cone Surface" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 162 "To calculate the torque, we first need the radius vect
or to a point on the surface of the cone.  The transformation from con
ical to cartesian body coordinates is " }{XPPEDIT 19 1 "[x = rho*sin(a
lpha)*cos(eta), y = rho*sin(alpha)*sin(eta), z = h-rho*cos(alpha)+a/ta
n(alpha)]:" "6#7%/%\"xG*(%$rhoG\"\"\"-%$sinG6#%&alphaGF(-%$cosG6#%$eta
GF(/%\"yG*(F'F(-F*6#F,F(-F*6#F0F(/%\"zG,(%\"hGF(*&F'F(-F.6#F,F(!\"\"*&
%\"aGF(-%$tanG6#F,F?F(" }{TEXT -1 3 ".  " }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "xyz := %:" "6#>%$xyzG%\"%G" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 105 "Hence, the radius vector from the center of mass to a po
int on the cone surface is, in the conical frame," }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "subs(xyz,`&*`(CartToCon(alpha,eta),mat(x,y,z)));
" "6#-%%subsG6$%$xyzG-%#&*G6$-%*CartToConG6$%&alphaG%$etaG-%$matG6%%\"
xG%\"yG%\"zG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "map(collect,evalm(%),[h
, a],simplify);" "6#-%$mapG6&%(collectG-%&evalmG6#%\"%G7$%\"hG%\"aG%)s
implifyG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "r_cone := %:" "6#>%'r_coneG
%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#&*G6$-%'matrixG6#7%7%*&-%$
sinG6#%&alphaG\"\"\"-%$cosG6#%$etaGF0*&F,\"\"\"-F-F3F0,$-F2F.!\"\"7%,$
F7F:F1\"\"!7%*&F9F0F1F6*&F9F6F7F6F,-F'6#7%7#*(%$rhoGF0F,F6F1F67#*(FFF6
F,F6F7F67#,(%\"hGF0*&FFF6F9F6F:*&%\"aGF6-%$tanGF.!\"\"F0" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7#,(*&-%$cosG6#%&alphaG\"\"\"%\"hG
F.!\"\"*&*&)F*\"\"#\"\"\"%\"aGF.F5-%$sinGF,!\"\"F0%$rhoGF.7#\"\"!7#,&*
&F7F.F/F5F.*&F*F5F6F5F." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "late
x(mat(xyz),\"d:/dynamics/precession/ConicalCoordinates.tex\");" "6#-%&
latexG6$-%$matG6#%$xyzGQNd:/dynamics/precession/ConicalCoordinates.tex
6\"" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "latex(r_cone,\"d:/dynamics/prece
ssion/r_cone.tex\");" "6#-%&latexG6$%'r_coneGQBd:/dynamics/precession/
r_cone.tex6\"" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 53 "Statement of t
he Torque Integral in the Conical Frame" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 56 "The torque integrated over the cone surface is therefore
" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs( F=K, subsop( [2,1,1]=
 'cross'(eval(r_cone),op([2,1,1],Fintegral)), Fintegral) );" "6#-%%sub
sG6$/%\"FG%\"KG-%'subsopG6$/7%\"\"#\"\"\"\"\"\"-.%&crossG6$-%%evalG6#%
'r_coneG-%#opG6$7%\"\"#\"\"\"\"\"\"%*FintegralGF@" }}{PARA 0 "" 0 "" 
{XPPEDIT 19 1 "torque_cone_integral := %:" "6#>%5torque_cone_integralG
%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"KG6#%$rh
oG7#&F*6#%$etaG7#&F*6#%&alphaG-%$IntG6$-F66$-%&crossG6$-F%6#7%7#,(*&-%
$cosGF3\"\"\"%\"hGFE!\"\"*&*&)FC\"\"#\"\"\"%\"aGFEFL-%$sinGF3!\"\"FGF,
FE7#\"\"!7#,&*&FNFEFFFLFE*&FCFLFMFLFE-F%6#7%FQFQ7#FE/F,;%\"fG,&FgnFE%
\"SGFE/F0;FR,$%#PiGFK" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "which ca
n be simplified as follows." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "
op([2, 1, 1],torque_cone_integral);" "6#-%#opG6$7%\"\"#\"\"\"\"\"\"%5t
orque_cone_integralG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%
\"\"!,(*&-%$cosG6#%&alphaG\"\"\"%\"hGF.F.*&*&)F*\"\"#\"\"\"%\"aGF.F4-%
$sinGF,!\"\"F.%$rhoG!\"\"F'" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "
factormat(mat(subs(A[C]-1 = -Q,eval(%))),collect,[Q, pi[alpha], pi[rho
], pi[eta], rho, A[C]],factor);" "6#-%*factormatG6&-%$matG6#-%%subsG6$
/,&&%\"AG6#%\"CG\"\"\"\"\"\"!\"\",$%\"QGF4-%%evalG6#%\"%G%(collectG7(F
6&%#piG6#%&alphaG&F>6#%$rhoG&F>6#%$etaGFC&F/6#F1%'factorG" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#*&*&,&*&%$rhoG\"\"\"-%$sinG6#%&alphaGF(F(*&-%$c
osGF+\"\"\",&*&F)F0%\"hGF(F(*&F.F(%\"aGF(F(F(!\"\"F(-%'matrixG6#7%7#\"
\"!7#*&*&,&F'F6*&*&F.F0F1F0F0F)!\"\"F(F(F)F0F0F%FCF;F(F0F)FC" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(sin(alpha)*h+cos(alpha)*a =
 U,%):" "6#-%%subsG6$/,&*&-%$sinG6#%&alphaG\"\"\"%\"hGF-F-*&-%$cosG6#F
,F-%\"aGF-F-%\"UG%\"%G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "facto
rmat(%,collect,[Q, pi[alpha], pi[rho], A[C]],factor);" "6#-%*factormat
G6&%\"%G%(collectG7&%\"QG&%#piG6#%&alphaG&F+6#%$rhoG&%\"AG6#%\"CG%'fac
torG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&,&*&%$rhoG\"\"\"-%$sinG6#%
&alphaGF(F(*&-%$cosGF+F(%\"UGF(!\"\"F(-%'matrixG6#7%7#\"\"!7#F1F6F(\"
\"\"F)!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "map( (x)->algsub
s(Q*U=V,x), map(expand,evalm(%)) ):" "6#-%$mapG6$R6#%\"xG7\"6$%)operat
orG%&arrowG6\"-%(algsubsG6$/*&%\"QG\"\"\"%\"UGF4%\"VGF(F-F-F--F$6$%'ex
pandG-%&evalmG6#%\"%G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "factor
mat(%,collect,[rho, V],factor);" "6#-%*factormatG6&%\"%G%(collectG7$%$
rhoG%\"VG%'factorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&,&*&%$rhoG\"
\"\"-%$sinG6#%&alphaGF(F(*&-%$cosGF+F(%\"UGF(!\"\"F(-%'matrixG6#7%7#\"
\"!7#*&*&,&F'F1*&*&F.\"\"\"F0F>F>F)!\"\"F(F(F)F>F>F%F?F6F(F>F)F?" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "arg := %:" "6#>%$argG%\"%G" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subslist := [Q = 1-A[C], U = sin
(alpha)*h+cos(alpha)*a, V = Q*U];" "6#>%)subslistG7%/%\"QG,&\"\"\"\"\"
\"&%\"AG6#%\"CG!\"\"/%\"UG,&*&-%$sinG6#%&alphaGF*%\"hGF*F**&-%$cosG6#F
7F*%\"aGF*F*/%\"VG*&F'F*F1F*" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "map( (x
,y)->lhs(x)=subs(y,rhs(x)), subslist, subslist ):" "6#-%$mapG6%R6$%\"x
G%\"yG7\"6$%)operatorG%&arrowG6\"/-%$lhsG6#F(-%%subsG6$F)-%$rhsG6#F(F.
F.F.%)subslistGF9" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "subslist := %:" "6
#>%)subslistG%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)subslistG7%/%
\"QG,&\"\"\"F)&%\"AG6#%\"CG!\"\"/%\"UG,&*&-%$sinG6#%&alphaGF)%\"hGF)F)
*&-%$cosGF5F)%\"aGF)F)/%\"VG*&F'F)F0F)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 6 "Check:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "map(simpl
ify,evalm(op([2, 1, 1],torque_cone_integral)-subs(subslist,arg)));" "6
#-%$mapG6$%)simplifyG-%&evalmG6#,&-%#opG6$7%\"\"#\"\"\"\"\"\"%5torque_
cone_integralG\"\"\"-%%subsG6$%)subslistG%$argG!\"\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#-%'matrixG6#7%7#\"\"!F'F'" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 16 "Finally, we have" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
19 1 "subsop([2, 1, 1] = arg,torque_cone_integral);" "6#-%'subsopG6$/7
%\"\"#\"\"\"\"\"\"%$argG%5torque_cone_integralG" }}{PARA 0 "" 0 "" 
{XPPEDIT 19 1 "torque_cone_integral := %:" "6#>%5torque_cone_integralG
%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"KG6#%$rh
oG7#&F*6#%$etaG7#&F*6#%&alphaG-%$IntG6$-F66$*&*&,&*&F,\"\"\"-%$sinGF3F
>F>*&-%$cosGF3F>%\"UGF>!\"\"F>-F%6#7%7#\"\"!7#*&*&,&F,FE*&*&FB\"\"\"FD
FQFQF?!\"\"F>F>F?FQFQF<FRFIF>FQF?FR/F,;%\"fG,&FUF>%\"SGF>/F0;FJ,$%#PiG
\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "where" }}}{EXCHG {PARA 0 
"" 0 "" {XPPEDIT 19 1 "mat(subslist);" "6#-%$matG6#%)subslistG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7#/%\"QG,&\"\"\"F+&%\"AG
6#%\"CG!\"\"7#/%\"UG,&*&-%$sinG6#%&alphaGF+%\"hGF+F+*&-%$cosGF8F+%\"aG
F+F+7#/%\"VG*&F*F+F4F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "and" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "cos(chi) = cos_chi;" "6#/-%$cosG
6#%$chiG%(cos_chiG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%$chiG
,(*&,(*(-F%6#%&alphaG\"\"\"-F%6#%$etaGF/,&*&-F%6#%&thetaGF/-F%6#%$phiG
F/F/*(-%$sinGF6F/-F%6#%$psiGF/-F=F9F/!\"\"F/F/*(F,\"\"\"-F=F1F/,&*&F<F
DF8FDFB*(F5FDF>FDFAFDFBF/F/*(-F=F-F/-F=F?F/FAFDF/F/&%#piG6#%\"XGF/FB*&
,(*(F,FDF0FD,&*&F5FDFAFDF/*(F<FDF>FDF8FDF/F/F/*(F,FDFEFD,&*&F<FDFAFDFB
*(F5FDF>FDF8FDF/F/F/*(FJFDFKFDF8FDFBF/&FM6#%\"YGF/FB*&,(**F,FDF0FDF<FD
FKFDF/**F,FDFEFDF5FDFKFDF/*&FJFDF>FDF/F/&FM6#%\"ZGF/FB" }}}}{SECT 1 
{PARA 5 "" 0 "" {TEXT -1 62 "Partial Evaluation of the Torque Integral
 in the Conical Frame" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "torque_
cone_integral;" "6#%5torque_cone_integralG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"KG6#%$rhoG7#&F*6#%$etaG7#&F*6#%&al
phaG-%$IntG6$-F66$*&*&,&*&F,\"\"\"-%$sinGF3F>F>*&-%$cosGF3F>%\"UGF>!\"
\"F>-F%6#7%7#\"\"!7#*&*&,&F,FE*&*&FB\"\"\"FDFQFQF?!\"\"F>F>F?FQFQF<FRF
IF>FQF?FR/F,;%\"fG,&FUF>%\"SGF>/F0;FJ,$%#PiG\"\"#" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 28 "We can do the integral over " }{XPPEDIT 18 0 "rho;
" "6#%$rhoG" }{TEXT -1 27 " right away.  We note that " }{XPPEDIT 18 
0 "f = a/sin(alpha);" "6#/%\"fG*&%\"aG\"\"\"-%$sinG6#%&alphaG!\"\"" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "S = (b-a)/sin(alpha);" "6#/%\"SG*&,
&%\"bG\"\"\"%\"aG!\"\"F(-%$sinG6#%&alphaGF*" }{TEXT -1 10 ".  Hence, \+
" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(f+S = b/sin(alpha),f =
 a/sin(alpha),%);" "6#-%%subsG6%/,&%\"fG\"\"\"%\"SGF)*&%\"bGF)-%$sinG6
#%&alphaG!\"\"/F(*&%\"aGF)-F.6#F0F1%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%'matrixG6#7%7#&%\"KG6#%$rhoG7#&F*6#%$etaG7#&F*6#%&alphaG-%$In
tG6$-F66$*&*&,&*&F,\"\"\"-%$sinGF3F>F>*&-%$cosGF3F>%\"UGF>!\"\"F>-F%6#
7%7#\"\"!7#*&*&,&F,FE*&*&FB\"\"\"FDFQFQF?!\"\"F>F>F?FQFQF<FRFIF>FQF?FR
/F,;*&%\"aGFQF?FR*&%\"bGFQF?FR/F0;FJ,$%#PiG\"\"#" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 27 "Evaluating the integral in " }{XPPEDIT 18 0 "rho;
" "6#%$rhoG" }{TEXT -1 9 ", we find" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "map(int,map(collect,evalm(op([2, 1, 1],subs(pi[rho] = P
rho,torque_cone_integral))),rho),rho = a/sin(alpha) .. b/sin(alpha)):
" "6#-%$mapG6%%$intG-F$6%%(collectG-%&evalmG6#-%#opG6$7%\"\"#\"\"\"\"
\"\"-%%subsG6$/&%#piG6#%$rhoG%%PrhoG%5torque_cone_integralGF;/F;;*&%\"
aG\"\"\"-%$sinG6#%&alphaG!\"\"*&%\"bGFB-FD6#FFFG" }}{PARA 0 "" 0 "" 
{XPPEDIT 19 1 "factormat(subs(Prho = pi[rho],V = Q*U,map(expand,evalm(
%))),collect,[cos(chi), pi[rho], pi[eta], pi[alpha], A[C], Q, U],facto
r):" "6#-%*factormatG6&-%%subsG6%/%%PrhoG&%#piG6#%$rhoG/%\"VG*&%\"QG\"
\"\"%\"UGF3-%$mapG6$%'expandG-%&evalmG6#%\"%G%(collectG7)-%$cosG6#%$ch
iG&F,6#F.&F,6#%$etaG&F,6#%&alphaG&%\"AG6#%\"CGF2F4%'factorG" }}{PARA 
0 "" 0 "" {XPPEDIT 19 1 "subsop([2, 1] = %,%%%);" "6#-%'subsopG6$/7$\"
\"#\"\"\"%\"%G%$%%%G" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "torque_cone := \+
%:" "6#>%,torque_coneG%\"%G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%'mat
rixG6#7%7#&%\"KG6#%$rhoG7#&F*6#%$etaG7#&F*6#%&alphaG-%$IntG6$,$*&*(,&%
\"bG!\"\"%\"aG\"\"\"\"\"\",(F>F=*&-%$cosGF3F?%\"UGF?\"\"#F<F=F?-F%6#7%
7#\"\"!7#,$*&*&,&*&*(FCF@F;F?FEF@F@*$)-%$sinGF3\"\"#F@!\"\"F=*&*&F;F@,
&F>F?F<F?F?F@*$)FU\"\"#F@FX#F?FFF?)FUFFF@F@*&F;\"\"\"FA\"\"\"FX!\"#FJF
?F@*$)FU\"\"#F@FX#F=FF/F0;FK,$%#PiGFF" }}}}{SECT 1 {PARA 5 "" 0 "" 
{TEXT -1 51 "Conversion of the Torque Integral to the Body Frame" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Define " }{XPPEDIT 19 1 "B[1] = (b+
a)*cos(alpha)*U/(2*sin(alpha)^2)-(b^2+b*a+a^2)/(3*sin(alpha)^2):" "6#/
&%\"BG6#\"\"\",&**,&%\"bG\"\"\"%\"aGF,F,-%$cosG6#%&alphaGF,%\"UGF,*&\"
\"#F,*$-%$sinG6#F1\"\"#F,!\"\"F,*&,(*$F+\"\"#F,*&F+F,F-F,F,*$F-\"\"#F,
F,*&\"\"$F,*$-F76#F1\"\"#F,F:F:" }{TEXT -1 5 " and " }{XPPEDIT 19 1 "B
[2] = (b+a)*U*Q/(2*sin(alpha));" "6#/&%\"BG6#\"\"#**,&%\"bG\"\"\"%\"aG
F+F+%\"UGF+%\"QGF+*&\"\"#F+-%$sinG6#%&alphaGF+!\"\"" }{TEXT -1 35 ".  \+
Then the torque integral becomes" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&
%\"BG6#\"\"#,$*&*(,&%\"aG\"\"\"%\"bGF-F-%\"UGF-%\"QGF-\"\"\"-%$sinG6#%
&alphaG!\"\"#F-F'" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Bsubs := [
%%, %]:" "6#>%&BsubsG7$%#%%G%\"%G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
19 1 "matrix([[K[rho]], [K[eta]], [K[alpha]]]) = Int((-a+b)*cos(chi)*a
bs(P)*matrix([[-B[2]*pi[eta]], [-2*B[1]*A[C]*cos(chi)+B[2]*pi[rho]+B[1
]*Q*pi[alpha]], [-B[1]*Q*pi[eta]]]),eta = 0 .. 2*Pi):" "6#/-%'matrixG6
#7%7#&%\"KG6#%$rhoG7#&F*6#%$etaG7#&F*6#%&alphaG-%$IntG6$**,&%\"aG!\"\"
%\"bG\"\"\"F=-%$cosG6#%$chiGF=-%$absG6#%\"PGF=-F%6#7%7#,$*&&%\"BG6#\"
\"#F=&%#piG6#F0F=F;7#,(**\"\"#F=&FM6#\"\"\"F=&%\"AG6#%\"CGF=-F?6#FAF=F
;*&&FM6#\"\"#F=&FQ6#F,F=F=*(&FM6#\"\"\"F=%\"QGF=&FQ6#F4F=F=7#,$*(&FM6#
\"\"\"F=FdoF=&FQ6#F0F=F;F=/F0;\"\"!*&\"\"#F=%#PiGF=" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 6 "Check:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 
"map(simplify,evalm(subs(Bsubs,op([2, 1],%))-op([2, 1],torque_cone)));
" "6#-%$mapG6$%)simplifyG-%&evalmG6#,&-%%subsG6$%&BsubsG-%#opG6$7$\"\"
#\"\"\"%\"%G\"\"\"-F06$7$\"\"#\"\"\"%,torque_coneG!\"\"" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7#,$*&*.-%$cosG6#%$chiG\"\"\"-%$ab
sG6#%\"PGF/%\"UGF/%\"QGF/&%#piG6#%$etaGF/,&*$)%\"bG\"\"#\"\"\"!\"\"*$)
%\"aGF>F?F/F/F?-%$sinG6#%&alphaG!\"\"#F/F>7#,$*&,>F;!\"$FA\"\"$*(-F,FF
F/F4F?F=F/\"\"'*(FQF?F4F?FCF/!\"'*,F0F?F+F?F5F?&F7FFF/)F=FOF?F>*.F0F?)
F+F>F?&%\"AG6#%\"CGF/FQF?F4F?FBF?FT**F0F?FYF?FZF?)FCFOF?\"\"%*0F0F?F+F
?F4F?F5F?&F76#%$rhoGF/FDF/FBF?FO*0F0F?F+F?F5F?FVF?FQF?F4F?FBF?FO*,F0F?
F+F?F5F?FVF?FinF?!\"#**F0F?FYF?FZF?FWF?!\"%*.F0F?FYF?FZF?FQF?F4F?F<F?F
R*0F0F?F+F?F4F?F5F?F\\oF?FDF?F<F?FN*0F0F?F+F?F5F?FVF?FQF?F4F?F<F?FNF?,
&F@F/*$)FQF>F?F/FH#F/FR7#,$*&*,F+F?F0F?F5F?F6F?,**(FQF?F4F?F<F?FN*$FWF
?F>*(FQF?F4F?FBF?FO*$FinF?FaoF/F?FgoFH#F@FR" }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "torque_cone := %%:" "6#>%,torque_coneG%#%%G" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "latex(torque_cone,\"d:/dynamics/
precession/ConeTorqueIntegralCone.tex\");" "6#-%&latexG6$%,torque_cone
GQRd:/dynamics/precession/ConeTorqueIntegralCone.tex6\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 55 "The pressure components in the cartesian \+
body frame are" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "mat(pi[x],pi[
y],pi[z]) = `&*`('R'(phi,psi,theta),mat(pi[X],pi[Y],pi[Z]));" "6#/-%$m
atG6%&%#piG6#%\"xG&F(6#%\"yG&F(6#%\"zG-%#&*G6$-.%\"RG6%%$phiG%$psiG%&t
hetaG-F%6%&F(6#%\"XG&F(6#%\"YG&F(6#%\"ZG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%'matrixG6#7%7#&%#piG6#%\"xG7#&F*6#%\"yG7#&F*6#%\"zG-%#&*G6$-%
\"RG6%%$phiG%$psiG%&thetaG-F%6#7%7#&F*6#%\"XG7#&F*6#%\"YG7#&F*6#%\"ZG
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Similarly, the conversion fro
m the cartesian body frame to the conical body frame is" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "mat(pi[rho], pi[eta], pi[alpha]) = `&*`
(CartToCon(alpha,eta),mat(pi[x], pi[y], pi[z]));" "6#/-%$matG6%&%#piG6
#%$rhoG&F(6#%$etaG&F(6#%&alphaG-%#&*G6$-%*CartToConG6$F0F--F%6%&F(6#%
\"xG&F(6#%\"yG&F(6#%\"zG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrix
G6#7%7#&%#piG6#%$rhoG7#&F*6#%$etaG7#&F*6#%&alphaG-%#&*G6$-F%6#7%7%*&-%
$sinGF3\"\"\"-%$cosGF/F?*&F=\"\"\"-F>F/F?,$-FAF3!\"\"7%,$FDFGF@\"\"!7%
*&FFF?F@FC*&FFFCFDFCF=-F%6#7%7#&F*6#%\"xG7#&F*6#%\"yG7#&F*6#%\"zG" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Hence, we have" }}}{EXCHG {PARA 0 
"" 0 "" {XPPEDIT 19 1 "subsop([2, 2] = rhs('%%'),%);" "6#-%'subsopG6$/
7$\"\"#\"\"#-%$rhsG6#.%#%%G%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-
%'matrixG6#7%7#&%#piG6#%$rhoG7#&F*6#%$etaG7#&F*6#%&alphaG-%#&*G6$-F%6#
7%7%*&-%$sinGF3\"\"\"-%$cosGF/F?*&F=\"\"\"-F>F/F?,$-FAF3!\"\"7%,$FDFGF
@\"\"!7%*&FFF?F@FC*&FFFCFDFCF=-F66$-%\"RG6%%$phiG%$psiG%&thetaG-F%6#7%
7#&F*6#%\"XG7#&F*6#%\"YG7#&F*6#%\"ZG" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "factormat(evalm(%),collect,[sin(eta), cos(eta)]);" "6#-
%*factormatG6%-%&evalmG6#%\"%G%(collectG7$-%$sinG6#%$etaG-%$cosG6#F/" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%#piG6#%$rhoG7#&F*
6#%$etaG7#&F*6#%&alphaG-F%6#7%7#,,*&,(*(-%$sinGF3\"\"\",&*&-F>6#%&thet
aGF?-%$cosG6#%$phiGF?!\"\"*(-FFFCF?-FF6#%$psiGF?-F>FGF?FIF?&F*6#%\"XGF
?F?*(F=\"\"\",&*&FBFTFOFTFI*(FKFTFLFTFEFTF?F?&F*6#%\"YGF?F?**F=FTFKFT-
F>FMF?&F*6#%\"ZGF?F?F?-F>F/F?F?*&,(*(F=FT,&*&FKFTFEFTF?*(FBFTFLFTFOFTF
IF?FPFTF?*(F=FT,&*&FKFTFOFTF?*(FBFTFLFTFEFTF?F?FXFTF?**F=FTFBFTFfnFTFg
nFTF?F?-FFF/F?F?**FPFT-FFF3F?FfnFTFOFTFI**FXFTFhoFTFfnFTFEFTF?*(FgnFTF
hoFTFLFTFI7#,&*&,(*&,&F_oFIF`oF?F?FPFTF?*&,&FcoFIFdoFIF?FXFTF?*(FBFTFf
nFTFgnFTFIF?FjnFTF?*&,(*&F@FTFPFTF?*&FUFTFXFTF?*(FgnFTFKFTFfnFTF?F?Ffo
FTF?7#,,*&,(*(FhoFTF@FTFPFTF?*(FhoFTFUFTFXFTF?**FhoFTFKFTFfnFTFgnFTF?F
?FjnFTF?*&,(*(FhoFTF^oFTFPFTF?*(FhoFTFboFTFXFTF?**FhoFTFBFTFfnFTFgnFTF
?F?FfoFTF?**FPFTF=FTFfnFTFOFTF?**FXFTF=FTFfnFTFEFTFI*(FgnFTF=FTFLFTF?
" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Psubs := [seq(lhs(%)[k,1] =
 rhs(%)[k,1],k = 1 .. 3)]:" "6#>%&PsubsG7#-%$seqG6$/&-%$lhsG6#%\"%G6$%
\"kG\"\"\"&-%$rhsG6#F.6$F0\"\"\"/F0;\"\"\"\"\"$" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 91 "We must transform the torque components from the con
ical frame to the body cartesian frame." }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "torque_cone;" "6#%,torque_coneG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"KG6#%$rhoG7#&F*6#%$etaG7#&F*6#%&al
phaG-%$IntG6$**,&%\"bG\"\"\"%\"aG!\"\"F;-%$cosG6#%$chiG\"\"\"-%$absG6#
%\"PGF;-F%6#7%7#,$*&&%\"BG6#\"\"#F;&%#piGF/F;F=7#,(*(&FN6#F;F;&%\"AG6#
%\"CGF;F>F;!\"#*&FMFB&FRF+F;F;*(FVFB%\"QGF;&FRF3F;F;7#,$*(FVFBFjnFBFQF
BF=F;/F0;\"\"!,$%#PiGFP" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs
op([2, 1] = `&*`(ConToCart(alpha,eta),op([2, 1],torque_cone)),torque_c
one):" "6#-%'subsopG6$/7$\"\"#\"\"\"-%#&*G6$-%*ConToCartG6$%&alphaG%$e
taG-%#opG6$7$\"\"#\"\"\"%,torque_coneGF8" }}{PARA 0 "" 0 "" {XPPEDIT 
19 1 "subs(K[rho] = K[x],K[eta] = K[y],K[alpha] = K[z],%);" "6#-%%subs
G6&/&%\"KG6#%$rhoG&F(6#%\"xG/&F(6#%$etaG&F(6#%\"yG/&F(6#%&alphaG&F(6#%
\"zG%\"%G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"KG6#
%\"xG7#&F*6#%\"yG7#&F*6#%\"zG-%$IntG6$-%#&*G6$-F%6#7%7%*&-%$sinG6#%&al
phaG\"\"\"-%$cosG6#%$etaGFD,$-FAFG!\"\"*&-FFFBFDFE\"\"\"7%*&F@FNFJFDFE
*&FMFNFJFN7%,$FMFK\"\"!F@**,&%\"bGFD%\"aGFKFD-FF6#%$chiGFN-%$absG6#%\"
PGFD-F%6#7%7#,$*&&%\"BG6#\"\"#FD&%#piGFGFDFK7#,(*(&Fao6#FDFD&%\"AG6#%
\"CGFDFYFD!\"#*&F`oFN&Feo6#%$rhoGFDFD*(FioFN%\"QGFD&FeoFBFDFD7#,$*(Fio
FNFepFNFdoFNFKFD/FH;FT,$%#PiGFco" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
19 1 "subsop([2, 1] = factormat(evalm(op([2, 1],%)),collect,[pi[rho], \+
pi[eta], pi[alpha]],factor),%);" "6#-%'subsopG6$/7$\"\"#\"\"\"-%*facto
rmatG6&-%&evalmG6#-%#opG6$7$\"\"#\"\"\"%\"%G%(collectG7%&%#piG6#%$rhoG
&F:6#%$etaG&F:6#%&alphaG%'factorGF6" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#/-%'matrixG6#7%7#&%\"KG6#%\"xG7#&F*6#%\"yG7#&F*6#%\"zG-%$IntG6$**,&%
\"bG!\"\"%\"aG\"\"\"F=-%$absG6#%\"PGF=-%$cosG6#%$chiG\"\"\"-F%6#7%7#,*
*(-%$sinG6#%$etaGF=&%\"BG6#\"\"#F=&%#piG6#%$rhoGF=F=*(-FCFOF=,&*&-FN6#
%&alphaGF=FQFFF=*(-FCFhnF=&FR6#F=F=%\"QGF=F=F=&FVFOF=F=**FMFFF\\oFFF^o
FF&FVFhnF=F=**FMFFF\\oFF&%\"AG6#%\"CGF=FBF=!\"#7#,**(FZFFFQFFFUFFF;*(F
MFFFenFFF_oFFF=**FZFFF\\oFFF^oFFFaoFFF;**FZFFF\\oFFFcoFFFBFFFT7#*&,&*&
F[oFFFQFFF;*(FgnFFF\\oFFF^oFFF=F=F_oFFF=/FP;\"\"!,$%#PiGFT" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "foo := %:" "6#>%$fooG%\"%G" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 7 "Define " }{XPPEDIT 19 1 "C[1] = sin(alpha)
*B[2]+cos(alpha)*B[1]*Q:" "6#/&%\"CG6#\"\"\",&*&-%$sinG6#%&alphaG\"\"
\"&%\"BG6#\"\"#F.F.*(-%$cosG6#F-F.&F06#\"\"\"F.%\"QGF.F." }{TEXT -1 5 
" and " }{XPPEDIT 19 1 "C[2] = -sin(alpha)*Q*B[1]+cos(alpha)*B[2]:" "6
#/&%\"CG6#\"\"#,&*(-%$sinG6#%&alphaG\"\"\"%\"QGF.&%\"BG6#\"\"\"F.!\"\"
*&-%$cosG6#F-F.&F16#\"\"#F.F." }{TEXT -1 7 ".  Then" }}}{EXCHG {PARA 
0 "" 0 "" {XPPEDIT 19 1 "Csubs := [%%, %]:" "6#>%&CsubsG7$%#%%G%\"%G" 
}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "invsubs(Csubs, foo);" "6#-%(i
nvsubsG6$%&CsubsG%$fooG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%'matrixG
6#7%7#&%\"KG6#%\"xG7#&F*6#%\"yG7#&F*6#%\"zG-%$IntG6$**,&%\"bG!\"\"%\"a
G\"\"\"F=-%$absG6#%\"PGF=-%$cosG6#%$chiG\"\"\"-F%6#7%7#,**(-%$sinG6#%$
etaGF=&%\"BG6#\"\"#F=&%#piG6#%$rhoGF=F=*(-FCFOF=&%\"CG6#F=F=&FVFOF=F=*
*FMFF&FRFgnF=%\"QGF=&FV6#%&alphaGF=F=**FMFFFjnFF&%\"AG6#FfnF=FBF=!\"#7
#,**(FZFFFQFFFUFFF;*(FMFFFenFFFhnFFF=**FZFFFjnFFF[oFFF\\oFFF;**FZFFFjn
FFF`oFFFBFFFT7#*&,&*&-FCF]oF=FQFFF;*(-FNF]oF=FjnFFF[oFFF=F=FhnFFF=/FP;
\"\"!,$%#PiGFT" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Finally, substi
tute for " }{XPPEDIT 18 0 "cos*chi;" "6#*&%$cosG\"\"\"%$chiGF%" }
{TEXT -1 57 " and the pressure components and perform the integration.
" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(Psubs,cos(chi) = cos_c
hi,%):" "6#-%%subsG6%%&PsubsG/-%$cosG6#%$chiG%(cos_chiG%\"%G" }}{PARA 
0 "" 0 "" {XPPEDIT 19 1 "map(int,map(collect,evalm(op([2, 1],%)),[sin(
eta), cos(eta), abs(P)]),eta):" "6#-%$mapG6%%$intG-F$6%%(collectG-%&ev
almG6#-%#opG6$7$\"\"#\"\"\"%\"%G7%-%$sinG6#%$etaG-%$cosG6#F8-%$absG6#%
\"PGF8" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "foo := %:" "6#>%$fooG%\"%G" }
}{PARA 0 "" 0 "" {XPPEDIT 19 1 "for k to 3 do \n  foo[k,1] := subs(eta
 = 2*Pi,foo[k,1])-subs(eta = 0,foo[k,1]);\n  foo[k,1] := factor(foo[k,
1]);\nod:" "6#?(%\"kG\"\"\"F%\"\"$%%trueGC$>&%$fooG6$F$\"\"\",&-%%subs
G6$/%$etaG*&\"\"#F%%#PiGF%&F+6$F$\"\"\"F%-F06$/F3\"\"!&F+6$F$\"\"\"!\"
\">&F+6$F$\"\"\"-%'factorG6#&F+6$F$\"\"\"" }}{PARA 0 "" 0 "" {XPPEDIT 
19 1 "factormat(foo,collect,[pi[X], pi[Y], pi[Z]]):" "6#-%*factormatG6
%%$fooG%(collectG7%&%#piG6#%\"XG&F*6#%\"YG&F*6#%\"ZG" }}{PARA 0 "" 0 "
" {XPPEDIT 19 1 "mat(K[x],K[y],K[z]) = %;" "6#/-%$matG6%&%\"KG6#%\"xG&
F(6#%\"yG&F(6#%\"zG%\"%G" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "torque_xyz_
cone := %:" "6#>%0torque_xyz_coneG%\"%G" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#/-%'matrixG6#7%7#&%\"KG6#%\"xG7#&F*6#%\"yG7#&F*6#%\"zG*.,&%\"bG!
\"\"%\"aG\"\"\"F:%#PiGF:-%$absG6#%\"PGF:,(*(-%$sinG6#%$psiGF:&%#piG6#%
\"YGF:-%$cosG6#%$phiGF:F8*&-FKFDF:&FG6#%\"ZGF:F:*(FB\"\"\"-FCFLF:&FG6#
%\"XGF:F:F:,,**-FC6#%&alphaGF:&%\"BG6#F:F:%\"QGF:-FKFfnF:\"\"#*&)FenF]
oFT&Fin6#F]oF:F:**FenFTFhnFT&%\"AG6#%\"CGF:F\\oFT\"\"%*&&FfoFjnF:FenFT
F:*&F`oFT)F\\oF]oFTF8F:-F%6#7%7#,(*&,&*(-FK6#%&thetaGF:FOFTFUFTF:*&-FC
FepF:FJFTF:F:FVFTF:*&,&*&FhpFTFUFTF:*(FdpFTFOFTFJFTF8F:FFFTF:*(FPFTFdp
FTFBFTF87#,(*&,&*&FdpFTFJFTF:*(FhpFTFOFTFUFTF8F:FVFTF:*&,&*&FdpFTFUFTF
:*(FhpFTFOFTFJFTF:F:FFFTF:*(FhpFTFBFTFPFTF:7#\"\"!F:" }}}{EXCHG {PARA 
0 "" 0 "" {XPPEDIT 19 1 "latex(torque_xyz_cone,\"d:/dynamics/precessio
n/ConeTorqueBody.tex\");" "6#-%&latexG6$%0torque_xyz_coneGQJd:/dynamic
s/precession/ConeTorqueBody.tex6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 16 "We can simplify " }{XPPEDIT 18 0 "C[1];" "6#&%\"CG6#\"\"\"" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "C[2];" "6#&%\"CG6#\"\"#" }{TEXT -1 
10 " somewhat:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(sin(alph
a)^2+cos(alpha)^2 = 1,collect(subs(Bsubs,Csubs[1]),[Q, U],factor)):" "
6#-%%subsG6$/,&*$-%$sinG6#%&alphaG\"\"#\"\"\"*$-%$cosG6#F,\"\"#F.\"\"
\"-%(collectG6%-F$6$%&BsubsG&%&CsubsG6#\"\"\"7$%\"QG%\"UG%'factorG" }}
{PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(subslist[1],collect(%,[sin(alpha)]
));" "6#-%%subsG6$&%)subslistG6#\"\"\"-%(collectG6$%\"%G7#-%$sinG6#%&a
lphaG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"*&*&,&*&,&%\"a
GF'%\"bGF'F'%\"UGF'#F'\"\"#*&-%$cosG6#%&alphaGF',(*$)F.F1\"\"\"F'*&F.F
'F-F'F'*$)F-F1F:F'F'#!\"\"\"\"$F',&F'F'&%\"AG6#F%F?F'F:*$)-%$sinGF5\"
\"#F:!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(subslist[1],
factor(subs(Bsubs,Csubs[2])));" "6#-%%subsG6$&%)subslistG6#\"\"\"-%'fa
ctorG6#-F$6$%&BsubsG&%&CsubsG6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/&%\"CG6#\"\"#,$*&*&,&\"\"\"F,&%\"AG6#F%!\"\"F,,(*$)%\"bGF'\"\"\"F,*
&F4F,%\"aGF,F,*$)F7F'F5F,F,F5-%$sinG6#%&alphaG!\"\"#F,\"\"$" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Csubs := [%%, %]:" "6#>%&CsubsG7
$%#%%G%\"%G" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 55 "Torque Due to R
adiation Pressure on the Flattop Surface" }}{EXCHG {PARA 0 "" 0 "" 
{OLE 1 6200 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
fyyyyyqyyyYJ<j<J=j=J>j>J?Jyky;::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::_lqvGcMJ<j\\nGcMsFI>:::::::JEf:yy
yxIF:>Z=B:<:;`:Z@[::JJ<yPLj_]rXAj;J:u:<:=ja^GE=;:::::::::N;?R:yyyyyy?Z
:vYxY:B:::::::Ih:<Z:J;vCJbNHEms>@[;;B:::::::JFN:yyyxI:;Z::::::J@Jch:<:
:::::::::::vYxI:;Z::::::::j:>:?J:V:^:f:;B:G:I:KJ:F;N;;j?Jyk@JAjA>:[Z:F
<N<V<^<f<n<;jEJF>:m:o:q:s:u:w:y:;[:F^:JKjKJLjLJMjMJNjNJOjOJPjPJQjQ>:[[
:F@N@V@^@f@n@v@>AFAnYvY::::::::::::::::::::::::::::::::::::::::::::::F
:Dj:B:=:;:j`j;H:<:;`:Z@G:i:lpjLQdxPrHnqxPsLpntpjLqlHQQTqsL`yyygpU>::::
DM;r:J:Bj>:nb:BCBKoCQCUXWURUUTAuVsDV_USAUSauVMuUWEQGDM=DOctMwUFWcOaDKx
IJ:::ZRI:d?HJ:L^tBQL`scU;RRp`csRHaVCVDAH`_sVHacsUt`VcKl`[CWL_iCVH^isUD
a^STt`[[_[hbWHa^cV=Z?BMR:cb:>Z:Bj>:nb=B:<:<rCLsrQrXpu:>cLOfdvbwH:_w;]Y
[uSi=JfJ:R:;J:<::rbL[YBrYDn;ctoST[bbkH<hLR<mHa[]^bt^e:J:::foc?>];;j[El
kuSB^fsoGJ:LnftnZpn^HncN>];;jeEL_DQB>hlgF:;BN_TKATKadY]KSN@;BPVd^nrmL]
l@:Zn;bKSTJL?];>:=>s:AJ:<JHjRjTJeB:e:A:as;]X>[B:;B:CTJcTJLNWRBb:;ZNB:<
:b:lOhPndln\\DOhLnZL?CUXfG^E:BQlpj::LAZyuyyY>ob:B:<F;:GD:;J:xIyIBiCYYx
_ITi=BJCX@uy_ITi<JS[wH:;J::::::::::yayY:^:F:vYxY;J:jZm;AbCN_Ej:F;HjDQ:
KJkJ:f:;B:Qb:B:E:Sb:uE;Eu:E:Mb:yyyyI:E:=b:UXmUP<J;@J;fZ:j::U;?T:A:K[:>
\\:B:KJ:FZ:F:;J:::::vYvyyAZ:fdOky;:::C:US:F[:B:oi:f:<J:JK^:f?=JAJLHj<v
;YE;it:YE;ilAv;Y:Ym>JwK:<:JK:UK;v:;B:si:=Z:>:^:f?AJ<JrAJ:^:j:f:F[<:E:=
b:yyy;B:C:?:E:=:f_:J[J:>?><;B:K:=B:=J:>:::::yqy?`jNg<>`jne<VZ=nc<V:qc:
[CFGc:[CFYb:Ar:gY:Ar:qX:C:US:;j><:oAAB:vy:B:?;f?=JkB:CkQVWiU:Uqq@Js=Jm
@jwwe;>[;>e;>];Fd;N_;Vc;va;nb;vd;Fb;>h;na;Nk;^a;fn;Va;>r;Va;Vu;^a;nx;n
a;v[<Fb;n^<nb;Va<Vc;nc<Fd;ne<>e;Ng<ve;Vh<nf;Fi<vg;Vi<vh;Fi<vi;VHIV:Oe:
YV:wd:gV:Wd:uV:qc:AW:Gc:KW:Yb:SW:gqQAjgAjRAJZAjSAjLAjSAJ?AjRAJrvo;vd;^
o;va;>o;N_;Vn;>];fm;>[;nl;VYYV:_q=Ajpwi;VWiM<jPN:C:V:C:[Q:^:j:f:F[<jyg
:FZ<>Z:B:C:U;M:oAA:;jyO>jP>:J;@j:>;f^=N:aW:it:Qc:QV:C:UK;^:>X;j<j>:j:B
:;B:C:Uk;:oAA:;:?;f?;J<J;:f>jsVp;vh;^:f??J<Jr;N:?j<j:B:;B:I:sQ:<Z::C:U
K:F[:Jwg:>:yyyN>jPF:G\\:^>=kSAjq@jRAjq@JRAjq@JQAJr@jPAjr@JPAJs@jOAJt@j
OAjt@jOAju@jOAJv@jOAJw@JPAjw@jPAJx@JQAjx@JRAJy@jRAJy@jSAjy@JTAJy@JUAJy
@jUAjx@JVAJx@jVAjw@JWAJw@JWAJv@jWAjuNq;fh;Nq;^h;Fq;Nh;>q;Fh;vp;>h;np;v
g;^PYU:aW:YM<jPN:C:V:E:Mb:j<j:B:<J<jPj>Jw?:A:;:?;f?CJ<J;F:K:E;vHaW:iL<
jPN:C:[Q<f:F;:F:<J:<J<jPV:Jwk;J:JK:UK<^:N::E;vDyluIJ<jPN:C:[AE:M::=Z:>
Z:^:f?A:NYV:>:N>jP^:C:?:jL:vZ=vx=^:f??J<Jrk<j>:j:B:;B:C:KR:;B:E:Ob:;Z:
^:N:^:n_;>=E:MJ:B:[J:<jwEjTyYPB:;JBA:DJ:<ZLVdsgg\\wgfC:C:UK<v:F^?F:Sm
\\?`;><<jw?<I:;JXE:<j:NZ\\VdsWhngfgcUC:Uk<f:F;yAj:>:<J<J>j<J?Z:^:NZ;F:
JQ:F;;Z:><;B:qqT_=<J:>LB:bBSTXCUT=eU^:f?Gj=jJP:KMmDjx@J<>:Uk<:Mjy;F:B:
C:Uk;J:NYV:<J:<B?C:jPv:C:?j:>;f>juv[;>m;^:f??J<Jrw:E:M::=Z:>Z:^:>;f:N;
B:C:?:C:W;j>>:<J<jPn:jJP:=M<Aj[@J<jPf:j>vY:=:<J<J>:N;B:C:?j::W;j>>:<J<
jPn:jJP:omj<jMAJ<jPf:j>vY:=:<J<J>:N;B:C:?j::W;j>>:<J<jPn:jJP:mMSLJBHJ<
jPf:j>vY:=:<J<J>:N;B:C:?j::W;j>>:<J<jPn:jJP:kmQLJV^Z:jPf:j>vY:=:<J<jPV
:Jwk;J:JK:Uk=^:N:=J>jL:V_<VKjPN:C:[AE:M::=Z:>Z:^:>;:O:<J<J;J<JQ:F;;Z:^
:f?G:F^?Jsfi;nx;^:f?E:F;yAj:Z:^:f?A:NYV:>:N>jPv:C:?j:>`:B:EKENt;^k;^s;
nk;nr;>l;>r;Fl;Fq;Nl;Np;Vl;No;Vl;VN_V:=W:_V:yV:_V:uV:]V:qV:]V:mV:[V:gV
:WV:aV:UV:]n?AJAAJ?Wk;>k;Fk;nj;vj;^j;fj;Vj;fj;Nj;^:f??J<Jr;N:?j<j:B:;B
:C:UK:^:j:n=<JLf[:Vv;>j;^t;Vm;^t;>m;Vt;fl;Nt;VL]X:[V:YX:WV:UX:SV:Nk;Fs
;>k;vr;nj;fr;fj;Vv;>JC:UK;^:j;f:F;:F:<Z:^:f?F;NY;j;J:JK:Uk=:?j:>;f>?jK
wj;vc;vTC:UK;^:>Xf:F;:F:<J:<jD>:qQcyYDJ:^QB:b>cTTUUSaEBWTSiEB_tUUURWE:
sK<jPv:C:f:C:K:E:O:<J<J;J<JQ:F;;Z:^:f?IJ?>:=S;KJ]`pmPQsPPJLmknf<^:f?Ej
;<:;::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::5:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
19 1 "eval(subs(Csubs,Bsubs,subslist,A[C] = A[T],a = 0,b = a,torque_xy
z_cone),alpha = Pi/2);" "6#-%%evalG6$-%%subsG6)%&CsubsG%&BsubsG%)subsl
istG/&%\"AG6#%\"CG&F.6#%\"TG/%\"aG\"\"!/%\"bGF5%0torque_xyz_coneG/%&al
phaG*&%#PiG\"\"\"\"\"#!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%'mat
rixG6#7%7#&%\"KG6#%\"xG7#&F*6#%\"yG7#&F*6#%\"zG,$*0)%\"aG\"\"#\"\"\"%#
PiG\"\"\"-%$absG6#%\"PGF<,(*(-%$sinG6#%$psiGF<&%#piG6#%\"YGF<-%$cosG6#
%$phiGF<!\"\"*&-FLFEF<&FH6#%\"ZGF<F<*(FCF:-FDFMF<&FH6#%\"XGF<F<F<%\"hG
F<,&F<F<&%\"AG6#%\"TGFOF<-F%6#7%7#,(*&,&*(-FL6#%&thetaGF<FQF:FVF:F<*&-
FDFcoF<FKF:F<F<FWF:F<*&,&*&FfoF:FVF:F<*(FboF:FQF:FKF:FOF<FGF:F<*(FRF:F
boF:FCF:FO7#,(*&,&*&FboF:FKF:F<*(FfoF:FQF:FVF:FOF<FWF:F<*&,&*&FboF:FVF
:F<*(FfoF:FQF:FKF:F<F<FGF:F<*(FfoF:FCF:FRF:F<7#\"\"!F<FO" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "torque_xyz_top := %:" "6#>%/torque_xyz_
topG%\"%G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "latex(torque_xyz_t
op,\"d:/dynamics/precession/FlatTopTorqueBody.tex\");" "6#-%&latexG6$%
/torque_xyz_topGQMd:/dynamics/precession/FlatTopTorqueBody.tex6\"" }}}
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "The Equations of Motion" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "The equations of motion are" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "mat(FirstOrderODEsK);" "6#-%$mat
G6#%0FirstOrderODEsKG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7
(7#/-%%diffG6$%$phiG%\"tG&%&OmegaG6#F,7#/-F*6$%$psiGF-&F/6#F57#/-F*6$%
&thetaGF-&F/6#F<7#/*&-%$sinG6#F5\"\"\"-F*6$F.F-FE,&*&,&*&,&FEFE%%betaG
!\"\"FEF=FEFE*(-%$cosGFDFE,&FEFEFMFEFEF.FEFNFEF6FEFE*&,&*&-FQ6#F<FE&%
\"KG6#%\"yGFEFE*&&FY6#%\"xGFE-FCFWFEFE\"\"\"&%\"IG6#%#xyG!\"\"FE7#/-F*
6$F6F-,&*&,&*(FPF[oFMFE)F.\"\"#F[oFE*(,&FNFEFMFEFEF=F[oF.F[oFEFEFBF[oF
E*&,&*&FXF[oFjnF[oFN*&FVF[oFgnF[oFEF[oF\\oF`oFE7#/*&FBF[o-F*6$F=F-FE,&
*&,&*&,&*&)FPFjoF[oFMF[oFEFEFEFEF.F[oFE*(F\\pF[oFPF[oF=F[oFEFEF6F[oFE*
&,&*&*&FBF[o&FY6#%\"zGFEF[oF\\pF`oFN*&FTFEFPF[oFNF[oF\\oF`oFE" }}}
{SECT 1 {PARA 5 "" 0 "" {TEXT -1 19 "Torque Combinations" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 33 "Assemble the torque combinations." }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Kcone := convert(evalm(rhs(torqu
e_xyz_cone)),vector):" "6#>%&KconeG-%(convertG6$-%&evalmG6#-%$rhsG6#%0
torque_xyz_coneG%'vectorG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "Ktop := co
nvert(evalm(rhs(torque_xyz_top)),vector):" "6#>%%KtopG-%(convertG6$-%&
evalmG6#-%$rhsG6#%/torque_xyz_topG%'vectorG" }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "(Kcone[1]+Ktop[1])*sin(theta)+(Kcone[2]+Ktop[2])*cos(
theta):" "6#,&*&,&&%&KconeG6#\"\"\"\"\"\"&%%KtopG6#\"\"\"F*F*-%$sinG6#
%&thetaGF*F**&,&&F'6#\"\"#F*&F,6#\"\"#F*F*-%$cosG6#F2F*F*" }}{PARA 0 "
" 0 "" {XPPEDIT 19 1 "applyrule(sin(theta)^2+cos(theta)^2 = 1,factor(%
)):" "6#-%*applyruleG6$/,&*$-%$sinG6#%&thetaG\"\"#\"\"\"*$-%$cosG6#F,
\"\"#F.\"\"\"-%'factorG6#%\"%G" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "map(c
ollect,%,[B[1], B[2], C[1]],factor);" "6#-%$mapG6&%(collectG%\"%G7%&%
\"BG6#\"\"\"&F*6#\"\"#&%\"CG6#\"\"\"%'factorG" }}{PARA 0 "" 0 "" 
{XPPEDIT 19 1 "torque[phi] := %:" "6#>&%'torqueG6#%$phiG%\"%G" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#*,%#PiG\"\"\"-%$absG6#%\"PGF%,&*&&%#pi
G6#%\"XGF%-%$cosG6#%$phiGF%F%*&-%$sinGF2F%&F-6#%\"YGF%F%F%,(*(-F66#%$p
siGF%F7\"\"\"F0F?!\"\"*&-F1F=F%&F-6#%\"ZGF%F%*(F<F?F5F?F,F?F%F%,**,-F1
6#%&alphaGF%-F6FJF%,&&%\"AG6#%\"CG\"\"#%\"QGF%F%,&%\"bGF@%\"aGF%F%&%\"
BG6#F%F%FR**,&FIF%FLF@F%,&FIF%FLF%F%FTF?&FX6#FRF%F@*(FLF?FTF?&FQFYF%F%
*()FVFRF?%\"hGF%,&F@F%&FO6#%\"TGF%F%F%F%" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 3 "and" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "(Kcone[1]+Kt
op[1])*cos(theta)-(Kcone[2]+Ktop[2])*sin(theta):" "6#,&*&,&&%&KconeG6#
\"\"\"\"\"\"&%%KtopG6#\"\"\"F*F*-%$cosG6#%&thetaGF*F**&,&&F'6#\"\"#F*&
F,6#\"\"#F*F*-%$sinG6#F2F*!\"\"" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "appl
yrule(sin(theta)^2+cos(theta)^2 = 1,factor(%)):" "6#-%*applyruleG6$/,&
*$-%$sinG6#%&thetaG\"\"#\"\"\"*$-%$cosG6#F,\"\"#F.\"\"\"-%'factorG6#%
\"%G" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "map(collect,%,[B[1], B[2], C[1]
],factor);" "6#-%$mapG6&%(collectG%\"%G7%&%\"BG6#\"\"\"&F*6#\"\"#&%\"C
G6#\"\"\"%'factorG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "torque[psi] := %:
" "6#>&%'torqueG6#%$psiG%\"%G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*,-
%$absG6#%\"PG\"\"\"%#PiGF),(*(-%$sinG6#%$psiGF)&%#piG6#%\"YGF)-%$cosG6
#%$phiGF)!\"\"*&-F6F/F)&F26#%\"ZGF)F)*(F-\"\"\"-F.F7F)&F26#%\"XGF)F)F)
,(*&F-F@F<F@F)*(FAF@FBF@F;F@F9*(F5F@F;F@F1F@F)F),**,-F66#%&alphaGF)-F.
FLF),&&%\"AG6#%\"CG\"\"#%\"QGF)F),&%\"bGF9%\"aGF)F)&%\"BG6#F)F)FT**,&F
KF)FNF9F),&FKF)FNF)F)FVF@&FZ6#FTF)F9*(FNF@FVF@&FSFenF)F)*()FXFTF@%\"hG
F),&F9F)&FQ6#%\"TGF)F)F)F)F9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "wh
ere" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Csubs;" "6#%&CsubsG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/&%\"CG6#\"\"\"*&*&,&*&,&%\"aGF(%\"b
GF(F(%\"UGF(#F(\"\"#*&-%$cosG6#%&alphaGF(,(*$)F/F2\"\"\"F(*&F/F(F.F(F(
*$)F.F2F;F(F(#!\"\"\"\"$F(,&F(F(&%\"AG6#F&F@F(F;*$)-%$sinGF6\"\"#F;!\"
\"/&F&6#F2,$*&*&FBF;F8F;F;FHFK#F(FA" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "Bsubs;" "6#%&BsubsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
7$/&%\"BG6#\"\"\",&*&*(,&%\"aGF(%\"bGF(F(-%$cosG6#%&alphaGF(%\"UGF(\"
\"\"*$)-%$sinGF1\"\"#F4!\"\"#F(\"\"#*&,(*$)F.F<F4F(*&F.F(F-F(F(*$)F-F<
F4F(F4*$)F7\"\"#F4F:#!\"\"\"\"$/&F&6#F<,$*&*(F,F4F3F4%\"QGF(F4F7F:F;" 
}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subslist;" "6#%)subslistG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/%\"QG,&\"\"\"F'&%\"AG6#%\"CG!\"\"/%
\"UG,&*&-%$sinG6#%&alphaGF'%\"hGF'F'*&-%$cosGF3F'%\"aGF'F'/%\"VG*&F&F'
F/F'" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 41 "Simplification of the T
orque Combinations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Let's check \+
the commonality between the two torque combinations." }}}{EXCHG {PARA 
0 "" 0 "" {XPPEDIT 19 1 "mat(K[x]*sin(theta)+K[y]*cos(theta),K[x]*cos(
theta)-K[y]*sin(theta)) = factormat(subs(subslist[1],mat(torque[phi],t
orque[psi])),collect,[B[1], B[2], C[1]],factor);" "6#/-%$matG6$,&*&&%
\"KG6#%\"xG\"\"\"-%$sinG6#%&thetaGF-F-*&&F*6#%\"yGF--%$cosG6#F1F-F-,&*
&&F*6#F,F--F76#F1F-F-*&&F*6#F5F--F/6#F1F-!\"\"-%*factormatG6&-%%subsG6
$&%)subslistG6#\"\"\"-F%6$&%'torqueG6#%$phiG&FR6#%$psiG%(collectG7%&%
\"BG6#\"\"\"&Fen6#\"\"#&%\"CG6#\"\"\"%'factorG" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/-%'matrixG6#7$7#,&*&-%$cosG6#%&thetaG\"\"\"&%\"KG6#%\"
yGF/F/*&&F16#%\"xGF/-%$sinGF-F/F/7#,&*&F0\"\"\"F8F=!\"\"*&F+F=F5F=F/*,
%#PiGF/-%$absG6#%\"PGF/,(*(-F96#%$psiGF/&%#piG6#%\"YGF/-F,6#%$phiGF/F>
*&-F,FIF/&FL6#%\"ZGF/F/*(FHF=-F9FPF/&FL6#%\"XGF/F/F/,**,-F,6#%&alphaGF
/-F9FinF/,&F/F/&%\"AG6#%\"CGF/F/,&%\"bGF>%\"aGF/F/&%\"BG6#F/F/\"\"#**,
&FhnF/F[oF>F/,&FhnF/F[oF/F/FaoF=&Feo6#FgoF/F>*(F[oF=FaoF=&F`oFfoF/F/*(
)FcoFgoF=%\"hGF/,&F>F/&F^o6#%\"TGF/F/F/F/-F%6#7$7#,&*&FYF=FOF=F/*&FXF=
FKF=F/7#,(*&FHF=FTF=F>*(FXF=FYF=FSF=F/*(FOF=FSF=FKF=F>F/" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "torque_terms := %:" "6#>%-torque_termsG
%\"%G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "select(has,rhs(%),a);
" "6#-%'selectG6%%$hasG-%$rhsG6#%\"%G%\"aG" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#,**,-%$cosG6#%&alphaG\"\"\"-%$sinGF'F),&F)F)&%\"AG6#%\"
CGF)F),&%\"bG!\"\"%\"aGF)F)&%\"BG6#F)F)\"\"#**,&F%F)F*F3F),&F%F)F*F)F)
F1\"\"\"&F66#F8F)F3*(F*F<F1F<&F0F7F)F)*()F4F8F<%\"hGF),&F3F)&F.6#%\"TG
F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "L := location(%%,%);
" "6#>%\"LG-%)locationG6$%#%%G%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"LG7$\"\"#\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(Csu
bs,Bsubs,subslist[1],%%);" "6#-%%subsG6&%&CsubsG%&BsubsG&%)subslistG6#
\"\"\"%#%%G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,**,-%$cosG6#%&alphaG\"
\"\"-%$sinGF'\"\"\",&F,F,&%\"AG6#%\"CGF,F,,&%\"bG!\"\"%\"aGF,F,,&*&*(,
&F5F,F3F,F,F%F,%\"UGF,F)*$)F*\"\"#F)!\"\"#F,\"\"#*&,(*$)F3F@F)F,*&F3F,
F5F,F,*$)F5F@F)F,F)*$)F*\"\"#F)F>#F4\"\"$F,F@*&*.,&F%F,F*F4F,,&F%F,F*F
,F,F2F)F9F)F:F),&F,F,F.F4F,F)F*F>#F4F@*&*(F2F),&*&F9F)F:F)F?*&F%F)FBF,
FKF,FQF)F)F*F>F,*(FGF)%\"hGF,,&F4F,&F/6#%\"TGF,F,F," }}}{EXCHG {PARA 
0 "" 0 "" {XPPEDIT 19 1 "bar := %:" "6#>%$barG%\"%G" }}}{EXCHG {PARA 
0 "" 0 "" {XPPEDIT 19 1 "latex(%,\"d:/dynamics/precession/torque_term_
to_simplify.tex\");" "6#-%&latexG6$%\"%GQSd:/dynamics/precession/torqu
e_term_to_simplify.tex6\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Co
llect(%,[b-a, b+a, b^2+b*a+a^2, U, A[C]],simplify,loc);" "6#-%(Collect
G6&%\"%G7',&%\"bG\"\"\"%\"aG!\"\",&F)F*F+F*,(*$F)\"\"#F**&F)F*F+F*F**$
F+\"\"#F*%\"UG&%\"AG6#%\"CG%)simplifyG%$locG" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#,(*(,&*&*&,**&)-%$cosG6#%&alphaG\"\"#\"\"\"%\"bG\"\"\"!
\"#*&%\"aGF2F*F0F/F1F2F5!\"\"F2&%\"AG6#%\"CGF2F0-%$sinGF-!\"\"F2*&,&F1
F6F5F2F0F;F=F2F2%\"UGF2,&F5F2F1F2F2F2*&,&*&*(F+F2F?F2F7F0F0F;F=#F6\"\"
$*&*&F+F0F?F0F0F;F=F6F2,(*$)F1F/F0F2*&F1F0F5F0F2*$)F5F/F0F2F2F2*(FOF0%
\"hGF2,&F6F2&F86#%\"TGF2F2F2" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 
"location(%,remove(has,select(has,%,U),b-a));" "6#-%)locationG6$%\"%G-
%'removeG6%%$hasG-%'selectG6%F*F&%\"UG,&%\"bG\"\"\"%\"aG!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "subsop(% = map(factor,op(%,%%)),%%);" "6#-%'subsopG6$/%
\"%G-%$mapG6$%'factorG-%#opG6$F'%#%%GF/" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#,(*&**,(*&&%\"AG6#%\"CG\"\"\")-%$cosG6#%&alphaG\"\"#\"\"\"F2F(!
\"\"F,F,F,,&%\"bGF4%\"aGF,F,%\"UGF,,&F7F,F6F,F,F3-%$sinGF0!\"\"F,*&,&*
&*(F.F,F5F3F(F3F3F:F<#F4\"\"$*&*&F.F3F5F3F3F:F<F4F,,(*$)F6F2F3F,*&F6F,
F7F,F,*$)F7F2F3F,F,F,*(FJF3%\"hGF,,&F4F,&F)6#%\"TGF,F,F," }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "latex(%,\"d:/dynamics/precession/torque
_term_simplified.tex\");" "6#-%&latexG6$%\"%GQRd:/dynamics/precession/
torque_term_simplified.tex6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "C
heck:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "simplify(%-bar);" "6#-
%)simplifyG6#,&%\"%G\"\"\"%$barG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(subslist,subso
p(L = %%,torque_terms));" "6#-%%subsG6$%)subslistG-%'subsopG6$/%\"LG%#
%%G%-torque_termsG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7$7
#,&*&-%$cosG6#%&thetaG\"\"\"&%\"KG6#%\"yGF/F/*&&F16#%\"xGF/-%$sinGF-F/
F/7#,&*&F0\"\"\"F8F=!\"\"*&F+F=F5F=F/*,%#PiGF/-%$absG6#%\"PGF/,(*(-F96
#%$psiGF/&%#piG6#%\"YGF/-F,6#%$phiGF/F>*&-F,FIF/&FL6#%\"ZGF/F/*(FHF=-F
9FPF/&FL6#%\"XGF/F/F/,(*&**,(*&&%\"AG6#%\"CGF/)-F,6#%&alphaG\"\"#F=Fco
F[oF>F/F/F/,&%\"bGF>%\"aGF/F/,&*&-F9FaoF/%\"hGF/F/*&F`oF/FfoF/F/F/,&Ff
oF/FeoF/F/F=Fio!\"\"F/*&,&*&*(F`oF=FdoF=F[oF=F=FioF]p#F>\"\"$*&*&F`oF=
FdoF=F=FioF]pF>F/,(*$)FeoFcoF=F/*&FeoF/FfoF=F/*$)FfoFcoF=F/F/F/*(F[qF=
FjoF=,&F>F/&F\\o6#%\"TGF/F/F/F/-F%6#7$7#,&*&FYF=FOF=F/*&FXF=FKF=F/7#,(
*&FHF=FTF=F>*(FXF=FYF=FSF=F/*(FOF=FSF=FKF=F>F/" }}}{EXCHG {PARA 0 "" 
0 "" {XPPEDIT 19 1 "torque_terms := %:" "6#>%-torque_termsG%\"%G" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "This looks good.  Now define" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "select(has,rhs(torque_terms),psi
);" "6#-%'selectG6%%$hasG-%$rhsG6#%-torque_termsG%$psiG" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#*&,(*(-%$sinG6#%$psiG\"\"\"&%#piG6#%\"YGF*-%$cos
G6#%$phiGF*!\"\"*&-F0F(F*&F,6#%\"ZGF*F**(F&\"\"\"-F'F1F*&F,6#%\"XGF*F*
F*-%'matrixG6#7$7#,&*&F<F:F/F:F**&F;F:F+F:F*7#,(*&F&F:F6F:F3*(F;F:F<F:
F5F:F**(F/F:F5F:F+F:F3F*" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "sel
ect(has,%,sin(psi)*pi[Z]);" "6#-%'selectG6%%$hasG%\"%G*&-%$sinG6#%$psi
G\"\"\"&%#piG6#%\"ZGF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#
7$7#,&*&&%#piG6#%\"XG\"\"\"-%$cosG6#%$phiGF.F.*&-%$sinGF1F.&F+6#%\"YGF
.F.7#,(*&-F56#%$psiGF.&F+6#%\"ZGF.!\"\"*(F4\"\"\"F*FD-F0F=F.F.*(F/FDFE
FDF6FDFB" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "gsubs := [g[0](phi,
psi) = remove(has,%%,sin(psi)*pi[Z]), g[1](phi,psi) = %[1,1], g[2](phi
,psi) = %[2,1], G(a,b,h,A[C],A[T],alpha) = select(has,remove(has,rhs(t
orque_terms),psi),\{a, P, Pi\})]:" "6#>%&gsubsG7&/-&%\"gG6#\"\"!6$%$ph
iG%$psiG-%'removeG6%%$hasG%#%%G*&-%$sinG6#F.\"\"\"&%#piG6#%\"ZGF8/-&F)
6#\"\"\"6$F-F.&%\"%G6$\"\"\"\"\"\"/-&F)6#\"\"#6$F-F.&FD6$\"\"#\"\"\"/-
%\"GG6(%\"aG%\"bG%\"hG&%\"AG6#%\"CG&FZ6#%\"TG%&alphaG-%'selectG6%F2-F0
6%F2-%$rhsG6#%-torque_termsGF.<%FV%\"PG%#PiG" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 17 "Then we can write" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
19 1 "invsubs(gsubs, remove(has,rhs(torque_terms),psi));" "6#-%(invsub
sG6$%&gsubsG-%'removeG6%%$hasG-%$rhsG6#%-torque_termsG%$psiG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%\"GG6(%\"aG%\"bG%\"hG&%\"AG6#%\"CG&F*6#%\"
TG%&alphaG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "invsubs(gsubs,sel
ect(has,rhs(torque_terms),psi));" "6#-%(invsubsG6$%&gsubsG-%'selectG6%
%$hasG-%$rhsG6#%-torque_termsG%$psiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#*&-&%\"gG6#\"\"!6$%$phiG%$psiG\"\"\"-%'matrixG6#7$7#-&F&6#F,F)7#-&F&
6#\"\"#F)F," }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "lhs(torque_terms
) = %*%%;" "6#/-%$lhsG6#%-torque_termsG*&%\"%G\"\"\"%#%%GF*" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7$7#,&*&-%$cosG6#%&thetaG\"\"\"
&%\"KG6#%\"yGF/F/*&&F16#%\"xGF/-%$sinGF-F/F/7#,&*&F0\"\"\"F8F=!\"\"*&F
+F=F5F=F/*(-&%\"gG6#\"\"!6$%$phiG%$psiGF/-F%6#7$7#-&FC6#F/FF7#-&FC6#\"
\"#FFF/-%\"GG6(%\"aG%\"bG%\"hG&%\"AG6#%\"CG&Ffn6#%\"TG%&alphaGF/" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "torque_terms_subs := %:" "6#>%2t
orque_terms_subsG%\"%G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "where" }
}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "mat(gsubs);" "6#-%$matG6#%&gsu
bsG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7#/-&%\"gG6#\"\"!
6$%$phiG%$psiG,(*(-%$sinG6#F0\"\"\"&%#piG6#%\"YGF6-%$cosG6#F/F6!\"\"*&
-F<F5F6&F86#%\"ZGF6F6*(F3\"\"\"-F4F=F6&F86#%\"XGF6F67#/-&F+6#F6F.,&*&F
GFEF;FEF6*&FFFEF7FEF67#/-&F+6#\"\"#F.,(*&F3FEFAFEF>*(FFFEFGFEF@FEF6*(F
;FEF@FEF7FEF>7#/-%\"GG6(%\"aG%\"bG%\"hG&%\"AG6#%\"CG&F_o6#%\"TG%&alpha
G*(%#PiGF6-%$absG6#%\"PGF6,(*&**,(*&F^oF6)-F<6#FeoFWFEFWF^oF>F6F6F6,&F
\\oF>F[oF6F6,&*&-F4FcpF6F]oF6F6*&FbpF6F[oF6F6F6,&F[oF6F\\oF6F6FEFgp!\"
\"F6*&,&*&*(FbpFEFdpFEF^oFEFEFgpFjp#F>\"\"$*&*&FbpFEFdpFEFEFgpFjpF>F6,
(*$)F\\oFWFEF6*&F\\oF6F[oFEF6*$)F[oFWFEF6F6F6*(FhqFEF]oFE,&F>F6FboF6F6
F6F6" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 19 "Equations of Motion" }}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs( seq( lhs(torque_terms_subs
)[k,1]=\n           evalm(rhs(torque_terms_subs))[k,1],\n           k=
1..2), \n\n      K[z]=0,\n      FirstOrderODEsK ):" "6#-%%subsG6%-%$se
qG6$/&-%$lhsG6#%2torque_terms_subsG6$%\"kG\"\"\"&-%&evalmG6#-%$rhsG6#F
.6$F0\"\"\"/F0;\"\"\"\"\"#/&%\"KG6#%\"zG\"\"!%0FirstOrderODEsKG" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "mat(%);" "6#-%$matG6#%\"%G" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7#/-%%diffG6$%$phiG%\"tG
&%&OmegaG6#F,7#/-F*6$%$psiGF-&F/6#F57#/-F*6$%&thetaGF-&F/6#F<7#/*&-%$s
inG6#F5\"\"\"-F*6$F.F-FE,&*&,&*&,&FEFE%%betaG!\"\"FEF=FEFE*(-%$cosGFDF
E,&FEFEFMFEFEF.FEFNFEF6FEFE*&*(-&%\"gG6#\"\"!6$F,F5FE-%\"GG6(%\"aG%\"b
G%\"hG&%\"AG6#%\"CG&F\\o6#%\"TG%&alphaGFE-&FW6#FEFZFE\"\"\"&%\"IG6#%#x
yG!\"\"FE7#/-F*6$F6F-,&*&,&*(FPFfoFMFE)F.\"\"#FfoFE*(,&FNFEFMFEFEF=Ffo
F.FfoFEFEFBFfoFE*&*(FUFfoFenFfo-&FW6#FepFZFEFfoFgoF[pFE7#/*&FBFfo-F*6$
F=F-FE,&*&,&*&,&*&)FPFepFfoFMFfoFEFEFEFEF.FfoFE*(FgpFfoFPFfoF=FfoFEFEF
6FfoFE*&**FUFfoFenFfoFcoFfoFPFfoFfoFgoF[pFN" }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "FirstOrderODEs := %:" "6#>%/FirstOrderODEsG%\"%G" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "where" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "mat(gsubs);" "6#-%$matG6#%&gsubsG" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7&7#/-&%\"gG6#\"\"!6$%$phiG%$psiG,(*(-%$sin
G6#F0\"\"\"&%#piG6#%\"YGF6-%$cosG6#F/F6!\"\"*&-F<F5F6&F86#%\"ZGF6F6*(F
3\"\"\"-F4F=F6&F86#%\"XGF6F67#/-&F+6#F6F.,&*&FGFEF;FEF6*&FFFEF7FEF67#/
-&F+6#\"\"#F.,(*&F3FEFAFEF>*(FFFEFGFEF@FEF6*(F;FEF@FEF7FEF>7#/-%\"GG6(
%\"aG%\"bG%\"hG&%\"AG6#%\"CG&F_o6#%\"TG%&alphaG*(%#PiGF6-%$absG6#%\"PG
F6,(*&**,(*&F^oF6)-F<6#FeoFWFEFWF^oF>F6F6F6,&F\\oF>F[oF6F6,&*&-F4FcpF6
F]oF6F6*&FbpF6F[oF6F6F6,&F[oF6F\\oF6F6FEFgp!\"\"F6*&,&*&*(FbpFEFdpFEF^
oFEFEFgpFjp#F>\"\"$*&*&FbpFEFdpFEFEFgpFjpF>F6,(*$)F\\oFWFEF6*&F\\oF6F[
oFEF6*$)F[oFWFEF6F6F6*(FhqFEF]oFE,&F>F6FboF6F6F6F6" }}}{EXCHG {PARA 0 
"" 0 "" {XPPEDIT 19 1 "latex(FirstOrderODEs,\"d:/dynamics/precession/F
irstOrderODEs.tex\"); latex(mat(gsubs),\"d:/dynamics/precession/gsubs.
tex\");" "6#C$-%&latexG6$%/FirstOrderODEsGQJd:/dynamics/precession/Fir
stOrderODEs.tex6\"-F%6$-%$matG6#%&gsubsGQAd:/dynamics/precession/gsubs
.tex6\"" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Fast Spin Approxima
tion" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "The equations of motion ar
e" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "eqs := subs(op(select(has,
gsubs,psi)),G(a,b,h,A[C],A[T],alpha) = G,[seq(FirstOrderODEs[k,1],k = \+
4 .. 6)]):" "6#>%$eqsG-%%subsG6%-%#opG6#-%'selectG6%%$hasG%&gsubsG%$ps
iG/-%\"GG6(%\"aG%\"bG%\"hG&%\"AG6#%\"CG&F96#%\"TG%&alphaGF37#-%$seqG6$
&%/FirstOrderODEsG6$%\"kG\"\"\"/FG;\"\"%\"\"'" }}{PARA 0 "" 0 "" 
{XPPEDIT 19 1 "mat(%);" "6#-%$matG6#%\"%G" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7#/*&-%$sinG6#%$psiG\"\"\"-%%diffG6$&%&Om
egaG6#%$phiG%\"tGF.,&*&,&*&,&F.F.%%betaG!\"\"F.&F36#%&thetaGF.F.*(-%$c
osGF,F.,&F.F.F<F.F.F2F.F=F.&F36#F-F.F.*&*(,(*(F*\"\"\"&%#piG6#%\"YGF.-
FC6#F5F.F=*&FBFK&FM6#%\"ZGF.F.*(F*FK-F+FQF.&FM6#%\"XGF.F.F.%\"GGF.,&*&
FXFKFPFKF.*&FWFKFLFKF.F.FK&%\"IG6#%#xyG!\"\"F.7#/-F06$FEF6,&*&,&*(FBFK
F<F.)F2\"\"#FKF.*(,&F=F.F<F.F.F>FKF2FKF.F.F*FKF.*&*(FIFKFenFK,(*&F*FKF
SFKF=*(FWFKFXFKFBFKF.*(FPFKFBFKFLFKF=F.FKFinF]oF.7#/*&F*FK-F06$F>F6F.,
&*&,&*&,&*&)FBFgoFKF<FKF.F.F.F.F2FKF.*(FioFKFBFKF>FKF.F.FEFKF.*&**FIFK
FenFKFfnFKFBFKFKFinF]oF=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "For p
urposes of substitution, which we'll use in a moment, divide by " }
{XPPEDIT 18 0 "sin(psi);" "6#-%$sinG6#%$psiG" }{TEXT -1 19 " where app
ropriate." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "select(has,eqs,dif
f(Omega[phi],t)):" "6#-%'selectG6%%$hasG%$eqsG-%%diffG6$&%&OmegaG6#%$p
hiG%\"tG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "L := location(eqs,op(%)):" 
"6#>%\"LG-%)locationG6$%$eqsG-%#opG6#%\"%G" }}{PARA 0 "" 0 "" 
{XPPEDIT 19 1 "subsop(% = op(%%)/sin(psi),eqs):" "6#-%'subsopG6$/%\"%G
*&-%#opG6#%#%%G\"\"\"-%$sinG6#%$psiG!\"\"%$eqsG" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{XPPEDIT 19 1 "select(has,%,diff(Omega[theta],t)):" "6#
-%'selectG6%%$hasG%\"%G-%%diffG6$&%&OmegaG6#%&thetaG%\"tG" }}{PARA 0 "
" 0 "" {XPPEDIT 19 1 "L := location(%%,op(%)):" "6#>%\"LG-%)locationG6
$%#%%G-%#opG6#%\"%G" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "subsop(% = op(%%
)/sin(psi),%%%):" "6#-%'subsopG6$/%\"%G*&-%#opG6#%#%%G\"\"\"-%$sinG6#%
$psiG!\"\"%$%%%G" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "odesubs := %:" "6#>
%(odesubsG%\"%G" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "mat(%);" "6#-%$matG6
#%\"%G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7#/-%%diffG6$&
%&OmegaG6#%$phiG%\"tG*&,&*&,&*&,&\"\"\"F7%%betaG!\"\"F7&F-6#%&thetaGF7
F7*(-%$cosG6#%$psiGF7,&F7F7F8F7F7F,F7F9F7&F-6#FAF7F7*&*(,(*(-%$sinGF@F
7&%#piG6#%\"YGF7-F?6#F/F7F9*&F>\"\"\"&FL6#%\"ZGF7F7*(FIFR-FJFPF7&FL6#%
\"XGF7F7F7%\"GGF7,&*&FXFRFOFRF7*&FWFRFKFRF7F7FR&%\"IG6#%#xyG!\"\"F7FRF
IF]o7#/-F*6$FCF0,&*&,&*(F>FRF8F7)F,\"\"#FRF7*(,&F9F7F8F7F7F:FRF,FRF7F7
FIFRF7*&*(FGFRFenFR,(*&FIFRFSFRF9*(FWFRFXFRF>FRF7*(FOFRF>FRFKFRF9F7FRF
inF]oF77#/-F*6$F:F0*&,&*&,&*&,&*&)F>FgoFRF8FRF7F7F7F7F,FRF7*(FioFRF>FR
F:FRF7F7FCFRF7*&**FGFRFenFRFfnFRF>FRFRFinF]oF9FRFIF]o" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 38 "Differentiate the equations of motion." }
}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "map(diff,eqs,t):" "6#-%$mapG6%
%%diffG%$eqsG%\"tG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "eqs2 := %:" "6#>%
%eqs2G%\"%G" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "mat(%);" "6#-%$matG6#%\"
%G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7#/,&*(-%$cosG6#%$
psiG\"\"\"-%%diffG6$F.%\"tGF/-F16$&%&OmegaG6#%$phiGF3F/F/*&-%$sinGF-F/
-F16$F6-%\"$G6$F3\"\"#F/F/,**&,(*&,&F/F/%%betaG!\"\"F/-F16$&F76#%&thet
aGF3F/F/**F;\"\"\"F0FP,&F/F/FHF/F/F6F/F/*(F+FPFQFPF4FPFIF/&F76#F.F/F/*
&,&*&FGFPFLF/F/*(F+FPFQFPF6FPFIF/-F16$FSF3F/F/*&*(,,**F+FPF0FP&%#piG6#
%\"YGF/-F,6#F9F/FI**F;FPFinFP-F<F^oF/-F16$F9F3F/F/*(F;FPF0FP&Fjn6#%\"Z
GF/FI**F+FPF0FPF`oFP&Fjn6#%\"XGF/F/**F;FPF]oFPFaoFPFhoFPF/F/%\"GGF/,&*
&FhoFPF]oFPF/*&F`oFPFinFPF/F/FP&%\"IG6#%#xyG!\"\"F/*&*(,(*(F;FPFinFPF]
oFPFI*&F+FPFdoFPF/*(F;FPF`oFPFhoFPF/F/F\\pFP,&*(FhoFPF`oFPFaoFPFI*(F]o
FPFaoFPFinFPF/F/FPF`pFdpF/7#/-F16$FSF?,**&,***F;FPF0FPFHF/)F6FBFPFI**F
+FPFHFPF6FPF4FPFB*(,&FIF/FHF/F/FJFPF6FPF/*(FiqFPFLFPF4FPF/F/F;FPF/*(,&
*(F+FPFHFPFfqFPF/*(FiqFPFLFPF6FPF/F/F+FPF0FPF/*&*(FgnFPF\\pFP,(*&F;FPF
doFPFI*(F`oFPFhoFPF+FPF/*(F]oFPF+FPFinFPFIF/FPF`pFdpF/*&*(FgpFPF\\pFP,
,*(F+FPF0FPFdoFPFI**F]oFPFaoFPFhoFPF+FPF/**F`oFPFhoFPF;FPF0FPFI**F`oFP
FaoFPF+FPFinFPF/**F]oFPF;FPF0FPFinFPF/F/FPF`pFdpF/7#/,&*(F+FPF0FPFJFPF
/*&F;FP-F16$FLF?F/F/,,*&,**,F+FPFHFPF;FPF0FPF6FP!\"#*&,&*&)F+FBFPFHFPF
/F/F/F/F4FPF/**FiqFPF;FPF0FPFLFPFI*(FiqFPF+FPFJFPF/F/FSFPF/*&,&*&FjsFP
F6FPF/*(FiqFPF+FPFLFPF/F/FYFPF/*&**FgnFPF\\pFPF]pFPF+FPFPF`pFdpFI*&**F
gpFPF\\pFPF[qFPF+FPFPF`pFdpFI*&*,FgpFPF\\pFPF]pFPF;FPF0FPFPF`pFdpF/" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Substitute for " }{XPPEDIT 18 0 
"diff(Omega[theta],t);" "6#-%%diffG6$&%&OmegaG6#%&thetaG%\"tG" }{TEXT 
-1 2 ", " }{XPPEDIT 18 0 "diff(Omega[phi],t);" "6#-%%diffG6$&%&OmegaG6
#%$phiG%\"tG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "diff(Omega[psi],t);
" "6#-%%diffG6$&%&OmegaG6#%$psiG%\"tG" }{TEXT -1 55 " using the origin
al equations of motion.  Then, assume " }{XPPEDIT 18 0 "Omega[theta]" 
"6#&%&OmegaG6#%&thetaG" }{TEXT -1 40 " is large and the pressure terms
 (i.e., " }{XPPEDIT 18 0 "G(a,b,h,A[C],A[T],alpha);" "6#-%\"GG6(%\"aG%
\"bG%\"hG&%\"AG6#%\"CG&F*6#%\"TG%&alphaG" }{TEXT -1 72 ") are small.  \+
We find the resulting second-order differential equations," }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "subsODEs := proc( ode, odesubs )\n  loc
al q, Lq, u, Lu, eqn;\n  eqn := copy(ode);\n  if not (isdiff(eqn) and \+
difforder(eqn)>1) then\n    for q in eqn do\n      Lq := location( eqn
, q );\n      if isdiff(q) and difforder(q)=1 then\n        q := subs(
 odesubs, q );\n      elif nops(q) > 1 and not isdiff(u) then\n       \+
 q := procname( q, odesubs );\n      fi;\n      eqn := subsop( Lq=q, e
qn );\n    od;\n  fi;\n  eval(eqn);\nend:" "6#>%)subsODEsGR6$%$odeG%(o
desubsG7'%\"qG%#LqG%\"uG%#LuG%$eqnG6\"F/C%>F.-%%copyG6#F'@$43-%'isdiff
G6#F.2\"\"\"-%*difforderG6#F.?&F*F.%%trueGC%>F+-%)locationG6$F.F*@&3-F
96#F*/-F>6#F*\"\"\">F*-%%subsG6$F(F*32\"\"\"-%%nopsG6#F*4-F96#F,>F*-%)
procnameG6$F*F(>F.-%'subsopG6$/F+F*F.-%%evalG6#F.F/F/F/" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "for ii to 3 do factor(subs(odesubs,rhs(eqs2[ii]))-subsO
DEs(rhs(eqs2[ii]),odesubs)) od;" "6#?(%#iiG\"\"\"F%\"\"$%%trueG-%'fact
orG6#,&-%%subsG6$%(odesubsG-%$rhsG6#&%%eqs2G6#F$F%-%)subsODEsG6$-F16#&
F46#F$F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}
}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subsODEs(eqs2,odesubs):" "6#-%)
subsODEsG6$%%eqs2G%(odesubsG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(di
ff(psi,t) = Omega[psi],diff(phi,t) = Omega[phi],%):" "6#-%%subsG6%/-%%
diffG6$%$psiG%\"tG&%&OmegaG6#F*/-F(6$%$phiGF+&F-6#F2%\"%G" }}{PARA 0 "
" 0 "" {XPPEDIT 19 1 "convert(expansion(%,[Omega[psi], Omega[phi], G],
1),diff):" "6#-%(convertG6$-%*expansionG6%%\"%G7%&%&OmegaG6#%$psiG&F,6
#%$phiG%\"GG\"\"\"%%diffG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 
19 1 "Collect(%,[G],factor):" "6#-%(CollectG6%%\"%G7#%\"GG%'factorG" }
}{PARA 0 "" 0 "" {XPPEDIT 19 1 "eqs2 := invsubs([op(gsubs), lhs(op(sel
ect(has,gsubs,G))) = G],%):" "6#>%%eqs2G-%(invsubsG6$7$-%#opG6#%&gsubs
G/-%$lhsG6#-F*6#-%'selectG6%%$hasGF,%\"GGF7%\"%G" }}{PARA 0 "" 0 "" 
{XPPEDIT 19 1 "mat(%);" "6#-%$matG6#%\"%G" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7#/*&-%$sinG6#%$psiG\"\"\"-%%diffG6$&%&Om
egaG6#%$phiG-%\"$G6$%\"tG\"\"#F.,&*&*,,&!\"\"F.%%betaGF.F.&F36#%&theta
GF.-&%\"gG6#\"\"!6$F5F-F.-&FF6#F:FIF.-%\"GG6(%\"aG%\"bG%\"hG&%\"AG6#%
\"CG&FT6#%\"TG%&alphaGF.\"\"\"&%\"IG6#%#xyG!\"\"F?**)F>F:Fen)FAF:FenF*
FenF2F.F?7#/-F06$&F36#F-F6,&*&*,F>FenFAFenFDFenFMFen-&FF6#F.FIF.FenFfn
FjnF.*(F\\oFenF]oFenFboF.F?7#/*&F*Fen-F06$FAF6F.,&*,F\\oFen-%$cosGF,F.
F]oFenF*FenF2FenF.*&*.F>FenFbpFenFAFenFDFenFMFenFJFenFenFfnFjnF." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Omega[theta
];" "6#&%&OmegaG6#%&thetaG" }{TEXT -1 65 " is large, we may also assum
e its rate of change is small, hence " }{XPPEDIT 18 0 "diff(Omega[thet
a],`$`(t,2)) = 0;" "6#/-%%diffG6$&%&OmegaG6#%&thetaG-%\"$G6$%\"tG\"\"#
\"\"!" }{TEXT -1 82 ".  The first consequence of this is that the angu
lar velocity of the inclination, " }{XPPEDIT 18 0 "Omega[psi];" "6#&%&
OmegaG6#%$psiG" }{TEXT -1 51 ", exhibits simple harmonic motion: the s
olution of " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "diff(Omega[psi],
`$`(t,2))+(1-beta)^2*Omega[theta]^2*Omega[psi]+(1-beta)*g[0](phi,psi)*
G(a,b,h,A[C],A[T],alpha)*g[1](phi,psi)*Omega[theta]/I[xy] = 0:" "6#/,(
-%%diffG6$&%&OmegaG6#%$psiG-%\"$G6$%\"tG\"\"#\"\"\"*(,&\"\"\"F1%%betaG
!\"\"\"\"#&F)6#%&thetaG\"\"#&F)6#F+F1F1*.,&\"\"\"F1F5F6F1-&%\"gG6#\"\"
!6$%$phiGF+F1-%\"GG6(%\"aG%\"bG%\"hG&%\"AG6#%\"CG&FO6#%\"TG%&alphaGF1-
&FC6#\"\"\"6$FGF+F1&F)6#F:F1&%\"IG6#%#xyGF6F1FE" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 7 "is just" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "col
lect(subs(omega = Omega[theta],g[0] = g[0](phi,psi),g[1] = g[1](phi,ps
i),dsolve(subs(Omega[theta] = omega,g[1](phi,psi) = g[1],g[0](phi,psi)
 = g[0],%),Omega[psi])),[_C1, _C2],factor);" "6#-%(collectG6%-%%subsG6
&/%&omegaG&%&OmegaG6#%&thetaG/&%\"gG6#\"\"!-&F16#F36$%$phiG%$psiG/&F16
#\"\"\"-&F16#\"\"\"6$F8F9-%'dsolveG6$-F'6&/&F,6#F.F*/-&F16#\"\"\"6$F8F
9&F16#\"\"\"/-&F16#F36$F8F9&F16#F3%\"%G&F,6#F97$%$_C1G%$_C2G%'factorG
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&OmegaG6#%$psiG,(*&*(-&%\"gG6#
\"\"!6$%$phiGF'\"\"\"-%\"GG6(%\"aG%\"bG%\"hG&%\"AG6#%\"CG&F:6#%\"TG%&a
lphaGF2-&F-6#F2F0F2\"\"\"*(&F%6#%&thetaG\"\"\",&!\"\"F2%%betaGF2\"\"\"
&%\"IG6#%#xyG\"\"\"!\"\"F2*&%$_C1GF2-%$sinG6#*(FJF2FFF2%\"tGF2F2F2*&%$
_C2GF2-%$cosGFXF2F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "which is s
imple harmonic motion with a small linear drift in " }{XPPEDIT 18 0 "p
si(t);" "6#-%$psiG6#%\"tG" }{TEXT -1 112 ".  The second consequence st
ems from either the first or the third equation (both say the same thi
ng).  Setting " }{XPPEDIT 18 0 "diff(Omega[theta],`$`(t,2)) = 0;" "6#/
-%%diffG6$&%&OmegaG6#%&thetaG-%\"$G6$%\"tG\"\"#\"\"!" }{TEXT -1 103 " \+
places a constraint on the value of the longitudinal motion \227 i.e.,
 the precession rate.  We then have" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "subs(select(has,gsubs,g),isolate(rhs(eqs2[3]),Omega[phi
]));" "6#-%%subsG6$-%'selectG6%%$hasG%&gsubsG%\"gG-%(isolateG6$-%$rhsG
6#&%%eqs2G6#\"\"$&%&OmegaG6#%$phiG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "p
recession_rate := %:" "6#>%0precession_rateG%\"%G" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/&%&OmegaG6#%$phiG,$*&*(,(*(-%$sinG6#%$psiG\"\"\"&%#piG
6#%\"YGF1-%$cosG6#F'F1!\"\"*&-F7F/F1&F36#%\"ZGF1F1*(F-\"\"\"-F.F8F1&F3
6#%\"XGF1F1F1-%\"GG6(%\"aG%\"bG%\"hG&%\"AG6#%\"CG&FL6#%\"TG%&alphaGF1,
(*&F-F@F<F@F9*(FAF@FBF@F;F@F1*(F6F@F;F@F2F@F9F1F@**,&F9F1%%betaGF1\"\"
\"&F%6#%&thetaG\"\"\"&%\"IG6#%#xyG\"\"\"F-\"\"\"!\"\"F9" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 122 "To make this expression a little bit cle
arer, further assume that the pressure is mainly along the Z axis \227
 that is, that " }{XPPEDIT 18 0 "pi[X];" "6#&%#piG6#%\"XG" }{TEXT -1 
5 " and " }{XPPEDIT 18 0 "pi[Y];" "6#&%#piG6#%\"YG" }{TEXT -1 41 " are
 small.  Then this expression becomes" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "Collect(expansion(%,[pi[X], pi[Y]],1),[G(a,b,h,A[C],A[T
],alpha), Pi, abs(P), 1/Omega[theta], 1/I[xy], -1+beta, pi[Z]],factor)
;" "6#-%(CollectG6%-%*expansionG6%%\"%G7$&%#piG6#%\"XG&F,6#%\"YG\"\"\"
7)-%\"GG6(%\"aG%\"bG%\"hG&%\"AG6#%\"CG&F;6#%\"TG%&alphaG%#PiG-%$absG6#
%\"PG*&\"\"\"\"\"\"&%&OmegaG6#%&thetaG!\"\"*&\"\"\"FI&%\"IG6#%#xyGFN,&
\"\"\"FN%%betaGFI&F,6#%\"ZG%'factorG" }}{PARA 12 "" 1 "" {XPPMATH 20 "
6#/&%&OmegaG6#%$phiG*&*&,&*&-%$cosG6#%$psiG\"\"\")&%#piG6#%\"ZG\"\"#\"
\"\"F0*&**,&F,F0-%$sinGF.!\"\"F0,&F,F0F;F0F0,&*&&F36#%\"YGF0-F-6#F'F0F
=*&&F36#%\"XGF0-F<FEF0F0F0F2F0F7F;!\"\"F=F0-%\"GG6(%\"aG%\"bG%\"hG&%\"
AG6#%\"CG&FS6#%\"TG%&alphaGF0F7*(,&F=F0%%betaGF0\"\"\"&%\"IG6#%#xyG\"
\"\"&F%6#%&thetaG\"\"\"FK" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Thu
s, we have a weak phase dependence on the non-axial pressure component
s.  For the purely axial case " }{XPPEDIT 18 0 "pi[Z] = 1" "6#/&%#piG6
#%\"ZG\"\"\"" }{TEXT -1 9 ", we have" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "subs(pi[Z] = -1,pi[X] = 0,pi[Y] = 0,%);" "6#-%%subsG6&/
&%#piG6#%\"ZG,$\"\"\"!\"\"/&F(6#%\"XG\"\"!/&F(6#%\"YGF2%\"%G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/&%&OmegaG6#%$phiG*&*&-%$cosG6#%$psiG\"\"\"-
%\"GG6(%\"aG%\"bG%\"hG&%\"AG6#%\"CG&F66#%\"TG%&alphaGF.\"\"\"*(,&!\"\"
F.%%betaGF.\"\"\"&%\"IG6#%#xyG\"\"\"&F%6#%&thetaG\"\"\"!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Finally, the precession for a flat
 disk of uniform albedo, " }{XPPEDIT 18 0 "alpha = Pi/2;" "6#/%&alphaG
*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "A[C] = A;" "
6#/&%\"AG6#%\"CGF%" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "A[T] = A;" "6
#/&%\"AG6#%\"TGF%" }{TEXT -1 4 ", is" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "factor(eval(subs(gsubs,alpha = Pi/2,A[C] = A,A[T] = A,%
)));" "6#-%'factorG6#-%%evalG6#-%%subsG6'%&gsubsG/%&alphaG*&%#PiG\"\"
\"\"\"#!\"\"/&%\"AG6#%\"CGF6/&F66#%\"TGF6%\"%G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%&OmegaG6#%$phiG*&*.-%$cosG6#%$psiG\"\"\"%#PiGF.-%$ab
sG6#%\"PGF.%\"hGF.)%\"bG\"\"#\"\"\",&!\"\"F.%\"AGF.F.F8*(,&F:F.%%betaG
F.\"\"\"&F%6#%&thetaG\"\"\"&%\"IG6#%#xyG\"\"\"!\"\"" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 36 "Let's look at the precession period " }
{XPPEDIT 18 0 "T[phi] = 2*Pi/Omega[phi];" "6#/&%\"TG6#%$phiG*(\"\"#\"
\"\"%#PiGF*&%&OmegaG6#F'!\"\"" }{TEXT -1 88 " as a function of cone an
gle and albedo.  For a pressure field along the Z axis, we have" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(pi[Z] = -1,pi[X] = 0,pi[Y] \+
= 0,precession_rate):" "6#-%%subsG6&/&%#piG6#%\"ZG,$\"\"\"!\"\"/&F(6#%
\"XG\"\"!/&F(6#%\"YGF2%0precession_rateG" }}{PARA 0 "" 0 "" {XPPEDIT 
19 1 "T[phi] = 2*Pi/rhs(%);" "6#/&%\"TG6#%$phiG*(\"\"#\"\"\"%#PiGF*-%$
rhsG6#%\"%G!\"\"" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "precession_period :
= %:" "6#>%2precession_periodG%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/&%\"TG6#%$phiG,$*&**%#PiG\"\"\",&!\"\"F,%%betaGF,F,&%\"IG6#%#xyGF,&%
&OmegaG6#%&thetaGF,\"\"\"*&-%$cosG6#%$psiG\"\"\"-%\"GG6(%\"aG%\"bG%\"h
G&%\"AG6#%\"CG&FF6#F%%&alphaG\"\"\"!\"\"\"\"#" }}}{SECT 1 {PARA 5 "" 
0 "" {TEXT -1 28 "3D Plot of Precession Period" }}{EXCHG {PARA 0 "" 0 
"" {XPPEDIT 19 1 "subs(select(has,gsubs,G),precession_period):" "6#-%%
subsG6$-%'selectG6%%$hasG%&gsubsG%\"GG%2precession_periodG" }}{PARA 0 
"" 0 "" {XPPEDIT 19 1 "subs(psi = inclin,abs(P) = P,A[C] = AC,A[T] = A
T,I[xy] = Ixy,Omega[theta] = 2*Pi/Tspin,rhs(%)):" "6#-%%subsG6)/%$psiG
%'inclinG/-%$absG6#%\"PGF-/&%\"AG6#%\"CG%#ACG/&F06#%\"TG%#ATG/&%\"IG6#
%#xyG%$IxyG/&%&OmegaG6#%&thetaG*(\"\"#\"\"\"%#PiGFF%&TspinG!\"\"-%$rhs
G6#%\"%G" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "optimize(%,tryhard):" "6#-%
)optimizeG6$%\"%G%(tryhardG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "Tphi := \+
makeproc([%],parameters = [alpha, a, b, h, AC, AT, inclin, Tspin, beta
, Ixy, P]):" "6#>%%TphiG-%)makeprocG6$7#%\"%G/%+parametersG7-%&alphaG%
\"aG%\"bG%\"hG%#ACG%#ATG%'inclinG%&TspinG%%betaG%$IxyG%\"PG" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Tmin := -100.0:" "6#>%%TminG,$$
\"%+5!\"\"!\"\"" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "Tmax := 100.0:" "6#>
%%TmaxG$\"%+5!\"\"" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "ACmin := .7:" "6#
>%&ACminG$\"\"(!\"\"" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "ACmax := .95:" 
"6#>%&ACmaxG$\"#&*!\"#" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "Ncontours := \+
30:" "6#>%*NcontoursG\"#I" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 
19 1 "Clevels := [seq(Tmin+(ii-1.0)*(Tmax-Tmin)/(Ncontours-1.0),ii = 1
 .. Ncontours)]:" "6#>%(ClevelsG7#-%$seqG6$,&%%TminG\"\"\"*(,&%#iiGF+$
\"#5!\"\"!\"\"F+,&%%TmaxGF+F*F2F+,&%*NcontoursGF+$\"#5!\"\"F2F2F+/F.;
\"\"\"F6" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "ConColors := [yellow, yello
w]:" "6#>%*ConColorsG7$%'yellowGF&" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "a
lphamin := 84:" "6#>%)alphaminG\"#%)" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 
"alphamax := 92:" "6#>%)alphamaxG\"##*" }}{PARA 0 "" 0 "" {XPPEDIT 19 
1 "ConPos := Tmin:" "6#>%'ConPosG%%TminG" }}{PARA 0 "" 0 "" {XPPEDIT 
19 1 "orient := [40, 60]:" "6#>%'orientG7$\"#S\"#g" }}}{EXCHG {PARA 0 
"" 0 "" {XPPEDIT 19 1 "surfplot := plot3d(Tphi(alpha*deg,1,3,1,AC,.8,4
5*deg,2*3600,.1,200.0,4.5*10^(-6))/3600.0, AC = ACmin .. ACmax, alpha \+
= alphamin .. alphamax, grid = [60, 60], style = patchcontour, contour
s = Clevels):" "6#>%)surfplotG-%'plot3dG6(*&-%%TphiG6-*&%&alphaG\"\"\"
%$degGF.\"\"\"\"\"$\"\"\"%#ACG$\"\")!\"\"*&\"#XF.F/F.*&\"\"#F.\"%+OF.$
\"\"\"!\"\"$\"%+?!\"\"*&$\"#X!\"\"F.)\"#5,$\"\"'!\"\"F.F.$\"&+g$!\"\"F
J/F3;%&ACminG%&ACmaxG/F-;%)alphaminG%)alphamaxG/%%gridG7$\"#g\"#g/%&st
yleG%-patchcontourG/%)contoursG%(ClevelsG" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "conplot := plots[contourplot](Tphi(alpha*deg,1,3,1,AC,.
8,45*deg,2*3600,.1,200.0,4.5*10^(-6))/3600.0,AC = ACmin .. ACmax,alpha
 = alphamin .. alphamax,grid = [40, 40],contours = Clevels,coloring = \+
ConColors,filled = true,view = Tmin .. Tmax):" "6#>%(conplotG-&%&plots
G6#%,contourplotG6**&-%%TphiG6-*&%&alphaG\"\"\"%$degGF1\"\"\"\"\"$\"\"
\"%#ACG$\"\")!\"\"*&\"#XF1F2F1*&\"\"#F1\"%+OF1$\"\"\"!\"\"$\"%+?!\"\"*
&$\"#X!\"\"F1)\"#5,$\"\"'!\"\"F1F1$\"&+g$!\"\"FM/F6;%&ACminG%&ACmaxG/F
0;%)alphaminG%)alphamaxG/%%gridG7$\"#S\"#S/%)contoursG%(ClevelsG/%)col
oringG%*ConColorsG/%'filledG%%trueG/%%viewG;%%TminG%%TmaxG" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "plot_xform := plottools[transform](proc
 (x, y) options operator, arrow; [x, y, ConPos] end):" "6#>%+plot_xfor
mG-&%*plottoolsG6#%*transformG6#R6$%\"xG%\"yG7\"6$%)operatorG%&arrowG6
\"7%F-F.%'ConPosGF3F3F3" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "disp
 := plots[display](\{surfplot, plot_xform(conplot)\},view = Tmin .. Tm
ax,projection = .9,orientation = orient,labels = [\"AC\", \"alpha\", T
],title = \"Precession Period vs. Cone Angle and Albedo\"):" "6#>%%dis
pG-&%&plotsG6#%(displayG6(<$%)surfplotG-%+plot_xformG6#%(conplotG/%%vi
ewG;%%TminG%%TmaxG/%+projectionG$\"\"*!\"\"/%,orientationG%'orientG/%'
labelsG7%Q#AC6\"Q&alphaFB%\"TG/%&titleGQLPrecession~Period~vs.~Cone~An
gle~and~AlbedoFB" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "disp;" "6#%
%dispG" }}{PARA 13 "" 1 "" {GLPLOT3D 626 513 513 {PLOTDATA 3 "6eo-%%GR
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