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.
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
If the rigid body is symmetric (has two equal moments of inertia), the vector
The angular kinetic energy may be expressed in terms of the moment of inertia tensor
Thus, the conservation of kinetic energy imposes a constraint on the three-dimensional angular velocity vector
is called the polhode (coined by Poinsot from Greek roots for "pole path") and is generally circular or taco-shaped.
; in absolute space, it must lie on the invariable plane defined by its dot product with the conserved vector
on the invariable plane is called the herpolhode (coined from Greek roots for "serpentine pole path").
[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.
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
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.
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.
These conservation laws are equivalent to two constraints to the three-dimensional angular momentum vector
These two surfaces intersect in two curves shaped like the edge of a taco that define the possible solutions for
, and the polhode, stay on a closed loop, in the object's moving frame of reference.
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.
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).
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.
but the intermediate axis is not aligned with the angular momentum, then the orientation will be some point on a topological open annulus.
It appears to have been developed by Jacques Philippe Marie Binet.
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.