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 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 . If the rigid rotor is symmetric (has two equal moments of inertia), the vector describes a cone (and its endpoint a circle). This is the torque-free precession of the rotation axis of the rotor.

Angular kinetic energy constraintEdit

The law of conservation of energy implies that in the absence of energy dissipation or applied torques, the angular kinetic energy   is conserved, so  .

The angular kinetic energy may be expressed in terms of the moment of inertia tensor   and the angular velocity vector  

 

where   are the components of the angular velocity vector   along the principal axes, and the   are the principal moments of inertia. Thus, the conservation of kinetic energy imposes a constraint on the three-dimensional angular velocity vector  ; 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   is called the polhode (coined by Poinsot from Greek roots for "pole path") and is generally circular or taco-shaped.

Angular momentum constraintEdit

The law of conservation of angular momentum states that in the absence of applied torques, the angular momentum vector   is conserved in an inertial reference frame, so  .

The angular momentum vector   can be expressed in terms of the moment of inertia tensor   and the angular velocity vector  

 

which leads to the equation

 

Since the dot product of   and   is constant, and   itself is constant, the angular velocity vector   has a constant component in the direction of the angular momentum vector  . This imposes a second constraint on the vector  ; in absolute space, it must lie on the invariable plane defined by its dot product with the conserved vector  . The normal vector to the invariable plane is aligned with  . The path traced out by the angular velocity vector   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).[1]

Tangency condition and constructionEdit

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   equals the angular momentum vector  

 

Hence, the normal vector to the kinetic-energy ellipsoid at   is proportional to  , 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   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 frameEdit

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   of the angular momentum and the kinetic energy   are both conserved

 
 

where the   are the components of the angular momentum vector along the principal axes, and the   are the principal moments of inertia.

These conservation laws are equivalent to two constraints to the three-dimensional angular momentum vector  . The kinetic energy constrains   to lie on an ellipsoid, whereas the angular momentum constraint constrains   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  . This shows that  , and the polhode, stay on a closed loop, in the object's moving frame of reference.

Dzhanibekov effect demonstration in microgravity, NASA.

If the body is set spinning on its intermediate principal axis, then the intersection of the ellipsoid and the sphere is like two loops that cross at two points, lined up with that axis.   will eventually move off this point along one of the four tracks that depart from this point, and head to the opposite point. This is reflected by   on the Poinsot ellipsoid. See video at right and Tennis racket theorem.

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

Special caseEdit

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 pancake). 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.)

Hyperion (a moon of Saturn), two moons of Pluto and many other small bodies of the Solar System have tumbling rotations.

ApplicationsEdit

One of the applications of Poinsot's construction is in visualizing the rotation of a spacecraft in orbit.[2]

See alsoEdit

ReferencesEdit

  1. ^ Jerry Ginsberg. "Gyroscopic Effects," Engineering Dynamics, Volume 10, p. 650, Cambridge University Press, 2007
  2. ^ F. Landis Markley and John L. Crassidis, Chapter 3.3, "Attitude Dynamics," p. 89; Fundamentals of Spacecraft Attitude Determination and Control, Springer Technology and Engineering Series, 2014.

SourcesEdit

  • Poinsot (1834) Theorie Nouvelle de la Rotation des Corps, Bachelier, Paris.
  • 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

External linksEdit