Milankowitch Cycles. 1 Coordinates. 2 Rotation matrix of Eulerian angles. (%i1) kill(all); (%o0) done
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1 369(9-rot.wxm 1 / 1 Milankowitch Cycles (%i1 kill(all; (%o one 1 Coorinates (%i1 epens([x,y,theta,phi,psi,theta_1,t,u,l,r,omega],t; (%o1 [x( t,y( t,( t,'( t,( t, 1 ( t,t( t,u( t,l( t,r( t,!( t ] (%i e_r: [sin(theta*cos(phi, sin(theta*sin(phi, cos(theta]; (%o [cos(' sin(,sin(' sin(,cos( ] Rotation matrix of Eulerian angles (%i3 Rz: matrix([cos(phi, sin(phi, ], [-sin(phi, cos(phi, ], [,, 1]; (%o3 cos( ' sin( ' cos( ' 1 (%i4 Ry: matrix([cos(theta,, sin(theta], [, 1, ], [-sin(theta,, cos(theta]; (%o4 1 (%i5 Rz1: matrix([cos(psi, sin(psi, ], [-sin(psi, cos(psi, ], [,, 1]; (%o5 cos( sin( sin( cos( 1 (%i6 R3: Rz1.Ry.Rz; (%o6 cos( ' cos( sin( cos( ' sin( cos( cos( ' sin( ' cos( cos( +cos( ' sin( cos( ' cos( sin( cos( sin(
2 369(9-rot.wxm / 1 3 bol r_1 (%i7 r_1: transpose([r11,r1,r13].r3; (%o7 cos( ' r13 sin( +r1 ( cos( ' sin( cos( +r11 (cos(' cos( r13 sin( +r1 ( cos( ' cos( sin( cos( +r11 (sin(' cos( cos( +cos( ' sin( r1 sin( +cos( r11 sin( +r13 (%i1 X_1: first(r_1[1]; Y_1: first(r_1[]; Z_1: first(r_1[3]; (%o8 cos( ' r13 sin( +r1 ( cos( ' cos( +r11 ( cos( ' (%o9 r13 sin( +r1 ( cos( ' cos( +r11 ( sin( ' cos( +cos( ' (%o1 r1 sin( + r11 sin( +r13 Check (%i13 r_1x: r11*r3[1,1] +r1*r3[,1] +r13*r3[3,1]; r_1y: r11*r3[1,] +r1*r3[,] +r13*r3[3,]; r_1z: r11*r3[1,3] +r1*r3[,3] +r13*r3[3,3]; (%o11 cos( ' r13 sin( +r1 ( cos( ' cos( +r11 ( cos( ' (%o1 r13 sin( +r1 ( cos( ' cos( +r11 ( sin( ' cos( +cos( ' (%o13 r1 sin( + r11 sin( +r13 (%i14 r_1-[r_1x, r_1y, r_1z]; sin sin (%o14 4 bol r (%i15 [X, Y, Z]: [r*cos(theta_1, r*sin(theta_1, ]; (%o15 [r cos( 1,r sin( 1,] (%i16 [X, Y, Z]: [,, ]; (%o16 [,,] 5 Kinetic energy
3 369(9-rot.wxm 3 / 1 (%i19 T[1]: m/*(iff(x+x_1,t^; T[]: m/*(iff(y+y_1,t^; T[3]: m/*(iff(z+z_1,t^; (%o17 (m (r1 (cos(' t cos( ' t cos( + sin( ' t ' cos( +sin( ' t cos( ' t ' cos( + r11 ( cos(' t ' r13 t + t cos( ' ' t ' r13 sin( ^ / t t ' t ' sin( cos(' (%o18 (m (r1 (sin(' t t cos( ' t ' cos( ' t t ' cos( + r11 ( sin(' t ' cos( +cos( ' r13 t t ' t ' r13 sin( ^ / t t cos( +cos( ' ' t ' sin( sin(' (%o19 (m ( r13 t sin( r1 t +cos( r11 cos( t r1 t r11 sin( ^ / t (%i T: (T[1]+T[]+T[3]$ 6 Potential energy (%i1 U: -m*m*g/r; (%o1 G M m r (%i U: ; (%o 7 Lagrange function
4 369(9-rot.wxm 4 / Simplify r_1: r1=r13=, a point on the equator (%i3 r1: r13: ; (%o3 (%i4 L: ev(t - U; (%o4 (m r11 ( sin(' cos( ' t ' cos( + / +(m r11 ( cos(' sin( ' m / + ' t ' cos( r11 t 8 Lagrange equations II 8.1 phi equation (%i5 D1: iff(l, iff(phi,t; t t t cos( ' t r11 t (%o5 m r11 ( cos( ' sin( ' sin( t t cos( +cos( ' ' t t ' sin( +m r11 ( cos( ' t cos( ' t ' ' t cos( + t ' sin( ^ t t ' sin( ^ ' ( sin(' t ' cos( +cos( ' ( cos(' t ' t cos( ' t
5 369(9-rot.wxm 5 / 1 (%i6 E1: (trigsimp(ratsimp(iff(d1,t - iff(l,phi = ; (%o6 m r11 t + ( m m +m t +m cos( t r11 r11 m t ' r11 t + m t ' t m t ' t r11 cos( +m t ' r11 = (%i7 E11: (solve(e1, iff(phi,t,; (%o7 [ '=( t + ( t t + 8. theta equation + t t ' (%i8 D: iff(l, iff(theta,t; t cos( /(cos( 1] t r11 t + t ' (%o8 m sin( ' r11 sin( ( sin(' t sin( ' t cos( +cos( ' t ' cos( +cos( ' t sin( ' t ' sin( m cos( ' r11 sin( ( cos(' t cos( ' t r11 t ' ' t ' sin( +m r11 cos( t t ' r11 t (%i9 E: trigsimp(ratsimp(iff(d,t - iff(l,theta = ; (%o9 m r11 t ' r11 cos( + m t ' m t t +m t r11 t +(m ' t r11 sin( =
6 369(9-rot.wxm 6 / 1 (%i3 E1: solve(e, iff(theta,t,; (%o3 [ =( sin( t t ' / cos( ] 8.3 psi equation t t + t ' (%i31 D3: iff(l, iff(psi,t; t (%o31 m r11 ( cos( ' sin( t t cos( +cos( ' ' t t ' sin( +m r11 ( cos( ' t cos( ' ' sin( t ( sin(' t ' cos( +cos( ' ( cos(' t ' t ' sin( m r11 sin( r11 t r11 t (%i3 E3: trigsimp(ratsimp(iff(d3,t - iff(l,psi = ; (%o3 m m r11 t m t ' r11 +m r11 = (%i33 E31: solve(e3, iff(psi,t,; (%o33 [ = t t ' 8.4 theta_1 equation t + t cos( ' t t ' r11 ' cos( ] t (%i34 D4: iff(l, iff(theta_1,t; (%o34 ' t r11 cos( +m t ' t t t +
7 369(9-rot.wxm 7 / 1 (%i35 E4: trigsimp(ratsimp(iff(d4,t - iff(l,theta_1 = ; (%o35 = (%i36 E41: solve(e4, iff(theta_1,t,; (%o36 all 8.5 r equation (%i37 D5: iff(l, iff(r,t; (%o37 (%i38 E5: trigsimp(ratsimp(iff(d5,t - iff(l,r = ; (%o38 = (%i39 E51: solve(e5, iff(r,t,; (%o39 all 9 Prepare solution (%i4 GC: first(solve([first(e1,first(e,first(e3,first(e4,first(e5], [iff(phi,t,, iff(theta,t,, iff(psi,t,, iff(theta_1,t, solve: epenent equations eliminate: (4 5 (%i41 transform(eq := (A: ratsubst(phi_, iff(phi,t, Eq, A: ratsubst(phi_, iff(phi,t,, A, A: ratsubst(theta_, iff(theta,t, A, A: ratsubst(theta_, iff(theta,t,, A, A: ratsubst(psi_, iff(psi,t, A, A: ratsubst(psi_, iff(psi,t,, A, A: ratsubst(theta_1, iff(theta_1,t, A, A: ratsubst(theta_1, iff(theta_1,t,, A, A: ratsubst(r_, iff(r,t, A, ratsubst(r_, iff(r,t,, A ; (%o41 transform( Eq :=(A:ratsubst ', ',Eq,A: t ratsubst phi_, ',A,A:ratsubst t,,a,a: t ratsubst theta_,,a,a:ratsubst t,,a,a: t ratsubst psi_,,a,a:ratsubst theta_1, t t 1,A,A: ratsubst ratsubst theta_1, r, t r,a t 1,A,A:ratsubst r, t r,a,
8 369(9-rot.wxm 8 / 1 (%i46 G1: transform(gc[1]$ G: transform(gc[]$ G3: transform(gc[3]$ G4: transform(gc[4]$ G5: transform(gc[5]$ 1 Solution (%i47 str: [r11=1, m=1, G=, M=]; (%o47 [r11=1,m=1,g =,M =] (%i57 Eq1: gfactor(ev(rhs(g1, str, eval$ Eq: gfactor(ev(rhs(g, str, eval$ Eq3: gfactor(ev(rhs(g3, str, eval$ Eq4: gfactor(ev(rhs(g4, str, eval$ Eq5: gfactor(ev(rhs(g5, str, eval$ Eq6: phi_; Eq7: theta_; Eq8: psi_; Eq9: theta_1; Eq1: r_; (%o53 ' (%o54 (%o55 (%o56 theta_1 (%o57 r (%i93 s: rk([eq1, Eq, Eq3, Eq6, Eq7, Eq8], [phi_, theta_, psi_, phi, theta, psi], [.1,.1,.1,., %pi/, ], [t,,.1,.1]$
9 369(9-rot.wxm 9 / 1 (%i15 Eq1;Eq;Eq3; (%o13 ( sin( + ( sin( (%o14 ( + +( ' ' 3 cos( + ' ' ' 3 ' +' cos( + ' sin( /( ( (%o15 ((( sin( +( ' ' ' +( + cos( + ' sin( + ' 3 ' ' ' /(( (%i116 trigsimp(enom(eq1;trigsimp(num(eq1; (%o115 (%o Graphics (%i99 c1: makelist([p[1],p[]],p,s$ c: makelist([p[1],p[3]],p,s$ c3: makelist([p[1],p[4]],p,s$ c4: makelist([p[1],p[5]],p,s$ c5: makelist([p[1],p[6]],p,s$ c6: makelist([p[1],p[7]],p,s$ 1. Plot theta, phi, psi (%i11 c1;c; (%o1 [[.,.1]] (%o11 [[.,.1]]
10 369(9-rot.wxm 1 / 1 (%i78 wxplot([[iscrete, c1], [iscrete, c]], [xlabel, "time"], [legen, "phi_{ot}", "theta_{ot}"]$ Message from maxima's sterr stream: Warning: empty x range [:], ajusting (%t78
11 369(9-rot.wxm 11 / 1 (%i79 wxplot([[iscrete, c3], [iscrete, c4]], [xlabel, "time"], [legen, "psi_{ot}", "theta_1_{ot}"]$ Message from maxima's sterr stream: Warning: empty x range [:], ajusting Warning: empty y range [:], ajusting to [-1:1] (%t79
12 369(9-rot.wxm 1 / 1 (%i8 wxplot([[iscrete, c5], [iscrete, c6]], [xlabel, "time"], [legen, "r_{ot}", "phi"]$ Message from maxima's sterr stream: Warning: empty x range [:], ajusting (%t8 ; ;
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