Poinsot's ellipsoid

In classical mechanics, Poinsot's construction (after Louis Poinsot) 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 ${\displaystyle {\boldsymbol {\omega }}}$ of the rigid rotor is not constant, but satisfies Euler's equations. The conservation of kinetic energy and angular momentum provide two constraints on the motion of ${\displaystyle {\boldsymbol {\omega }}}$.

Without explicitly solving these equations, the motion ${\displaystyle {\boldsymbol {\omega }}}$ can be described geometrically as follows:^[1]

• The rigid body's motion is entirely determined by the motion of its inertia ellipsoid, which is rigidly fixed to the rigid body like a coordinate frame.
• Its inertia ellipsoid rolls, without slipping, on the invariable plane, with the center of the ellipsoid a constant height above the plane.
• At all times, ${\displaystyle {\boldsymbol {\omega }}}$ is the point of contact between the ellipsoid and the plane.

The motion is periodic, so ${\displaystyle {\boldsymbol {\omega }}}$ traces out two closed curves, one on the ellipsoid, another on the plane.

• The closed curve on the ellipsoid is the polhode.
• The closed curve on the plane is the herpolhode.

If the rigid body is symmetric (has two equal moments of inertia), the vector ${\displaystyle {\boldsymbol {\omega }}}$ describes a cone (and its endpoint a circle). This is the torque-free precession of the rotation axis of the rotor.

The law of conservation of energy implies that in the absence of energy dissipation or applied torques, the angular kinetic energy ${\displaystyle T\ }$ is conserved, so ${\textstyle {\frac {dT}{dt}}=0}$.

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

${\displaystyle T={\frac {1}{2}}{\boldsymbol {\omega }}\cdot \mathbf {I} \cdot {\boldsymbol {\omega }}={\frac {1}{2}}I_{1}\omega _{1}^{2}+{\frac {1}{2}}I_{2}\omega _{2}^{2}+{\frac {1}{2}}I_{3}\omega _{3}^{2}}$

where ${\displaystyle \omega _{k}\ }$ are the components of the angular velocity vector ${\displaystyle {\boldsymbol {\omega }}}$, and the ${\displaystyle I_{k}\ }$ are the principal moments of inertia when both are in the body frame. Thus, the conservation of kinetic energy imposes a constraint on the three-dimensional angular velocity vector ${\displaystyle {\boldsymbol {\omega }}}$; in the principal axis frame, it must lie on the ellipsoid defined by the above equation, called the inertia ellipsoid.

The path traced out on this ellipsoid by the angular velocity vector ${\displaystyle {\boldsymbol {\omega }}}$ is called the polhode (coined by Poinsot from Greek roots for "pole path") and is generally circular or taco-shaped.

The law of conservation of angular momentum states that in the absence of applied torques, the angular momentum vector ${\displaystyle \mathbf {L} }$ is conserved in an inertial reference frame, so ${\textstyle {\frac {d\mathbf {L} }{dt}}=0}$.

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

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

which leads to the equation

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

Since the dot product of ${\displaystyle {\boldsymbol {\omega }}}$ and ${\displaystyle \mathbf {L} }$ is constant, and ${\displaystyle \mathbf {L} }$ itself is constant, the angular velocity vector ${\displaystyle {\boldsymbol {\omega }}}$ has a constant component in the direction of the angular momentum vector ${\displaystyle \mathbf {L} }$. This imposes a second constraint on the vector ${\displaystyle {\boldsymbol {\omega }}}$; in absolute space, it must lie on the invariable plane defined by its dot product with the conserved vector ${\displaystyle \mathbf {L} }$. The normal vector to the invariable plane is aligned with ${\displaystyle \mathbf {L} }$. The path traced out by the angular velocity vector ${\displaystyle {\boldsymbol {\omega }}}$ on the invariable plane is called the herpolhode (coined from Greek roots for "serpentine pole path").

The herpolhode is generally an open curve, which means that the rotation does not perfectly repeat, but the polhode is a closed curve (see below).^[2]

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

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

Hence, the normal vector to the kinetic-energy ellipsoid at ${\displaystyle {\boldsymbol {\omega }}}$ is proportional to ${\displaystyle \mathbf {L} }$, 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 ${\displaystyle {\boldsymbol {\omega }}}$ 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.

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 ${\displaystyle L\ }$of the angular momentum and the kinetic energy ${\displaystyle T\ }$are both conserved

${\displaystyle {\begin{aligned}L^{2}&=L_{1}^{2}+L_{2}^{2}+L_{3}^{2}\\[2pt]T&={\frac {L_{1}^{2}}{2I_{1}}}+{\frac {L_{2}^{2}}{2I_{2}}}+{\frac {L_{3}^{2}}{2I_{3}}}\end{aligned}}}$

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

These conservation laws are equivalent to two constraints to the three-dimensional angular momentum vector ${\displaystyle \mathbf {L} }$. The kinetic energy constrains ${\displaystyle \mathbf {L} }$ to lie on an ellipsoid, whereas the angular momentum constraint constrains ${\displaystyle \mathbf {L} }$ to lie on a sphere. These two surfaces intersect in two curves shaped like the edge of a taco that define the possible solutions for ${\displaystyle \mathbf {L} }$. This shows that ${\displaystyle \mathbf {L} }$, and the polhode, stay on a closed loop, in the object's moving frame of reference.

The orientation of the body in space thus has two degrees of freedom. Firstly, some point on the "taco edge" has to align with ${\displaystyle \mathbf {L} ,}$ which is a constant vector in absolute space. Secondly, with the vector in the body frame that goes through this point fixed, the body can have any amount of rotation around that vector. So in principle, the body's orientation is some point on a toroidal 2-manifold inside the 3-manifold of all orientations. In general, the object will follow a non-periodic path on this torus, but it may follow a periodic path. The time taken for ${\displaystyle \mathbf {L} }$ to complete one cycle around its track in the body frame is constant, but after a cycle the body will have rotated by an amount that may not be a rational number of degrees, in which case the orientation will not be periodic, but almost periodic.

In general a torus is almost determined by three parameters: the ratios of the second and third moments of inertia to the highest of the three moments of inertia, and the ratio ${\displaystyle L^{2}/(TI_{3})}$ relating the angular momentum to the energy times the highest moment of inertia. But for any such a set of parameters there are two tori, because there are two "tacos" (corresponding to two polhodes). A set of 180° rotations carries any orientation of one torus into an orientation of the other with the opposite point aligned with the angular momentum vector. If the angular momentum is exactly aligned with a principal axes, the torus degenerates into a single loop. If exactly two moments of inertia are equal (a so-called symmetric body), then in addition to tori there will be an infinite number of loops, and if all three moments of inertia are equal, there will be loops but no tori. If the three moments of inertia are all different and ${\displaystyle L^{2}=TI_{2}}$ but the intermediate axis is not aligned with the angular momentum, then the orientation will be some point on a topological open annulus.

In the general case of rotation of an unsymmetric body, which has different values of the moment of inertia about the three principal axes, the rotational motion can be quite complex unless the body is rotating around a principal axis. As described in the tennis racket theorem, rotation of an object around its first or third principal axis is stable, while rotation around its second principal axis (or intermediate axis) is not. The motion is simplified in the case of an axisymmetric body, in which the moment of inertia is the same about two of the principal axes. These cases include rotation of a prolate spheroid (the shape of an American football), or rotation of an oblate spheroid (the shape of a flattened sphere). In this case, the angular velocity describes a cone, and the polhode is a circle. This analysis is applicable, for example, to the axial precession of the rotation of a planet (the case of an oblate spheroid.)

One of the applications of Poinsot's construction is in visualizing the rotation of a spacecraft in orbit.^[3]