Lagrange-dumbbell.wxm 1 / 9 Spinning top - represented by ... · Lagrange-dumbbell.wxm 1 / 9 Spinning top - represented by dumbbell (%i1) kill (all ); (%o0) done 1 Coordinates (%i1)

Lagrange-dumbbell.wxm 1 / 9

Spinning top - represented by

dumbbell

(%i1) kill(all);

(%o0) done

1 Coordinates

(%i1) depends([x,y,z,X,Y,Z,theta,phi,theta_1,phi_1,r,T,U,L],t);

(%o1) [x( )t ,y( )t ,z( )t ,X( )t ,Y( )t ,Z( )t ,�( )t ,'( )t ,�1( )t ,'1( )t ,r( )t ,

T( )t ,U( )t ,L( )t ]

(%i2) h_1: h*[sin(theta)*cos(phi), sin(theta)*sin(phi), cos(theta)];

(%o2) [h cos( )' sin( )� ,h sin( )' sin( )� ,h cos( )� ]

(%i3) h_2: h*[sin(%pi-theta)*cos(phi+%pi), sin(%pi-theta)*sin(phi+%pi), cos

(%o3) [−h cos( )' sin( )� ,−h sin( )' sin( )� ,−h cos( )� ]

(%i4) R: r*[sin(theta_1)*cos(phi_1), sin(theta_1)*sin(phi_1), cos(theta_1)]

(%o4) [cos( )'1 r sin( )�1 ,sin( )'1 r sin( )�1 ,r cos( )�1 ]

(%i5) r_1: R+h_1;

(%o5) [cos( )'1 r sin( )�1 +h cos( )' sin( )� ,sin( )'1 r sin( )�1 +h sin( )'

sin( )� ,r cos( )�1 +h cos( )� ]

(%i6) r_2: R+h_2;

(%o6) [cos( )'1 r sin( )�1 −h cos( )' sin( )� ,sin( )'1 r sin( )�1 −h sin( )' sin( )�

,r cos( )�1 −h cos( )� ]

2 Kinetic energy


(%i7) T[1]: m/2*(diff(r_1[1],t)^2+diff(r_1[2],t)^2+diff(r_1[3],t)^2);

(%o7) (m (

� �

−r sin( )�1

� �

d

d t�1 +

� �

d

d tr cos( )�1 −h sin( )�

� �

d

d t�

2

+(

sin( )'1 r cos( )�1

� �

d

d t�1 +sin( )'1

� �

d

d tr sin( )�1 +cos( )'1

� �

d

d t'1 r sin( )�1

+h sin( )' cos( )��

d

d t� +h cos( )'

� �

d

d t' sin( )� )^ 2 +(cos( )'1 r cos( )�1

� �

d

d t�1 +cos( )'1

� �

d

d tr sin( )�1 −sin( )'1

� �

d

d t'1 r sin( )�1 +h cos( )' cos( )�

� �

d

d t� −h sin( )'

� �

d

d t' sin( )� )^ 2 ))/ 2

(%i8) T[2]: m/2*(diff(r_2[1],t)^2+diff(r_2[2],t)^2+diff(r_2[3],t)^2);

(%o8) (m (

� �

−r sin( )�1

� �

d

d t�1 +

� �

d

d tr cos( )�1 +h sin( )�

� �

d

d t�

2

+(

sin( )'1 r cos( )�1

� �

d

d t�1 +sin( )'1

� �

d

d tr sin( )�1 +cos( )'1

� �

d

d t'1 r sin( )�1

−h sin( )' cos( )��

d

d t� −h cos( )'

� �

d

d t' sin( )� )^ 2 +(cos( )'1 r cos( )�1

� �

d

d t�1 +cos( )'1

� �

d

d tr sin( )�1 −sin( )'1

� �

d

d t'1 r sin( )�1 −h cos( )' cos( )�

� �

d

d t� +h sin( )'

� �

d

d t' sin( )� )^ 2 ))/ 2

(%i9) T: factor(trigsimp(ratsimp(T[1]+T[2])));

(%o9) m (r2

� �

d

d t�1

2

+

� �

d

d t'1

2

r2 sin( )�12+h2

� �

d

d t�

2

+h2

� �

d

d t'

2

sin( )� 2+

� �

d

d tr

2

)

3 Potential energy

➔ /*U[1]: trigsimp(-M*m*G/(r_1[1]^2+r_1[2]^2+r_1[3]));

U[2]: trigsimp(-M*m*G/(r_1[1]^2+r_1[2]^2+r_1[3]));*/;

(%i11) U[1]: trigsimp(-M*m*G/r);

U[2]: trigsimp(-M*m*G/r);

(%o10) −G M m

r

(%o11) −G M m

r


(%i12) U: gfactor(trigsimp(U[1] + U[2]));

(%o12) −2 G M m

r

4 Lagrange function

(%i13) T;

(%o13) m (r2

� �

d

d t�1

2

+

� �

d

d t'1

2

r2 sin( )�12+h2

� �

d

d t�

2

+h2

� �

d

d t'

2

sin( )� 2+

� �

d

d tr

2

)

(%i14) U;

(%o14) −2 G M m

r

(%i15) L: T - U;

(%o15) m (r2

� �

d

d t�1

2

+

� �

d

d t'1

2

r2 sin( )�12+h2

� �

d

d t�

2

+h2

� �

d

d t'

2

sin( )� 2+

� �

d

d tr

2

)+2 G M m

r

5 Lagrange equations II

5.1 theta equation

(%i16) D1: diff(L, diff(theta,t));

(%o16) 2 h2 m

� �

d

d t�

(%i17) E1: expand(ratsimp(diff(D1,t) - diff(L,theta) = 0));

(%o17) 2 h2 m

� �

d2

d t2� −2 h2 m

� �

d

d t'

2

cos( )� sin( )� =0

(%i18) E11: trigsimp(solve(E1, diff(theta,t,2)));

(%o18) [d2

d t2�=

� �

d

d t'

2

cos( )� sin( )� ]

5.2 phi equation

(%i19) D2: diff(L, diff(phi,t));

(%o19) 2 h2 m

� �

d

d t' sin( )� 2


(%i20) E2: trigsimp(ratsimp(diff(D2,t) - diff(L,phi) = 0));

(%o20) 4 h2 m

� �

d

d t' cos( )� sin( )�

� �

d

d t� +2 h2 m

� �

d2

d t2' sin( )� 2=0

(%i21) E21: trigsimp(solve(E2, diff(phi,t,2)));

(%o21) [d2

d t2'=−

2

� �

d

d t' cos( )�

� �

d

d t�

sin( )�]

5.3 theta_1 equation

(%i22) D3: diff(L, diff(theta_1,t));

(%o22) 2 m r2

� �

d

d t�1

(%i23) E3: expand(ratsimp(diff(D3,t) - diff(L,theta_1) = 0));

(%o23) 2 m r2

� �

d2

d t2�1 +4 m r

� �

d

d tr

� �

d

d t�1 −2 m

� �

d

d t'1

2

r2 cos( )�1

sin( )�1 =0

(%i24) E31: trigsimp(solve(E3, diff(theta_1,t,2)));

(%o24) [d2

d t2�1=−

2

� �

d

d tr

� �

d

d t�1 −

� �

d

d t'1

2

r cos( )�1 sin( )�1

r]

5.4 phi_1 equation

(%i25) D4: diff(L, diff(phi_1,t));

(%o25) 2 m

� �

d

d t'1 r2 sin( )�1

2

(%i26) E4: expand(ratsimp(diff(D4,t) - diff(L,phi_1) = 0));

(%o26) 4 m

� �

d

d t'1 r2 cos( )�1 sin( )�1

� �

d

d t�1 +4 m

� �

d

d t'1 r

� �

d

d tr

sin( )�12+2 m

� �

d2

d t2'1 r2 sin( )�1

2=0

(%i27) E41: trigsimp(solve(E4, diff(phi_1,t,2)));

(%o27) [d2

d t2'1=−

2

� �

d

d t'1 r cos( )�1

� �

d

d t�1 +2

� �

d

d t'1

� �

d

d tr sin( )�1

r sin( )�1]

5.5 r equation


(%i28) D5: diff(L, diff(r,t));

(%o28) 2 m

� �

d

d tr

(%i29) E5: expand(ratsimp(diff(D5,t) - diff(L,r) = 0));

(%o29) −2 m r

� �

d

d t�1

2

−2 m

� �

d

d t'1

2

r sin( )�12+2 m

� �

d2

d t2r +

2 G M m

r2=0

(%i30) E51: trigsimp(solve(E5, diff(r,t,2)));

(%o30) [d2

d t2r=

r3

� �

d

d t�1

2

+

� �

d

d t'1

2

r3 sin( )�12−G M

r2]

(%i31) expand(%);

(%o31) [d2

d t2r=r

� �

d

d t�1

2

+

� �

d

d t'1

2

r sin( )�12−

G M

r2]

6 Prepare solution

(%i39) GC: trigsimp(first(solve([first(E11),first(E21),first(E31),first(E41

[diff(theta,t,2), diff(phi,t,2), diff(theta_1,t,2), diff(phi_1,t

(%o39) [d2

d t2�=

� �

d

d t'

2

cos( )� sin( )� ,d2

d t2'=−

2

� �

d

d t' cos( )�

� �

d

d t�

sin( )�,

d2

d t2�1

=−

2

� �

d

d tr

� �

d

d t�1 −

� �

d

d t'1

2

r cos( )�1 sin( )�1

r,

d2

d t2'1=−

2

� �

d

d t'1 r cos( )�1

� �

d

d t�1 +2

� �

d

d t'1

� �

d

d tr sin( )�1

r sin( )�1,

d2

d t2r=

r3

� �

d

d t�1

2

+

� �

d

d t'1

2

r3 sin( )�12−G M

r2]

(%i41) transform(Eq) := (A: ratsubst(phi_d, diff(phi,t), Eq),

A: ratsubst(phi_dd, diff(phi,t,2), A),

A: ratsubst(theta_d, diff(theta,t), A),

A: ratsubst(theta_dd, diff(theta,t,2), A),

A: ratsubst(phi_1d, diff(phi_1,t), A),

A: ratsubst(phi_1dd, diff(phi_1,t,2), A),

A: ratsubst(theta_1d, diff(theta_1,t), A),

A: ratsubst(theta_1dd, diff(theta_1,t,2), A),

A: ratsubst(r_d, diff(r,t), A),

ratsubst(r_dd, diff(r,t,2), A)

)$


(%i46) G1: transform(GC[1])$

G2: transform(GC[2])$




7 Solution

(%i231) str: [h=0.25, m=1, G=1, M=10];

(%o231) [h=0.25,m=1,G=1,M=10]

(%i241) Eq1: gfactor(ev(rhs(G1), str, eval))$

Eq2: gfactor(ev(rhs(G2), str, eval))$




Eq6: theta_d;

Eq7: phi_d;

Eq8: theta_1d;

Eq9: phi_1d;

Eq10: r_d;

(%o237) �d(%o238) 'd

(%o239) theta_1d

(%o240) phi_1d

(%o241) rd

➔ /*kill(theta, phi, theta_d, phi_d);*/;

(%i338) s: rk([Eq1, Eq2, Eq3, Eq4, Eq5, Eq6, Eq7, Eq8, Eq9, Eq10],

[theta_d, phi_d, theta_1d, phi_1d, r_d, theta, phi, theta_1, phi_1

[.03, .02, 0.0, 0.02, 0., %pi/2, %pi/2, %pi/2, 0, 12], [t,0,92

➔ /*theta: phi: theta_d: phi_d: 0.;*/;

/*ev(Eq3);ev(Eq4);ev(Eq5);*/;

/*s: rk([ev(Eq3), ev(Eq4), ev(Eq5), ev(Eq8), ev(Eq9), ev(Eq10)],

[theta_1d, phi_1d, r_d, theta_1, phi_1, r],

[.0, 0.18, 0., 0.01, 0., 10], [t,0,240.,1])$*/;

7.1 Graphics


(%i348) c1: makelist([p[1],p[2]],p,s)$

c2: makelist([p[1],p[3]],p,s)$








c10:makelist([p[1],p[11]],p,s)$

7.2 Plot theta, phi, psi

➔ /*c1;c2;c3;c4;c5;c6;*/;

(%i349) wxplot2d([[discrete, c1], [discrete, c2]], [y,-0.1, 0.1],

[xlabel, "time"],

[legend, "theta^{dot}", "phi^{dot}"])$

(%t349)


(%i350) wxplot2d([[discrete, c3], [discrete, c4]],

[xlabel, "time"],

[legend, "theta_1^{dot}", "phi_1^{dot}"])$

(%t350)


[xlabel, "time"],

[legend, "r^{dot}", "theta"])$

(%t351)



[xlabel, "time"],

[legend, "phi", "theta_1"])$

(%t352)


[xlabel, "time"],

[legend, "phi_1", "r"])$

(%t353)

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Lagrange-dumbbell.wxm 1 / 9 Spinning top - represented by ... · Lagrange-dumbbell.wxm 1 / 9 Spinning top - represented by dumbbell (%i1) kill (all ); (%o0) done 1 Coordinates (%i1)