3550:
38:
655:
22:
1369:
168:
30:
1046:
2363:
1895:
650:{\displaystyle {\begin{aligned}I_{1}{\dot {\omega }}_{1}&=-(I_{3}-I_{2})\omega _{3}\omega _{2}~~~~~~~~~~~~~~~~~~~~{\text{(1)}}\\I_{2}{\dot {\omega }}_{2}&=-(I_{1}-I_{3})\omega _{1}\omega _{3}~~~~~~~~~~~~~~~~~~~~{\text{(2)}}\\I_{3}{\dot {\omega }}_{3}&=-(I_{2}-I_{1})\omega _{2}\omega _{1}~~~~~~~~~~~~~~~~~~~~{\text{(3)}}\end{aligned}}}
2109:
2925:
1673:
1364:{\displaystyle {\begin{aligned}I_{2}{\ddot {\omega }}_{2}&=-(I_{1}-I_{3})\omega _{1}{\dot {\omega }}_{3}\\I_{3}I_{2}{\ddot {\omega }}_{2}&=(I_{1}-I_{3})(I_{2}-I_{1})(\omega _{1})^{2}\omega _{2}\\{\text{i.e. }}~~~~{\ddot {\omega }}_{2}&={\text{(negative quantity)}}\cdot \omega _{2}\end{aligned}}}
2517:
A visualization of the instability of the intermediate axis. The magnitude of the angular momentum and the kinetic energy of a spinning object are both conserved. As a result, the angular velocity vector remains on the intersection of two ellipsoids. Here, the yellow ellipsoid is the angular momentum
4843:
When the body is not exactly rigid, but can flex and bend or contain liquid that sloshes around, it can dissipate energy through its internal degrees of freedom. In this case, the body still has constant angular momentum, but its energy would decrease, until it reaches the minimal point. As analyzed
4874:
In general, celestial bodies large or small would converge to a constant rotation around its axis of maximal moment of inertia. Whenever a celestial body is found in a complex rotational state, it is either due to a recent impact or tidal interaction, or is a fragment of a recently disrupted
2514:
2716:
106:
in the diagram), and then catch the handle. In almost all cases, during that rotation the face will also have completed a half rotation, so that the other face is now up. By contrast, it is easy to throw the racket so that it will rotate around the handle axis
2653:
131:
2358:{\displaystyle {\frac {d}{dt}}{\begin{bmatrix}\omega _{1}\\\omega _{2}\end{bmatrix}}={\begin{bmatrix}0&-\omega _{3}(I_{3}-I_{2})/I_{1}\\-\omega _{3}(I_{1}-I_{3})/I_{2}&0\end{bmatrix}}{\begin{bmatrix}\omega _{1}\\\omega _{2}\end{bmatrix}}}
101:
This can be demonstrated by the following experiment: hold a tennis racket at its handle, with its face being horizontal, and throw it in the air such that it performs a full rotation around its horizontal axis perpendicular to the handle
1890:{\displaystyle {\begin{aligned}I_{1}I_{3}{\ddot {\omega }}_{1}&=(I_{3}-I_{2})(I_{2}-I_{1})(\omega _{2})^{2}\omega _{1}\\{\text{i.e.}}~~~~{\ddot {\omega }}_{1}&={\text{(positive quantity)}}\cdot \omega _{1}\end{aligned}}}
896:
2077:
2525:
4694:
3470:. Thus the angular momentum ellipsoid is both flatter and sharper, as visible in the animation. In general, the angular momentum ellipsoid is always more "exaggerated" than the energy ellipsoid.
1678:
1051:
816:
173:
4584:
134:
138:
137:
133:
132:
4425:
4260:
The above analysis is all done in the perspective of an observer which is rotating with the body. An observer watching the body's motion in free space would see its angular momentum vector
3468:
139:
4523:
4305:
4626:
Consequently, there are two possibilities: either the rigid body's second major axis is in the same direction, or it has reversed direction. If it is still in the same direction, then
2413:
3364:
763:
118:
The experiment can be performed with any object that has three different moments of inertia, for instance with a book, remote control, or smartphone. The effect occurs whenever the
970:
1666:
1597:
1039:
4343:
4026:
3760:
4816:
4165:
3991:
3848:
3725:
3666:
2920:{\displaystyle {\begin{cases}\sum _{i}I_{i}\omega _{i}^{2}=\sum _{i}I_{i}\omega _{i}(0)^{2}\\\sum _{i}I_{i}^{2}\omega _{i}^{2}=\sum _{i}I_{i}^{2}\omega _{i}(0)^{2}\end{cases}}}
2503:
2458:
710:
1463:
1417:
3930:
starts anywhere on the diagonal curves, it would approach one of the points, distance exponentially decreasing, but never actually reach the point. In other words, we have 4
4109:
3899:
3610:
3155:
4760:
4235:
3088:
2963:
1627:
1000:
2104:
1986:
1959:
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1493:
136:
4723:
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3209:
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931:
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3544:
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3290:
4255:
3179:
3292:
also varies—thus giving us a varying ellipsoid of constant energy. This is shown in the animation as a fixed orange ellipsoid and increasing blue ellipsoid.
156:
5160:
111:) without accompanying half-rotation around another axis; it is also possible to make it rotate around the vertical axis perpendicular to the handle (ê
4930:
933:. To determine the nature of equilibrium, assume small initial angular velocities along the other two axes. As a result, according to equation (1),
821:
135:
33:
Composite video of a tennis racquet rotated around the three axes – the intermediate one flips from the light edge to the dark edge
4996:
4922:
4844:
geometrically above, this happens when the body's angular velocity is exactly aligned with its axis of maximal moment of inertia.
1999:
4200:
is very close to a saddle point. The body would linger near the saddle point, then rapidly move to the other saddle point, near
4629:
3549:
88:
768:
5179:
3558:
For small energy, there is no intersection, since we need a minimum of energy to stay on the angular momentum ellipsoid.
62:
37:
5102:
4528:
5159:
The "Dzhanibekov effect" - an exercise in mechanics or fiction? Explain mathematically a video from a space station,
4377:
3417:
2648:{\displaystyle {\begin{cases}2E=\sum _{i}I_{i}\omega _{i}^{2}\\L^{2}=\sum _{i}I_{i}^{2}\omega _{i}^{2}\end{cases}}}
4696:
viewed in the rigid body's reference frame are also mostly in the same direction. However, we have just seen that
4459:
4263:
5184:
2522:
During motion, both the energy and angular momentum-squared are conserved, thus we have two conserved quantities:
2372:
3298:
715:
2994:
are zero—that is, the object is exactly spinning around one of the principal axes. In all other situations,
122:
differs only slightly from the object's second principal axis; air resistance or gravity are not necessary.
936:
4896:
1635:
1566:
1008:
4310:
3996:
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4114:
3940:
3797:
3674:
3615:
2463:
2418:
662:
1422:
1376:
5031:
83:
whilst in space in 1985. The effect was known for at least 150 years prior, having been described by
76:
4828:
The body rapidly undergoes a complicated motion, until its second major axis has reversed direction.
4063:
3853:
3564:
3119:
2725:
2534:
80:
54:
4728:
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3058:
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1602:
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1964:
1937:
1902:
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4031:
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3243:
3214:
3029:
2997:
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2658:
21:
3369:
5189:
5131:
4992:
4988:
Classical
Mechanics with Calculus of Variations and Optimal Control: An Intuitive Introduction
4986:
4890:
4859:
in 1958. The elongated body of the spacecraft had been designed to spin about its long (least-
95:
94:
The theorem describes the following effect: rotation of an object around its first and third
5039:
4589:
3553:
All intersection curves of the angular momentum ellipsoid with energy ellipsoid (not shown).
3093:
2366:
1536:
119:
98:
is stable, whereas rotation around its second principal axis (or intermediate axis) is not.
3476:
3187:
1501:
909:
29:
4430:
4348:
4893: – Scalar measure of the rotational inertia with respect to a fixed axis of rotation
2415:
is a stable rotation around the origin—a neutral equilibrium point. Similarly, the point
5035:
5012:
Efroimsky, Michael (March 2002). "Euler, Jacobi, and
Missions to Comets and Asteroids".
3526:
3503:
3272:
2513:
4374:
undergoing complicated motions in space. At the beginning, the observer would see both
4240:
3164:
16:
A rigid body with 3 distinct axes of inertia is unstable rotating about the middle axis
5043:
906:
Consider the situation when the object is rotating around axis with moment of inertia
5173:
5123:
4856:
84:
4821:
Qualitatively, then, this is what an observer watching in free space would observe:
4167:. This is when the body rotates around its axis with the smallest moment of inertia.
1934:. Therefore, even a small disturbance, in the form of a very small initial value of
5051:
4957:
4884:
3668:. This is when the body rotates around its axis with the largest moment of inertia.
144:
70:
5127:
5154:
4868:
3546:
increases from zero to infinity. We can see that the curves evolve as follows:
3116:. In particular, the motion of the body in free space (obtained by integrating
5165:
5142:
5109:
5086:
5070:
4961:
4905: – Curve produced by the angular velocity vector on the inertia ellipsoid
4864:
4848:
891:{\displaystyle {\dot {\omega }}_{1},{\dot {\omega }}_{2},{\dot {\omega }}_{3}}
58:
4852:
1930:
opposed (and therefore will grow) and so rotation around the second axis is
1495:
is being opposed and so rotation around this axis is stable for the object.
73:
2106:
does not vary much, and write the equations of motion as a matrix equation:
5135:
4237:, linger again for a long time, and so on. The motion repeats with period
3184:
Consequently, we can analyze the geometry of motion with a fixed value of
5026:
4456:. After a while, the body performs a complicated motion and ends up with
1498:
Similar reasoning gives that rotation around axis with moment of inertia
155:
The tennis racket theorem can be qualitatively analysed with the help of
5138: : historically, the first mathematical description of this effect.
5071:"Slow motion Dzhanibekov effect demonstration with table tennis rackets"
4831:
The body rotates around its second major axis again for a while. Repeat.
41:
Title page of "Théorie
Nouvelle de la Rotation des Corps", 1852 printing
5147:
4902:
4860:
3794:
They intersect at two "diagonal" curves that intersects at the points
2713:
must stay on the intersection curve between two ellipsoids defined by
765:. The angular velocities around the object's three principal axes are
160:
4835:
This can be easily seen in the video demonstration in microgravity.
3366:, then the angular momentum ellipsoid's major axes are in ratios of
2518:
ellipsoid, and the expanding blue ellipsoid is the energy ellipsoid.
4060:
The energy ellipsoid last intersects the momentum ellipsoid when
3561:
The energy ellipsoid first intersects the momentum ellipsoid when
3548:
2512:
129:
36:
28:
20:
3240:
on the fixed ellipsoid of constant squared angular momentum. As
712:
denote the object's principal moments of inertia, and we assume
148:
5099:
4899: – Geometric method for visualizing a rotating rigid body
2072:{\displaystyle |\omega _{3}|\gg |\omega _{1}|,|\omega _{2}|}
87:
in 1834 and included in standard physics textbooks such as (
2913:
2641:
1533:
Now apply the same analysis to axis with moment of inertia
4825:
The body rotates around its second major axis for a while.
1996:
If the object is mostly rotating along its third axis, so
5143:"Ellipsoids and The Bizarre Behaviour of Rotating Bodies"
4689:{\displaystyle {\vec {\omega }}(0),{\vec {\omega }}(T/2)}
3414:, and the energy ellipsoid's major axes are in ratios of
902:
Stable rotation around the first and third principal axis
4887: – Description of the orientation of a rigid body
2320:
2176:
2133:
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2002:
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1049:
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978:
939:
912:
824:
771:
718:
665:
171:
5164:
The
Bizarre Behavior of Rotating Bodies, Veritasium
4991:. American Mathematical Society. pp. 151–152.
1632:Now, differentiating equation (1) and substituting
1005:Now, differentiating equation (2) and substituting
811:{\displaystyle \omega _{1},\omega _{2},\omega _{3}}
4963:The Bizarre Behavior of Rotating Bodies, Explained
4810:
4754:
4717:
4688:
4615:
4578:
4517:
4448:
4419:
4366:
4337:
4307:conserved, while both its angular velocity vector
4299:
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4229:
4192:
4159:
4103:
4049:
4020:
3985:
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2705:
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2407:
2357:
2098:
2071:
1980:
1953:
1918:
1889:
1660:
1621:
1591:
1555:
1529:Unstable rotation around the second principal axis
1517:
1487:
1457:
1411:
1363:
1033:
994:
964:
925:
890:
810:
757:
704:
649:
4863:) axis but refused to do so, and instead started
4586:are mostly aligned with the second major axis of
1599:is very small. Therefore, the time dependence of
972:is very small. Therefore, the time dependence of
4579:{\displaystyle {\vec {L}},{\vec {\omega }}(T/2)}
163:–free conditions, they take the following form:
3937:They intersect at two cycles around the points
3671:They intersect at two cycles around the points
4420:{\displaystyle {\vec {\omega }}(0),{\vec {L}}}
3993:. Since each cycle contains no point at which
3727:. Since each cycle contains no point at which
3500:its intersection curves with the ellipsoid of
4427:mostly aligned with the second major axis of
3463:{\displaystyle 1:1/{\sqrt {2}}:1/{\sqrt {3}}}
2930:By inspecting Euler's equations, we see that
8:
4518:{\displaystyle I(T/2),{\vec {\omega }}(T/2)}
4300:{\displaystyle {\vec {L}}=I{\vec {\omega }}}
4057:must be a periodic motion around each cycle.
3791:must be a periodic motion around each cycle.
2927:This is shown on the animation to the left.
818:and their time derivatives are denoted by
115:) without any accompanying half-rotation.
5128:Théorie nouvelle de la rotation des corps
5110:"Djanibekov effect modeled in Mathcad 14"
5108:Viacheslav Mezentsev (7 September 2011).
5025:
4944:Theorie Nouvelle de la Rotation des Corps
4793:
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2589:
2575:
2570:
2560:
2550:
2529:
2527:
2480:
2465:
2429:
2420:
2408:{\displaystyle (\omega _{1},\omega _{2})}
2396:
2383:
2374:
2341:
2327:
2315:
2296:
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2018:
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379:
356:
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344:
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258:
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235:
222:
199:
188:
187:
180:
172:
170:
4915:
3359:{\displaystyle I_{1}=1,I_{2}=2,I_{3}=3}
758:{\displaystyle I_{1}<I_{2}<I_{3}}
4923:Эффект Джанибекова (гайка Джанибекова)
3161:, just completed faster by a ratio of
4171:The tennis racket effect occurs when
3473:Now inscribe on a fixed ellipsoid of
965:{\displaystyle ~{\dot {\omega }}_{1}}
7:
2460:is a neutral equilibrium point, but
1661:{\displaystyle {\dot {\omega }}_{3}}
1592:{\displaystyle {\dot {\omega }}_{2}}
1034:{\displaystyle {\dot {\omega }}_{3}}
143:Dzhanibekov effect demonstration in
4871:from flexible structural elements.
4338:{\displaystyle {\vec {\omega }}(t)}
2367:zero trace and positive determinant
79:, who noticed one of the theorem's
5087:"Dzhanibekov effect demonstration"
4021:{\displaystyle {\dot {\omega }}=0}
3755:{\displaystyle {\dot {\omega }}=0}
57:which describes the movement of a
25:Principal axes of a tennis racket.
14:
5153:- intuitive video explanation by
5130:, Paris, Bachelier, 1834, 170 p.
4929:. The software can be downloaded
4811:{\displaystyle (0,\pm L/I_{2},0)}
4160:{\displaystyle (\pm L/I_{1},0,0)}
3986:{\displaystyle (\pm L/I_{1},0,0)}
3843:{\displaystyle (0,\pm L/I_{2},0)}
3720:{\displaystyle (0,0,\pm L/I_{3})}
3661:{\displaystyle (0,0,\pm L/I_{3})}
2655:and so for any initial condition
2498:{\displaystyle (0,\omega _{2},0)}
2453:{\displaystyle (\omega _{1},0,0)}
705:{\displaystyle I_{1},I_{2},I_{3}}
4762:are near opposite saddle points
1458:{\displaystyle I_{1}-I_{3}<0}
1412:{\displaystyle I_{2}-I_{1}>0}
2965:implies that two components of
1988:, causes the object to 'flip'.
91:) throughout the 20th Century.
4805:
4769:
4749:
4735:
4712:
4706:
4683:
4669:
4663:
4651:
4645:
4639:
4610:
4596:
4573:
4559:
4553:
4538:
4512:
4498:
4492:
4480:
4466:
4443:
4437:
4411:
4399:
4393:
4387:
4361:
4355:
4332:
4326:
4320:
4291:
4273:
4224:
4210:
4187:
4181:
4154:
4118:
4104:{\displaystyle 2E=L^{2}/I_{1}}
4044:
4038:
3980:
3944:
3934:between the two saddle points.
3917:
3911:
3894:{\displaystyle 2E=L^{2}/I_{2}}
3837:
3801:
3778:
3772:
3714:
3678:
3655:
3619:
3605:{\displaystyle 2E=L^{2}/I_{3}}
3256:
3250:
3227:
3221:
3150:{\displaystyle c\omega (ct)dt}
3138:
3129:
3077:
3068:
3042:
3036:
3010:
3004:
2981:
2975:
2946:
2940:
2901:
2894:
2803:
2796:
2700:
2694:
2671:
2665:
2492:
2467:
2447:
2422:
2402:
2376:
2284:
2258:
2223:
2197:
2065:
2050:
2042:
2027:
2019:
2004:
1799:
1785:
1782:
1756:
1753:
1727:
1273:
1259:
1256:
1230:
1227:
1201:
1119:
1093:
555:
529:
398:
372:
241:
215:
1:
5044:10.1016/S0273-1177(02)00017-0
53:, is a kinetic phenomenon of
5085:zapadlovsky (16 June 2010).
5069:Dan Russell (5 March 2010).
4755:{\displaystyle \omega (T/2)}
4230:{\displaystyle \omega (T/2)}
3083:{\displaystyle c\omega (ct)}
2958:{\displaystyle \omega (t)=0}
1622:{\displaystyle ~\omega _{2}}
995:{\displaystyle ~\omega _{1}}
63:principal moments of inertia
5103:International Space Station
3295:For concreteness, consider
2099:{\displaystyle \omega _{3}}
1981:{\displaystyle \omega _{3}}
1954:{\displaystyle \omega _{1}}
1919:{\displaystyle \omega _{1}}
1488:{\displaystyle \omega _{2}}
5206:
5014:Advances in Space Research
4718:{\displaystyle \omega (0)}
4345:and its moment of inertia
4193:{\displaystyle \omega (0)}
4050:{\displaystyle \omega (t)}
3923:{\displaystyle \omega (t)}
3784:{\displaystyle \omega (t)}
3262:{\displaystyle \omega (0)}
3233:{\displaystyle \omega (0)}
3055:is a solution, then so is
3048:{\displaystyle \omega (t)}
3016:{\displaystyle \omega (t)}
2987:{\displaystyle \omega (t)}
2706:{\displaystyle \omega (t)}
2677:{\displaystyle \omega (0)}
3407:{\displaystyle 1:1/2:1/3}
3026:By Euler's equations, if
2369:, implying the motion of
65:. It has also dubbed the
51:intermediate axis theorem
3023:must remain in motion.
4960:(September 19, 2019).
4812:
4756:
4719:
4690:
4617:
4616:{\displaystyle I(T/2)}
4580:
4519:
4450:
4421:
4368:
4339:
4301:
4251:
4231:
4194:
4161:
4105:
4051:
4022:
3987:
3924:
3895:
3844:
3785:
3756:
3721:
3662:
3606:
3554:
3540:
3517:
3494:
3464:
3408:
3360:
3286:
3263:
3234:
3205:
3175:
3151:
3110:
3109:{\displaystyle c>0}
3084:
3049:
3017:
2988:
2959:
2921:
2707:
2678:
2649:
2519:
2499:
2454:
2409:
2359:
2100:
2073:
1982:
1955:
1920:
1891:
1662:
1623:
1593:
1557:
1556:{\displaystyle I_{2}.}
1519:
1489:
1459:
1413:
1365:
1035:
996:
966:
927:
892:
812:
759:
706:
651:
152:
42:
34:
26:
4813:
4757:
4720:
4691:
4618:
4581:
4520:
4451:
4422:
4369:
4340:
4302:
4252:
4232:
4195:
4162:
4106:
4052:
4023:
3988:
3925:
3896:
3845:
3786:
3757:
3722:
3663:
3607:
3552:
3541:
3518:
3495:
3493:{\displaystyle L^{2}}
3465:
3409:
3361:
3287:
3269:varies, the value of
3264:
3235:
3206:
3204:{\displaystyle L^{2}}
3176:
3152:
3111:
3085:
3050:
3018:
2989:
2960:
2922:
2708:
2679:
2650:
2516:
2500:
2455:
2410:
2360:
2101:
2074:
1983:
1956:
1921:
1892:
1663:
1624:
1594:
1558:
1520:
1518:{\displaystyle I_{3}}
1490:
1460:
1414:
1366:
1036:
997:
967:
928:
926:{\displaystyle I_{1}}
893:
813:
760:
707:
652:
142:
47:tennis racket theorem
40:
32:
24:
5119:– via YouTube.
5096:– via YouTube.
5080:– via YouTube.
4766:
4729:
4700:
4630:
4590:
4529:
4460:
4449:{\displaystyle I(0)}
4431:
4378:
4367:{\displaystyle I(t)}
4349:
4311:
4264:
4241:
4204:
4175:
4115:
4064:
4032:
3997:
3941:
3905:
3854:
3798:
3766:
3731:
3675:
3616:
3565:
3527:
3504:
3477:
3418:
3370:
3299:
3273:
3244:
3215:
3188:
3165:
3120:
3094:
3059:
3030:
2998:
2969:
2934:
2717:
2688:
2684:, the trajectory of
2659:
2526:
2464:
2419:
2373:
2110:
2083:
2000:
1965:
1938:
1903:
1674:
1636:
1603:
1567:
1537:
1502:
1472:
1423:
1377:
1047:
1009:
976:
937:
910:
822:
769:
716:
663:
169:
81:logical consequences
77:Vladimir Dzhanibekov
61:with three distinct
5180:Classical mechanics
5036:2002AdSpR..29..725E
4985:Levi, Mark (2014).
4897:Poinsot's ellipsoid
3932:heteroclinic orbits
2883:
2855:
2840:
2762:
2637:
2622:
2580:
2505:is a saddle point.
1867:(positive quantity)
1668:from equation (3),
1341:(negative quantity)
1041:from equation (3),
55:classical mechanics
4947:, Bachelier, Paris
4808:
4752:
4715:
4686:
4613:
4576:
4515:
4446:
4417:
4364:
4335:
4297:
4247:
4227:
4190:
4157:
4101:
4047:
4018:
3983:
3920:
3891:
3840:
3781:
3752:
3717:
3658:
3602:
3555:
3539:{\displaystyle 2E}
3536:
3516:{\displaystyle 2E}
3513:
3490:
3460:
3404:
3356:
3285:{\displaystyle 2E}
3282:
3259:
3230:
3201:
3171:
3147:
3106:
3080:
3045:
3013:
2984:
2955:
2917:
2912:
2869:
2868:
2841:
2826:
2825:
2775:
2748:
2737:
2703:
2674:
2645:
2640:
2623:
2608:
2607:
2566:
2555:
2520:
2509:Geometric analysis
2495:
2450:
2405:
2355:
2349:
2309:
2162:
2096:
2069:
1978:
1951:
1916:
1887:
1885:
1658:
1629:may be neglected.
1619:
1589:
1553:
1515:
1485:
1455:
1409:
1361:
1359:
1031:
1002:may be neglected.
992:
962:
923:
888:
808:
755:
702:
647:
645:
153:
67:Dzhanibekov effect
43:
35:
27:
4891:Moment of inertia
4847:This happened to
4818:. Contradiction.
4666:
4642:
4556:
4541:
4525:, and again both
4495:
4414:
4390:
4323:
4294:
4276:
4250:{\displaystyle T}
4009:
3743:
3458:
3440:
3174:{\displaystyle c}
3090:for any constant
2859:
2816:
2766:
2728:
2598:
2546:
2126:
1868:
1849:
1838:
1835:
1832:
1829:
1825:
1711:
1649:
1608:
1580:
1342:
1323:
1312:
1309:
1306:
1303:
1299:
1185:
1142:
1074:
1022:
981:
953:
942:
879:
857:
835:
641:
637:
634:
631:
628:
625:
622:
619:
616:
613:
610:
607:
604:
601:
598:
595:
592:
589:
586:
583:
580:
510:
484:
480:
477:
474:
471:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
426:
423:
353:
327:
323:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
196:
157:Euler's equations
140:
5197:
5185:Physics theorems
5152:
5120:
5118:
5116:
5097:
5095:
5093:
5081:
5079:
5077:
5056:
5055:
5029:
5027:astro-ph/0112054
5009:
5003:
5002:
4982:
4976:
4975:
4973:
4971:
4954:
4948:
4939:
4933:
4928:
4920:
4855:launched by the
4839:With dissipation
4817:
4815:
4814:
4809:
4798:
4797:
4788:
4761:
4759:
4758:
4753:
4745:
4724:
4722:
4721:
4716:
4695:
4693:
4692:
4687:
4679:
4668:
4667:
4659:
4644:
4643:
4635:
4622:
4620:
4619:
4614:
4606:
4585:
4583:
4582:
4577:
4569:
4558:
4557:
4549:
4543:
4542:
4534:
4524:
4522:
4521:
4516:
4508:
4497:
4496:
4488:
4476:
4455:
4453:
4452:
4447:
4426:
4424:
4423:
4418:
4416:
4415:
4407:
4392:
4391:
4383:
4373:
4371:
4370:
4365:
4344:
4342:
4341:
4336:
4325:
4324:
4316:
4306:
4304:
4303:
4298:
4296:
4295:
4287:
4278:
4277:
4269:
4256:
4254:
4253:
4248:
4236:
4234:
4233:
4228:
4220:
4199:
4197:
4196:
4191:
4166:
4164:
4163:
4158:
4141:
4140:
4131:
4111:, at the points
4110:
4108:
4107:
4102:
4100:
4099:
4090:
4085:
4084:
4056:
4054:
4053:
4048:
4028:, the motion of
4027:
4025:
4024:
4019:
4011:
4010:
4002:
3992:
3990:
3989:
3984:
3967:
3966:
3957:
3929:
3927:
3926:
3921:
3900:
3898:
3897:
3892:
3890:
3889:
3880:
3875:
3874:
3849:
3847:
3846:
3841:
3830:
3829:
3820:
3790:
3788:
3787:
3782:
3762:, the motion of
3761:
3759:
3758:
3753:
3745:
3744:
3736:
3726:
3724:
3723:
3718:
3713:
3712:
3703:
3667:
3665:
3664:
3659:
3654:
3653:
3644:
3612:, at the points
3611:
3609:
3608:
3603:
3601:
3600:
3591:
3586:
3585:
3545:
3543:
3542:
3537:
3522:
3520:
3519:
3514:
3499:
3497:
3496:
3491:
3489:
3488:
3469:
3467:
3466:
3461:
3459:
3454:
3452:
3441:
3436:
3434:
3413:
3411:
3410:
3405:
3400:
3386:
3365:
3363:
3362:
3357:
3349:
3348:
3330:
3329:
3311:
3310:
3291:
3289:
3288:
3283:
3268:
3266:
3265:
3260:
3239:
3237:
3236:
3231:
3210:
3208:
3207:
3202:
3200:
3199:
3180:
3178:
3177:
3172:
3159:exactly the same
3156:
3154:
3153:
3148:
3115:
3113:
3112:
3107:
3089:
3087:
3086:
3081:
3054:
3052:
3051:
3046:
3022:
3020:
3019:
3014:
2993:
2991:
2990:
2985:
2964:
2962:
2961:
2956:
2926:
2924:
2923:
2918:
2916:
2915:
2909:
2908:
2893:
2892:
2882:
2877:
2867:
2854:
2849:
2839:
2834:
2824:
2811:
2810:
2795:
2794:
2785:
2784:
2774:
2761:
2756:
2747:
2746:
2736:
2712:
2710:
2709:
2704:
2683:
2681:
2680:
2675:
2654:
2652:
2651:
2646:
2644:
2643:
2636:
2631:
2621:
2616:
2606:
2594:
2593:
2579:
2574:
2565:
2564:
2554:
2504:
2502:
2501:
2496:
2485:
2484:
2459:
2457:
2456:
2451:
2434:
2433:
2414:
2412:
2411:
2406:
2401:
2400:
2388:
2387:
2364:
2362:
2361:
2356:
2354:
2353:
2346:
2345:
2332:
2331:
2314:
2313:
2301:
2300:
2291:
2283:
2282:
2270:
2269:
2257:
2256:
2240:
2239:
2230:
2222:
2221:
2209:
2208:
2196:
2195:
2167:
2166:
2159:
2158:
2145:
2144:
2127:
2125:
2114:
2105:
2103:
2102:
2097:
2095:
2094:
2079:, we can assume
2078:
2076:
2075:
2070:
2068:
2063:
2062:
2053:
2045:
2040:
2039:
2030:
2022:
2017:
2016:
2007:
1987:
1985:
1984:
1979:
1977:
1976:
1960:
1958:
1957:
1952:
1950:
1949:
1925:
1923:
1922:
1917:
1915:
1914:
1896:
1894:
1893:
1888:
1886:
1882:
1881:
1869:
1866:
1857:
1856:
1851:
1850:
1842:
1836:
1833:
1830:
1827:
1826:
1823:
1817:
1816:
1807:
1806:
1797:
1796:
1781:
1780:
1768:
1767:
1752:
1751:
1739:
1738:
1719:
1718:
1713:
1712:
1704:
1700:
1699:
1690:
1689:
1667:
1665:
1664:
1659:
1657:
1656:
1651:
1650:
1642:
1628:
1626:
1625:
1620:
1618:
1617:
1606:
1598:
1596:
1595:
1590:
1588:
1587:
1582:
1581:
1573:
1562:
1560:
1559:
1554:
1549:
1548:
1525:is also stable.
1524:
1522:
1521:
1516:
1514:
1513:
1494:
1492:
1491:
1486:
1484:
1483:
1464:
1462:
1461:
1456:
1448:
1447:
1435:
1434:
1418:
1416:
1415:
1410:
1402:
1401:
1389:
1388:
1370:
1368:
1367:
1362:
1360:
1356:
1355:
1343:
1340:
1331:
1330:
1325:
1324:
1316:
1310:
1307:
1304:
1301:
1300:
1297:
1291:
1290:
1281:
1280:
1271:
1270:
1255:
1254:
1242:
1241:
1226:
1225:
1213:
1212:
1193:
1192:
1187:
1186:
1178:
1174:
1173:
1164:
1163:
1150:
1149:
1144:
1143:
1135:
1131:
1130:
1118:
1117:
1105:
1104:
1082:
1081:
1076:
1075:
1067:
1063:
1062:
1040:
1038:
1037:
1032:
1030:
1029:
1024:
1023:
1015:
1001:
999:
998:
993:
991:
990:
979:
971:
969:
968:
963:
961:
960:
955:
954:
946:
940:
932:
930:
929:
924:
922:
921:
897:
895:
894:
889:
887:
886:
881:
880:
872:
865:
864:
859:
858:
850:
843:
842:
837:
836:
828:
817:
815:
814:
809:
807:
806:
794:
793:
781:
780:
764:
762:
761:
756:
754:
753:
741:
740:
728:
727:
711:
709:
708:
703:
701:
700:
688:
687:
675:
674:
656:
654:
653:
648:
646:
642:
639:
635:
632:
629:
626:
623:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
590:
587:
584:
581:
578:
577:
576:
567:
566:
554:
553:
541:
540:
518:
517:
512:
511:
503:
499:
498:
485:
482:
478:
475:
472:
469:
466:
463:
460:
457:
454:
451:
448:
445:
442:
439:
436:
433:
430:
427:
424:
421:
420:
419:
410:
409:
397:
396:
384:
383:
361:
360:
355:
354:
346:
342:
341:
328:
325:
321:
318:
315:
312:
309:
306:
303:
300:
297:
294:
291:
288:
285:
282:
279:
276:
273:
270:
267:
264:
263:
262:
253:
252:
240:
239:
227:
226:
204:
203:
198:
197:
189:
185:
184:
141:
120:axis of rotation
5205:
5204:
5200:
5199:
5198:
5196:
5195:
5194:
5170:
5169:
5141:
5114:
5112:
5107:
5091:
5089:
5084:
5075:
5073:
5068:
5065:
5060:
5059:
5011:
5010:
5006:
4999:
4984:
4983:
4979:
4969:
4967:
4956:
4955:
4951:
4941:Poinsot (1834)
4940:
4936:
4926:
4925:, 23 July 2009
4921:
4917:
4912:
4881:
4841:
4789:
4764:
4763:
4727:
4726:
4698:
4697:
4628:
4627:
4588:
4587:
4527:
4526:
4458:
4457:
4429:
4428:
4376:
4375:
4347:
4346:
4309:
4308:
4262:
4261:
4239:
4238:
4202:
4201:
4173:
4172:
4132:
4113:
4112:
4091:
4076:
4062:
4061:
4030:
4029:
3995:
3994:
3958:
3939:
3938:
3903:
3902:
3881:
3866:
3852:
3851:
3821:
3796:
3795:
3764:
3763:
3729:
3728:
3704:
3673:
3672:
3645:
3614:
3613:
3592:
3577:
3563:
3562:
3525:
3524:
3502:
3501:
3480:
3475:
3474:
3416:
3415:
3368:
3367:
3340:
3321:
3302:
3297:
3296:
3271:
3270:
3242:
3241:
3213:
3212:
3191:
3186:
3185:
3163:
3162:
3118:
3117:
3092:
3091:
3057:
3056:
3028:
3027:
2996:
2995:
2967:
2966:
2932:
2931:
2911:
2910:
2900:
2884:
2813:
2812:
2802:
2786:
2776:
2738:
2721:
2715:
2714:
2686:
2685:
2657:
2656:
2639:
2638:
2585:
2582:
2581:
2556:
2530:
2524:
2523:
2511:
2476:
2462:
2461:
2425:
2417:
2416:
2392:
2379:
2371:
2370:
2348:
2347:
2337:
2334:
2333:
2323:
2316:
2308:
2307:
2302:
2292:
2274:
2261:
2248:
2242:
2241:
2231:
2213:
2200:
2187:
2182:
2172:
2161:
2160:
2150:
2147:
2146:
2136:
2129:
2118:
2108:
2107:
2086:
2081:
2080:
2054:
2031:
2008:
1998:
1997:
1994:
1992:Matrix analysis
1968:
1963:
1962:
1941:
1936:
1935:
1906:
1901:
1900:
1884:
1883:
1873:
1858:
1839:
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5063:External links
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5057:
5020:(5): 725–734.
5004:
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195:
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127:
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96:principal axes
15:
13:
10:
9:
6:
4:
3:
2:
5202:
5191:
5188:
5186:
5183:
5181:
5178:
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5163:
5161:
5158:
5156:
5150:
5149:
5144:
5140:
5137:
5133:
5129:
5125:
5124:Louis Poinsot
5122:
5111:
5106:
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5101:
5088:
5083:
5072:
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5066:
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5053:
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4998:9781470414443
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4872:
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4857:United States
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2023:
2013:
2009:
1991:
1989:
1973:
1969:
1946:
1942:
1933:
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1911:
1907:
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1238:
1234:
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1218:
1214:
1209:
1205:
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1196:
1189:
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1179:
1170:
1166:
1160:
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1146:
1139:
1136:
1127:
1123:
1114:
1110:
1106:
1101:
1097:
1090:
1087:
1085:
1078:
1071:
1068:
1059:
1055:
1042:
1026:
1019:
1016:
1003:
987:
983:
957:
950:
947:
918:
914:
901:
899:
883:
876:
873:
866:
861:
854:
851:
844:
839:
832:
829:
803:
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795:
790:
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777:
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720:
697:
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689:
684:
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671:
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657:
573:
569:
563:
559:
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546:
542:
537:
533:
526:
523:
521:
514:
507:
504:
495:
491:
416:
412:
406:
402:
393:
389:
385:
380:
376:
369:
366:
364:
357:
350:
347:
338:
334:
259:
255:
249:
245:
236:
232:
228:
223:
219:
212:
209:
207:
200:
193:
190:
181:
177:
164:
162:
158:
150:
146:
125:
123:
121:
116:
99:
97:
92:
90:
86:
85:Louis Poinsot
82:
78:
75:
72:
68:
64:
60:
56:
52:
48:
39:
31:
23:
19:
5146:
5113:. Retrieved
5090:. Retrieved
5074:. Retrieved
5017:
5013:
5007:
4987:
4980:
4970:February 16,
4968:. Retrieved
4966:. Veritasium
4962:
4958:Derek Muller
4952:
4943:
4937:
4927:(in Russian)
4918:
4885:Euler angles
4875:progenitor.
4873:
4851:, the first
4846:
4842:
4834:
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4869:dissipation
3211:, and vary
5174:Categories
5115:2 February
5092:2 February
5076:2 February
4910:References
4865:precessing
4849:Explorer 1
2365:which has
1899:Note that
1563:This time
1468:Note that
1298:i.e.
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1932:unstable
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