2004 2

1 3 1.1..................... 3 1.1.1................... 3 1.1.2.................... 4 1.2................... 6 1.3........................ 8 1.4................... 9 1.4.1..................... 9 1.4.2................... 11 2 13 3 14 4 16 5 22 6 25 6.1.............. 25 6.2.... 29 7 35 1

H.K.Moffatt Spinning eggs a paradox resolved Nature [1] 1950 Moffatt 2

1 [6] 1.1 1.1.1 r i r j n x 1, y 1, z 1, x 2, y 2, z 2,.., x n, y n, z n 3n 3n 6 6 3 3 0 MG 3

1.1.2 m 1, m 2, m 3,... r 1, r 2, r 3,... i j F ij, i F i m 1 r 1 = F 1 + F 21 + F 31 +... m 2 r 2 = F 2 + F 22 + F 32 +...... (1.1) F ij + F ji = 0 m 1 r 1 + m 2 r 2 +... = F 1 + F 2 +... (1.2) M = m 1 + m 2 +... (1.3) MR = m 1 r 1 + m 2 r 2 +... (1.4) M d2 R dt 2 = F 1 + F 2 +... = i F i (1.5) M 1 R i F i 1 1.1 dp dt = i F i = 0 (1.6) 4

L 0 d dt r i = d dt i i r i m i r i = i r i F i = 0 (1.7) L G 0 L G + L = 0 dl dt = i r i F i = 0 (1.8) (1.2) (1.3) 2 6 5

1.2 1 z z zx θ t i m i, x i, y i, z i r i = x 2 i + y2 i =, z i = (1.9) θ i θ t x i = r i cos θ i, y i = r i sin θ i (1.10) θ i = θ, θi = θ, (1.11) x i = r i sin θ i θ i = y i θ, yi = x i θ (1.12) θ(t) 1.3 d dt i m i (r i r i ) = j z r j F j (1.13) (r i r i ) z = x i y i y i x i = x 2 θ i + yi 2 θ = r 2 θ i (1.14) r 2 i t d dt i m i (r i r i ) z = ( i m i r 2 i ) θ (1.15) I z = i m i (x 2 i + y 2 i ) (1.16) z 6

z I z θ θ 1.3 z I z d 2 θ dt 2 = j (x j F iy y i F jx ) (1.17) 7

1.3 M R I = 2 5 MR2 (1.18) M x R x y R y z R z I x = 1 5 M(R2 y + R 2 z) (1.19) I y = 1 5 M(R2 z + R 2 x) (1.20) I z = 1 5 M(R2 x + R 2 y) (1.21) 8

1.4 1.4.1 O O xyz, O ξηδ O ω ω, ω ω ω dt ω ωdt r i ω r i ( r i sin γ i )ωdt γ i r i ω ṙ i = ω r i (1.22) m i i P = i m i r i = ω i m i r i = ω MR (1.23) M = m i, R O L = i m i (r i r i ) = i m i r i (ω r i ) (1.24) (r (ω r)) x = y(ω x y ω y x) z(ω z x ω x z) = (y 2 + z 2 )ω x xyω y zxω z (1.25) L x = i m i (y 2 i + x 2 i )ω x i m i x i y i ω y i m i z i x i ω z (1.26) L ξ = i m i (η 2 i + ζ 2 i )ω ξ i m i ξ i η i ω η i m i ζ i ξ i ω ζ (1.27) 9

L ξ = I ξ ω ξ I ξη ω η I ζξ ω η (1.28) L η = I ξη ω η + I ζ ω ζ I ζξ ω ζ (1.29) L ζ = I ζξ ω ξ I ηζ ω η + I ζ ω ζ (1.30) L ξ L η L ζ = I ξ I ξη I ζξ I ξη I η I ηζ I ζξ I ηζ I ζ ω ξ ω η (1.31) ω ζ 3 3 0 L 0 ξ I L 0 η = ξ 0 0 0 ω 0 Iη 0 ξ 0 0 ω 0 η (1.32) L 0 ζ 0 0 Iζ 0 I 0 ξ, I0 η, I 0 ζ ω 0 ζ 10

1.4.2 B(t) dt db,(1) O ξηζ B (2)B O ξηζ B r i (1) (2) (ω r i ) B (2) (ω B)dt (1) ξ, η, ζ (db ξ, db η, db ζ ) db dt ( db dt ) ξ = db ξ dt ( db dt ) η = db η dt ( db dt ) ζ = db ζ dt + (ω B) ξ (1.33) + (ω B) η (1.34) + (ω B) ζ (1.35) B = r i O L (1.31) (1.32) ξ, η, ζ (1.32) L ω 0 L ξ = I 0 ξ ω ξ, L η = I 0 ηω η, L ξ = I 0 ζ ω ζ I 0 ξ, I0 η, I 0 ζ dl ξ dt = I 0 ξ dω ξ dt, dl η dt = I 0 η dη ξ dt, dl ζ dt = I 0 ζ dω ζ dt (1.4) (1.6) L ( dl dt ) ξ = Iξ 0 dω ξ dt + (ω ηl ζ ω ζ L η ) = I 0 ξ dω ξ dt (I0 η I 0 ζ )ω η ω ζ I 0 ξ dω ξ dt (I0 η I 0 ζ )ω η ω ζ = ( i r i F i ) ξ 11

Iη 0 dω η dt (I0 ζ Iξ 0 )ω ζ ω ξ = ( i I 0 ζ dω ζ dt (I0 ξ I 0 η)ω ζ ω η = ( i r i F i ) η r i F i ) ζ 12

2 1 J.H Jellet(1872) H X J = H X p (2.1) J 2 E.J.Routh(1905) 3 C.M.Brams, N.M.Hugonholts(1952) 4 M.K. Moffatt and Y.Shimomura(2002) 13

3 [7] ω 0 dl dt 0 L N 0 N = dl dt + ω L (3.1) L Iω (3.2) (3.1) ω L 0 L N L N (3.1) k (3.1) k N = k dl dt = d(k L) dt (3.3) N (3.3) k N = N sin θk L = L cos θ (3.4) 14

θ L k (3.3) (3.4) R θ = N L µmgr Iω (3.5) θ 15

4 [3,8] H.K.Moffatt G [3] 3 G P GZ GX GY G Z P GXY Z GZ,Z θ Gz Gx,Gy G h(θ) P GXYZ X P = 16

(X P, 0, Z P ) X P = dh dθ, Z P = h(θ) (4.1) (4.1) 1 [8] < 2 > 17

θ θ X, Y h LN LN X P θ (4.2) LN = OL ON = h h cos( θ) h h = h (4.3) X P = dh dθ X P = dh dθ (4.4) M, g P R = Mg F µ F Y F = (0, F, 0) F = µmg (4.5) 0 ω GZ Ω Gz ψ ω x = Ω sin θ (4.6) ω y = θ (4.7) ω z = Ω cos θ + ψ = n (4.8) I x = I y = A, I z = C (4.9) L x = Aω x = AΩ sin θ (4.10) L y = AΩ y = A θ (4.11) L z = Cn (4.12) 18

GXY Z Ω Ω = (0, 0, Ω) (4.13) Ω GXY Z d L dt = Ω L = N (4.14) N L x = L x cos θ + L z sin θ = (Cn AΩ cos θ) sin θ (4.15) L y = L y = A θ (4.16) L z = L x sin 2 θ + L z cos θ = AΩ sin θ + Cn cos θ Ω L = AΩ + (Cn AΩ cos θ) cos θ (4.17) (Ω L) X = Ω Y L z Ω Z L Y = AΩ θ (Ω L) Y = Ω Z L X Ω X L Z (4.18) = Ω(Cn AΩ cos θ) sin θ (Ω L) Z = Ω X L Y Ω Y L X = 0 (4.19) R + F = (0, F, R) = (0, µmg, Mg) (4.20) X P = (X P, Y P, Z P ) = (4.14) ( dh(θ) dθ, 0, h(θ) ) (4.21) d dt [(Cn AΩ cos θ) sin θ] AΩ θ = F Z P (4.22) A θ + Ω(Cn AΩ cos θ) sin θ = RX P (4.23) A Ω + d dt [(Cn AΩ cos θ) cos θ] = F X P (4.24) 19

sin θ,cos θ y Ω 2 Ω 2 3 RX P (4.23) (Cn AΩ cos θ) cos θ=0 sin θ 0 Cn = AΩ cos θ (4.25) gyroscopic balance X Y AΩ θ = F Z P (4.26) A Ω = F X P (4.27) (4.27) (4.26) Ω Ω = X P dh θ = dθ Z P h dθ dt = ḣ h (4.28) t Ωh = e c (4.29) AΩh = Ae c = J (4.30) J Jellet Ω h Ω h Ω h (4.26) 1 AΩ θ = F Z P (4.1) Z P = h(θ) (4.30) J θ = F h 2 (θ) (4.31) h(θ) θ t 20

z 2 a 2 + x2 + y 2 b 2 = 1 (4.32) h(θ) h 2 (θ) = a 2 (a 2 b 2 ) sin 2 θ (4.33) (4.31) A = M 5 (a2 + b 2 ), C = 2 5 Mb2 (4.34) θ = µmga2 (1 (1 b2 J a 2 ) sin2 θ) (4.35) a b arctan( b a (t o ) J tan θ) = µmga2 (t t 0 ) (4.36) J tan θ = a b tan µq(t t 0) (4.37) a b q = Mgab a b J (4.38) (a > b g > 0) θ π 2 t = π 2µq (4.39) θ = 0 θ = π 2 t = π 2 µq θ = π 4 Ω Ω Jellet J J = AΩ ( a2 + b 2 ) 1 2 (4.40) 2 21

5 x y z ξ, η, ζ M : I i : i F : N : R : Ṙ V e ξ, e ζ, e η e i e j = δ ij (i, j = ξ, η, ζ) e ξ e η = e ζ,e η e ζ = e ξ,e ζ e ξ = e η ω ω ξ, ω η, ω ζ ω ξ = e ξ ω (5.1) ω η = e η ω (5.2) ω ζ = e ζ ω (5.3) ω = ω ξ e ξ + ω η e η + ω ζ e ζ (5.4) ω e ξ = ω ζ e η ω η e ζ (5.5) ω e η = ω ξ e ζ ω ζ e ξ (5.6) ω e ζ = ω η e ξ ω ξ e η (5.7) 22

e ξ = ω e ξ (5.8) e η = ω e η (5.9) e ζ = ω e ζ (5.10) ω = f(ω, e ξ, e η, e ζ ) + N(R, V, e ξ, e η, e ζ, ω) (5.11) Ṙ = V (5.12) V = 1 M F (R, V, e ξ, e η, e ζ, ω) (5.13) (5.14) N F (5.11) f ω ξ = ω e ξ (5.15) ω η = ω e η (5.16) ω ζ = ω e ζ (5.17) N ξ = N e ξ (5.18) N η = N e η (5.19) N ζ = N e ζ (5.20) ω ξ = f ξ ω η ω ζ + N ξ (5.21) ω η = f η ω ζ ω ξ + N η (5.22) ω ζ = f ζ ω ξ ω η + N ζ (5.23) ω = ω ξ e ξ + ω η e η + ω ζ e ζ + ω ξ e ξ + ω η e η + ω ζ e ζ (5.24) f ξ = I η I ζ I ξ, f η = I ζ I ξ I η, f ζ = I ξ I η I ζ (5.25) 23

e ξ, e η, e ζ 6 e ξ = e η = e ζ = 1, e x e y = e y e z = e z e x = 0 (5.26) d dt (e ξ e ξ ) = 2e ξ e ξ = 2(ω e ξ ) e ξ (5.27) = 2ω (e ξ e ξ ) = 0 (5.28) d dt (e ξ e η ) = e ξ e η + e ξ e η = (ω e ξ ) e η + e ξ (ω e η ) = ω (e ξ e η + e η e ξ ) = 0 (5.29) [ ] 3 3 6 2 12 18 12 6 24

6 6.1 dω(t) dt = f(t) (6.1) ω(t + dt) = ω(t) + t dω dt (t) + θ(( t)2 ) (6.2) ω dt = 0.001 1000 25

R0 ζ ζ 2 R0 ζ P ξ ξ 3.1 radius ratio x P η η 2.9 radius ratio y P ζ ζ 5 radius ratio z ρ 1 density P ξ ξ 3 angular-velocity-ratio-x P η η 2 angular-velocity-ratio-y P ζ ζ 1 angular-velocity-ratio-z I ξ ξ moi1 I η η moi2 I ζ ζ moi3 ω 0 ξ ξ omg01 ω 0 η η omg02 ω 0 ζ ζ omg03 ω ξ ξ omg1 ω η η omg2 ω ζ ζ omg3 e ξx ξ x 0 e1x e ξy ξ y 1 e1y e ξz ξ z 0 e1z e ηx η x 0 e2x e ηy η y 0 e2y e ηz η z -1 e2z e ζx ζ x 1 e3x e ζy ζ y 0 e3y e ζz ζ z 0 e3z 26

V R0 ξ = P ξ P ζ R0 ζ (6.3) R0 η = P η P ζ R0 ζ (6.4) V = 4 3 π P ξp η R0 ζ 3 P ζ 2 (6.5) I ξ = 1 5 ρv (R02 η + R0 2 ζ) (6.6) I η = 1 5 ρv (R02 ζ + R0 2 ξ) (6.7) I ζ = 1 5 ρv (R02 ξ + R0 2 η) (6.8) ω ξ = P ξ ωo 2 P 2 ξ P 2 2 (6.9) η P ζ ω η = P η ωo 2 P 2 ξ P 2 2 (6.10) η P ζ ω ζ = P ζ ωo 2 P 2 ξ P 2 2 (6.11) η P ζ 27

80 60 omega-1 omega-2 omega-3 angular velocity (1/sec) 40 20 0-20 -40-60 -80 0 0.2 0.4 0.6 0.8 1 time (sec) 6.1: t[sec] [1/s] 6.1 ξ η ζ 3.1 : 2.9 : 5.0 ξ η ζ Moffatt 28

6.2 4 N N = 0 (z 0) (6.12) N = kz (z < 0) (6.13) z V r v = V G + ω r (6.14) F = µ v v N (6.15) R ξ = R η = 1.8cm (6.16) ω = 50( / ) 0.5 R ζ = 2.4cm (6.17) 4 29

400 300 ω 1 ω 2 ω 3 200 angular velocity[1/sec] 100 0-100 -200-300 -400 0 1 2 3 4 5 time[sec] 6.2: t[sec] [1/sec] 6.2 t( ) 3 (1/sec) ω3( ω ζ ) ω3 ω1(ω ξ ) ω2(ω η ) 0 30

2.5 2.4 2.3 2.2 z[cm] 2.1 2 1.9 1.8 0 1 2 3 4 5 time[sec] 6.3: t[sec] z[cm] 6.3 t( ), z(cm) ζ 2 3 31

6.2e+06 6e+06 5.8e+06 5.6e+06 energy[erg] 5.4e+06 5.2e+06 5e+06 4.8e+06 4.6e+06 0 1 2 3 4 5 time[sec] 6.4: [sec] energy[erg] 6.4 t( ) E(erg) 3 32

20000 18000 L x L y L z 16000 14000 12000 L[g.cm 2 /s] 10000 8000 6000 4000 2000 0-2000 0 1 2 3 4 5 time[sec] 6.5: t[sec] L[g cm 2 /s] 6.5 t( ) L z x y 0 33

328 326 324 ω[rad/sec] 322 320 318 316 314 0 1 2 3 4 5 time[sec] 6.6: t[sec] ω[rad/s] 6.6 t( ) ξ ω(rad/s) ζ ζ 34

7 1, 2, 4 ( ) 35

[1] H.K. Moffatt and Y. Simomura: Spiningeggs-a paradox resolved, Nature 416,385-386(2002). [2] H.K. Moffatt: Euler s disk and its finite-time singularity, Nature 404,833-834(2000). [3] : ( ), 72,932-939 (2002). [4] :, 8, 288-293(1953). [5] : ( ), 3, 28-32(1967); 4, 38-42 (1967). 6, 32-34 (1967). [6] :, pp.90-165 (1987). [7] V.D., M.G. :, pp.221-223 (1997). [8] :, 18,52-56 (2003). 36

37

: C #include<stdio.h> #include<math.h> main(){ kaiten4(); } int kaiten4(){ double radius_ratio_x=3.1; /* relative length of prinxipal axis */ double radius_ratio_y=2.9; double radius_ratio_z=5; double R00=2.0 ; /* radius for shperical shape [cm] */ double density=1; /* density [g/cm^3] */ double angnlar_velocity_ratio_x=3; /* relative size of initial ang. vel. */ double angnlar_velocity_ratio_y=2; double angnlar_velocity_ratio_z=1; double moi1, moi2, moi3 ; /* moment of inertia */ double omg01, omg02, omg03; /* initial angular velocity vector, B-frame */ double omg1, omg2, omg3 ; /* angular velocity vector, B-frame */ double R0x,R0y,R0z,R0 ; double omg0; double a1,a2,a3 ; double dt,t,pi, b,v; double e1x,e1y,e1z,e2x,e2y,e2z,e3x,e3y,e3z; double e1dx,e1dy,e1dz,e2dx,e2dy,e2dz,e3dx,e3dy,e3dz; double L1x,L1y,L1z,L2x,L2y,L2z,L3x,L3y,L3z,Lx,Ly,Lz; pi=4*atan(1.0); fprintf(stderr,"check : pi = %20.16f\n",pi); dt=0.0001; /* time step size [sec] */ omg0=20*pi; /* size of initial ang. vel. [radian/sec] */ R0=R00/pow(radius_ratio_x*radius_ratio_y*radius_ratio_z,1.0/3.0); R0x=R0*radius_ratio_x; R0y=R0*radius_ratio_y; R0z=R0*radius_ratio_z; fprintf(stderr,"r0=(%f %f %f)\n",r0x,r0y,r0z); V=4*pi*R0x*R0y*R0z/3; /* volume of the rigid body [cm^3] */ moi1=density*v*(r0y*r0y+r0z*r0z)/5; /* moment inertia [g cm^2] */ moi2=density*v*(r0z*r0z+r0x*r0x)/5; moi3=density*v*(r0x*r0x+r0y*r0y)/5; fprintf(stderr,"moi=(%f %f %f)\n",moi1,moi2,moi3); b=1/sqrt(angnlar_velocity_ratio_x*angnlar_velocity_ratio_x +angnlar_velocity_ratio_y*angnlar_velocity_ratio_y 38

+angnlar_velocity_ratio_z*angnlar_velocity_ratio_z); omg01=omg0*angnlar_velocity_ratio_x*b; omg02=omg0*angnlar_velocity_ratio_y*b; omg03=omg0*angnlar_velocity_ratio_z*b; omg1=omg01; omg2=omg02; omg3=omg03; e1x=1, e1y=0, e1z=0; e2x=0, e2y=-1, e2z=0; e3x=0, e3y=0, e3z=1; for(t=0;t<=1;t=t+dt) { L1x=moi1*omg1*e1x; L2x=moi2*omg2*e2x; L3x=moi3*omg3*e3x; L1y=moi1*omg1*e1y; L2y=moi2*omg2*e2y; L3y=moi3*omg3*e3y; L1z=moi1*omg1*e1z; L2z=moi2*omg2*e2z; L3z=moi3*omg3*e3z; Lx=L1x+L2x+L3x; Ly=L1y+L2y+L3y; Lz=L1z+L2z+L3z; /*printf("%f %f %f %f\n",t,lx,ly,lz);*/ a1=(moi2-moi3)*omg2*omg3/moi1; a2=(moi3-moi1)*omg3*omg1/moi2; a3=(moi1-moi2)*omg1*omg2/moi3; e1dx=dt*(omg3*e2x-omg2*e3x); e1dy=dt*(omg3*e2y-omg2*e3y); e1dz=dt*(omg3*e2z-omg2*e3z); e2dx=dt*(omg1*e3x-omg3*e1x); e2dy=dt*(omg1*e3y-omg3*e1y); e2dz=dt*(omg1*e3z-omg3*e1z); e3dx=dt*(omg2*e1x-omg1*e2x); e3dy=dt*(omg2*e1y-omg1*e2y); e3dz=dt*(omg2*e1z-omg1*e2z); printf("%f %f %f %f %f %f %f %f %f %f %f %f\n",t,omg1,omg2,omg3, e1x,e1y,e1z,e2x,e2y,e2z,e3x,e3y,e3z); omg1=omg1+dt*a1; omg2=omg2+dt*a2; 39

omg3=omg3+dt*a3; } } e1x=e1x+e1dx; e1y=e1y+e1dy; e1z=e1z+e1dz; e2x=e2x+e2dx; e2y=e2y+e2dy; e2z=e2z+e2dz; e3x=e3x+e3dx; e3y=e3y+e3dy; e3z=e3z+e3dz; 40