Spinning top - represented by dumbbell

Transcription

1 Lagange-umbbell.wxm 1 / 10 Spinning top - epesente by umbbell (%i1) kill(all); (%o0) one 1 Cooinates (%i1) epens([x,y,z,x,y,z,theta,phi,theta_1,phi_1,,t,u,l],t); (%o1) [x( t ),y( t ),z( t ),X( t ),Y( t ),Z( t ),( t ),'( t ), 1 ( t ),' 1 ( t ),( t ), T( t ),U( t ),L( t )] (%i) h_1: h*[sin(theta)*cos(phi), sin(theta)*sin(phi), cos(theta)]; (%o) [h cos( ') sin( ),h sin( ') sin( ),h cos( )] (%i3) h_: h*[sin(%pi-theta)*cos(phi+%pi), sin(%pi-theta)*sin(phi+%pi), co (%o3) [ h cos( ') sin( ), h sin( ') sin( ), h cos( )] (%i4) R: *[sin(theta_1)*cos(phi_1), sin(theta_1)*sin(phi_1), cos(theta_1) (%o4) [cos(' 1 ) sin( 1 ),sin(' 1 ) sin( 1 ), cos( 1 )] (%i5) _1: R+h_1; (%o5) [cos(' 1 ) sin( 1 ) +h cos( ') sin( ),sin(' 1 ) sin( 1 ) +h sin( ') sin( ), cos( 1 ) +h cos( )] (%i6) _: R+h_; (%o6) [cos(' 1 ) sin( 1 ) h cos( ') sin( ),sin(' 1 ) sin( 1 ) h sin( '), cos( 1 ) h cos( )] Kinetic enegy

2 Lagange-umbbell.wxm / 10 (%i7) T[1]: m/*(iff(_1[1],t)^+iff(_1[],t)^+iff(_1[3],t)^); (%o7) (m ( sin( 1 ) t 1 + t cos ( 1 ) h t +( sin( ' 1 ) cos( 1 ) t 1 +sin( ' 1 ) t sin ( 1 ) +cos( ' 1 ) t ' 1 sin( 1 ) +h sin( ') cos( ) +h cos( ) t ' t ' sin( ))^ +(cos (' 1 ) cos( 1 ) t 1 +cos( ' 1 ) t sin ( 1 ) sin( ' 1 ) t ' 1 sin( 1 ) +h cos( ') cos( ) h t ' t ' sin( ) )^ ))/ (%i8) T[]: m/*(iff(_[1],t)^+iff(_[],t)^+iff(_[3],t)^); (%o8) (m ( sin( 1 ) t 1 + t cos ( 1 ) +h t +( sin( ' 1 ) cos( 1 ) t 1 +sin( ' 1 ) t sin ( 1 ) +cos( ' 1 ) t ' 1 sin( 1 ) h sin( ') cos( ) h cos( ) t ' t ' sin( ))^ +(cos (' 1 ) cos( 1 ) t 1 +cos( ' 1 ) t sin ( 1 ) sin( ' 1 ) t ' 1 sin( 1 ) h cos( ') cos( ) +h t ' t ' sin( ) )^ ))/ (%i9) T: facto(tigsimp(atsimp(t[1]+t[]))); (%o9) m ( t 1 + t ' 1 sin( 1 ) +h t + t ) 3 Potential enegy +h t ' /*U[1]: tigsimp(-m*m*g/(_1[1]^+_1[]^+_1[3])); U[]: tigsimp(-m*m*g/(_1[1]^+_1[]^+_1[3]));*/; (%i11) U[1]: tigsimp(-m*m*g/); U[]: tigsimp(-m*m*g/); (%o10) G M m (%o11) G M m

3 Lagange-umbbell.wxm 3 / 10 (%i1) U: gfacto(tigsimp(u[1] + U[])); (%o1) G M m 4 Lagange function (%i13) T; (%o13) m ( + (%i14) U; t 1 t ) (%o14) G M m (%i15) L: T - U; (%o15) m ( t t ' 1 + t )+ G M m t ' 1 5 Lagange equations II 5.1 theta equation (%i16) D1: iff(l, iff(theta,t)); (%o16) h m t 1 +h t 1 +h t +h t ' +h t ' (%i17) E1: expan(atsimp(iff(d1,t) - iff(l,theta) = 0)); (%o17) h m t h m t ' cos( ) sin( ) =0 (%i18) E11: tigsimp(solve(e1, iff(theta,t,))); (%o18) [ = t t ' 5. phi equation cos( ) sin( )] (%i19) D: iff(l, iff(phi,t)); (%o19) h m t '

4 Lagange-umbbell.wxm 4 / 10 (%i0) E: tigsimp(atsimp(iff(d,t) - iff(l,phi) = 0)); (%o0) 4 h m t ' cos( ) t + h m (%i1) E1: tigsimp(solve(e, iff(phi,t,))); (%o1) [ t '= t ' cos( ) 5.3 theta_1 equation t (%i) D3: iff(l, iff(theta_1,t)); (%o) m t 1 ] ' =0 t (%i3) E3: expan(atsimp(iff(d3,t) - iff(l,theta_1) = 0)); (%o3) m t 1 +4 m t t 1 m t ' 1 cos( 1 ) sin( 1 ) =0 (%i4) E31: tigsimp(solve(e3, iff(theta_1,t,))); (%o4) [ t t 1 t ' 1 cos( 1 ) sin( 1 ) t 1= ] 5.4 phi_1 equation (%i5) D4: iff(l, iff(phi_1,t)); (%o5) m t ' 1 sin( 1 ) (%i6) E4: expan(atsimp(iff(d4,t) - iff(l,phi_1) = 0)); (%o6) 4 m t ' 1 cos( 1 ) sin( 1 ) t 1 +4 m t ' 1 sin( 1 ) + m ' t 1 sin( 1 ) =0 t (%i7) E41: tigsimp(solve(e4, iff(phi_1,t,))); (%o7) [ ' t ' 1 cos( 1 ) t 1 + t ' 1 t sin ( 1 ) t 1= ] sin( 1 ) 5.5 equation

5 Lagange-umbbell.wxm 5 / 10 (%i8) D5: iff(l, iff(,t)); (%o8) m t (%i9) E5: expan(atsimp(iff(d5,t) - iff(l,) = 0)); (%o9) m t 1 m t ' 1 sin( 1 ) + m (%i30) E51: tigsimp(solve(e5, iff(,t,))); 3 (%o30) [ t = t 1 + t ' 1 3 sin( 1 ) G M ] (%i31) expan(%); (%o31) [ t = t 1 6 Pepae solution + t ' 1 sin( 1 ) G M ] t + G M m =0 (%i3) GC: tigsimp(fist(solve([fist(e11),fist(e1),fist(e31),fist(e41 [iff(theta,t,), iff(phi,t,), iff(theta_1,t,), iff(phi_1,t (%o3) [ t ' cos( ) sin( ), '= t t t 1 t ' 1 cos( 1 ) sin( 1 ) = t ' 1 cos( 1 ) t 1 + t ' t 1 = t + t ' 1 sin( 1 ) 3 sin( 1 ) G M ], t ' 1= t sin ( ) t ' cos( ), t = t (%i33) tansfom(eq) := (A: atsubst(phi_, iff(phi,t), Eq), A: atsubst(phi_, iff(phi,t,), A), A: atsubst(theta_, iff(theta,t), A), A: atsubst(theta_, iff(theta,t,), A), A: atsubst(phi_1, iff(phi_1,t), A), A: atsubst(phi_1, iff(phi_1,t,), A), A: atsubst(theta_1, iff(theta_1,t), A), A: atsubst(theta_1, iff(theta_1,t,), A), A: atsubst(_, iff(,t), A), atsubst(_, iff(,t,), A) )$, t 1

6 Lagange-umbbell.wxm 6 / 10 (%i38) G1: tansfom(gc[1])$ G: tansfom(gc[])$ G3: tansfom(gc[3])$ G4: tansfom(gc[4])$ G5: tansfom(gc[5])$ 7 Solution (%i39) st: [h=0.5, m=1, G=1, M=10]; (%o39) [h=0.5,m=1,g =1,M =10] (%i49) Eq1: gfacto(ev(hs(g1), st, eval))$ Eq: gfacto(ev(hs(g), st, eval))$ Eq3: gfacto(ev(hs(g3), st, eval))$ Eq4: gfacto(ev(hs(g4), st, eval))$ Eq5: gfacto(ev(hs(g5), st, eval))$ Eq6: theta_; Eq7: phi_; Eq8: theta_1; Eq9: phi_1; Eq10: _; (%o45) (%o46) ' (%o47) theta_1 (%o48) phi_1 (%o49) /*kill(theta, phi, theta_, phi_);*/; (%i50) kill(); (%o50) one (%i51) s: k([eq1, Eq, Eq3, Eq4, Eq5, Eq6, Eq7, Eq8, Eq9, Eq10], [theta_, phi_, theta_1, phi_1, _, theta, phi, theta_1, phi [.045,.16, 0.0, 0.06, 0., %pi/, %pi/, %pi/, 0, 1], [t,0, /*theta: phi: theta_: phi_: 0.;*/; /*ev(eq3);ev(eq4);ev(eq5);*/; /*s: k([ev(eq3), ev(eq4), ev(eq5), ev(eq8), ev(eq9), ev(eq10)], [theta_1, phi_1, _, theta_1, phi_1, ], [.0, 0.8, 0., 0.01, 0., 10], [t,0,40.,1])$*/; 7.1 Gaphics

7 Lagange-umbbell.wxm 7 / 10 (%i61) 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)$ c7: makelist([p[1],p[8]],p,s)$ c8: makelist([p[1],p[9]],p,s)$ c9: makelist([p[1],p[10]],p,s)$ c10:makelist([p[1],p[11]],p,s)$ 7. Plot theta, phi, psi /*c1;c;c3;c4;c5;c6;*/; (%i6) wxplot([[iscete, c1], [iscete, c]], [xlabel, "time"], [legen, "theta^{ot}", "phi^{ot}"])$ (%t6)

8 Lagange-umbbell.wxm 8 / 10 (%i63) wxplot([[iscete, c3], [iscete, c4]],[y,-0.05, 0.3], [xlabel, "time"], [legen, "theta_1^{ot}", "phi_1^{ot}"])$ (%t63) (%i64) wxplot([[iscete, c5], [iscete, c6]], [xlabel, "time"], [legen, "^{ot}", "theta"])$ (%t64)

9 Lagange-umbbell.wxm 9 / 10 (%i65) wxplot([[iscete, c7], [iscete, c8]], [xlabel, "time"], [legen, "phi", "theta_1"])$ (%t65) (%i66) wxplot([[iscete, c9], [iscete, c10]], [xlabel, "time"], [legen, "phi_1", ""])$ (%t66) 7.3 Plot space cuve of cente of mass

10 Lagange-umbbell.wxm 10 / 10 (%i67) tansf() := ( block([i,t,h,,theta,phi,x,y,z,x1,y1,z1], h: 0.5, cl1: [], cl: [], fo i:1 thu length(c1) step 4 o ( [t,theta,phi]: [c1[i][1],c6[i][],c7[i][]], x: h*sin(theta)*cos(phi), y: h*sin(theta)*sin(phi), z: h*cos(theta), /*pint(t,x,y,z),*/ [theta,phi,]: [c8[i][],c9[i][],c10[i][]], x1: *sin(theta)*cos(phi), y1: *sin(theta)*sin(phi), z1: *cos(theta), /*pint(t,x,y,z),*/ cl: appen(cl, [[x1,y1,z1]]), cl1: appen(cl1, [[x+x1,y+y1,z+z1]]) )/*, etun(cl1,cl)*/))$ (%i70) tansf()$ c1: points(cl1)$ c: points(cl)$ (%i71) wxaw3(use_peamble="set ticslevel 0", zange=[-0.4,0.4],line_wi view=[76, 8], c, colo=e, c1)$ 0 eos, 0 wanings (%t71)