Poinsot's ellipsoid


Poinsot's ellipsoid

In classical mechanics, Poinsot's construction is a geometrical method for visualizing the torque-free motion of a rotating rigid body, that is, the motion of a rigid body on which no external forces are acting. This motion has four constants: the kinetic energy of the body and the three components of the angular momentum, expressed with respect to a laboratory frame. The angular velocity vector oldsymbolomega of the rigid rotor is "not constant", but satisfies Euler's equations. Without explicitly solving these equations, Louis Poinsot was able to visualize the motion of the endpoint of the angular velocity vector. To this end he used the conservation of kinetic energy and angular momentum as constraints on the motion of the angular velocity vector oldsymbolomega. If the rigid rotor is symmetric (has two equal moments of inertia), the vector oldsymbolomega describes a cone (and its endpoint a circle). This is the torque-free precession of the rotation axis of the rotor.

Kinetic energy constraint

In the absence of applied torques, the kinetic energy T is conserved so frac{dT}{dt} = 0.

The kinetic energy may be expressed in terms of the moment of inertia tensor mathbf{I} and the angular velocity vector oldsymbolomega

:T = frac{1}{2} oldsymbolomega cdot mathbf{I} cdot oldsymbolomega = frac{1}{2} I_{1} omega_{1}^{2} + frac{1}{2} I_{2} omega_{2}^{2} + frac{1}{2} I_{3} omega_{3}^{2}

where omega_{k} are the components of the angular velocity vector oldsymbolomega along the principal axes, and the I_{k} are the principal moments of inertia. Thus, the conservation of kinetic energy imposes a constraint on the three-dimensional angular velocity vector oldsymbolomega; in the principal axis frame, it must lie on an ellipsoid, called inertia ellipsoid.

:T =frac{1}{2} I_{1} omega_{1}^{2} + frac{1}{2} I_{2} omega_{2}^{2} + frac{1}{2} I_{3} omega_{3}^{2}

The ellipsoid axes values are the half of the principal moments of inertia. The path traced out on this ellipsoid by the angular velocity vector oldsymbolomega is called the polhode (Greek for "pole path") and is generally circular or taco-shaped.

Angular momentum constraint

In the absence of applied torques, the angular momentum vector mathbf{L} is conserved in an inertial reference framefrac{dmathbf{L{dt} = 0.

The angular momentum vector mathbf{L} can also be expressedin terms of the moment of inertia tensor mathbf{I} and the angular velocity vector oldsymbolomega

:mathbf{L} = mathbf{I} cdot oldsymbolomega

which leads to the equation

:T = frac{1}{2} oldsymbolomega cdot mathbf{L}

showing that the angular velocity vector oldsymbolomegahas a constant component in the direction of the angular momentum vector mathbf{L}. This imposes a second constraint on the vector oldsymbolomega; in absolute space, it must lie on an invariable plane defined by its dot product with the conserved vector mathbf{L}. The normal vector to the invariable plane is aligned with mathbf{L}. The path traced out by the angular velocity vector oldsymbolomega on the invariable plane is called the herpolhode (Greek for "serpentine path"). The etymology of "herpolhode" is obscure, since most herpolhodes are simple circles, not sinusoidal; Poinsot may have been thinking of the ouroboros, the symbol of endless cycling.

Tangency condition and construction

These two constraints operate in different reference frames; the ellipsoidal constraint holds in the (rotating) principal axis frame, whereas the invariable plane constant operates in absolute space. To relate these constraints, we note that the gradient vector of the kinetic energy with respect to angular velocity vector oldsymbolomega equals the angular momentum vector mathbf{L}

:frac{dT}{doldsymbolomega} = mathbf{I} cdot oldsymbolomega = mathbf{L}

Hence, the normal vector to the kinetic-energy ellipsoid at oldsymbolomega is proportional to mathbf{L}, which is also true of the invariable plane. Since their normal vectors point in the same direction, these two surfaces will intersect tangentially.

Taken together, these results show that, in an absolute reference frame, the instantaneous angular velocity vector oldsymbolomega is the point of intersection between a fixed invariable plane and a kinetic-energy ellipsoid that is tangent to it and rolls around on it without slipping. This is Poinsot's construction.

Derivation of the polhodes

In the principal axis frame (which is rotating in absolute space), the angular momentum vector is "not" conserved even in the absence of applied torques, but varies as described by Euler's equations. However, in the absence of applied torques, the magnitude L of the angular momentum and the kinetic energy T are both conserved

:L^{2} = L_{1}^{2} + L_{2}^{2} + L_{3}^{2}

:T = frac{L_{1}^{2{2I_{1 + frac{L_{2}^{2{2I_{2 + frac{L_{3}^{2{2I_{3

where the L_{k} are the components of the angular momentum vector along the principal axes, and the I_{k} are the principal moments of inertia.

These conservation laws are equivalent to two constraints to the three-dimensional angular momentum vector mathbf{L}. The kinetic energy constrains mathbf{L} to lie on anellipsoid, whereas the angular momentum constraint constrainsmathbf{L} to lie on a sphere. These two surfaces intersect in taco-shaped curves that define the possible solutionsfor mathbf{L}.

This construction differs from Poinsot's construction because it considersthe angular momentum vector mathbf{L} rather than the angular velocity vector oldsymbolomega. It appears to have been developed by Jacques Philippe Marie Binet.

References

* Poinsot (1834) "Theorie Nouvelle de la Rotation des Corps"'.

* Landau LD and Lifshitz EM (1976) "Mechanics", 3rd. ed., Pergamon Press. ISBN 0-08-021022-8 (hardcover) and ISBN 0-08-029141-4 (softcover).

* Goldstein H. (1980) "Classical Mechanics", 2nd. ed., Addison-Wesley. ISBN 0-201-02918-9

* Symon KR. (1971) "Mechanics", 3rd. ed., Addison-Wesley. ISBN 0-201-07392-7

ee also

* polhode
* precession
* Principal axes
* Moment of inertia
* Tait-Bryan rotations
* Euler angles


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