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 an inertial laboratory frame. The angular velocity vector \boldsymbol\omega 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 \boldsymbol\omega. If the rigid rotor is symmetric (has two equal moments of inertia), the vector \boldsymbol\omega describes a cone (and its endpoint a circle). This is the torque-free precession of the rotation axis of the rotor.

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Angular kinetic energy constraint

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

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

T = \frac{1}{2} \boldsymbol\omega \cdot \mathbf{I} \cdot \boldsymbol\omega = \frac{1}{2} I_{1} \omega_{1}^{2} %2B \frac{1}{2} I_{2} \omega_{2}^{2} %2B \frac{1}{2} I_{3} \omega_{3}^{2}

where \omega_{k}\ are the components of the angular velocity vector \boldsymbol\omega 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 \boldsymbol\omega; in the principal axis frame, it must lie on an ellipsoid, called inertia ellipsoid.

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 \boldsymbol\omega is called the polhode (coined by Poinsot from Greek roots 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 frame \frac{d\mathbf{L}}{dt} = 0.

The angular momentum vector \mathbf{L} can be expressed in terms of the moment of inertia tensor \mathbf{I} and the angular velocity vector \boldsymbol\omega

\mathbf{L} = \mathbf{I} \cdot \boldsymbol\omega

which leads to the equation

T = \frac{1}{2} \boldsymbol\omega \cdot \mathbf{L}

Since the dot product of \boldsymbol\omega and \mathbf{L} is constant, and \mathbf{L} itself is constant, the angular velocity vector \boldsymbol\omega has a constant component in the direction of the angular momentum vector \mathbf{L}. This imposes a second constraint on the vector \boldsymbol\omega; 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 \boldsymbol\omega on the invariable plane is called the herpolhode (coined from Greek roots for "serpentine pole path").

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 \boldsymbol\omega equals the angular momentum vector \mathbf{L}

\frac{dT}{d\boldsymbol\omega} = \mathbf{I} \cdot \boldsymbol\omega = \mathbf{L}

Hence, the normal vector to the kinetic-energy ellipsoid at \boldsymbol\omega 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 \boldsymbol\omega 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 body frame

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} %2B L_{2}^{2} %2B L_{3}^{2}
T = \frac{L_{1}^{2}}{2I_{1}} %2B \frac{L_{2}^{2}}{2I_{2}} %2B \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 an ellipsoid, whereas the angular momentum constraint constrains \mathbf{L} to lie on a sphere. These two surfaces intersect in taco-shaped curves that define the possible solutions for \mathbf{L}.

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

References

See also