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. 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 ${\displaystyle {\boldsymbol {\omega }}}$. If the rigid rotor 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.

Angular kinetic energy constraint

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

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 ${\displaystyle {\boldsymbol {\omega }}}$ along the principal axes, and the ${\displaystyle I_{k}\ }$ are the principal moments of inertia. 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 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 ${\displaystyle {\boldsymbol {\omega }}}$ 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 ${\displaystyle \mathbf {L} }$ is conserved in an inertial reference frame ${\displaystyle {\frac {d\mathbf {L} }{dt}}=0}$.

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 an 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").

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 ${\displaystyle {\boldsymbol {\omega }}}$ 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.

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

${\displaystyle L^{2}=L_{1}^{2}+L_{2}^{2}+L_{3}^{2}}$

${\displaystyle T={\frac {L_{1}^{2}}{2I_{1}}}+{\frac {L_{2}^{2}}{2I_{2}}}+{\frac {L_{3}^{2}}{2I_{3}}}}$

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 taco-shaped curves that define the possible solutions for ${\displaystyle \mathbf {L} }$.

This construction differs from Poinsot's construction because it considers the angular momentum vector ${\displaystyle \mathbf {L} }$ rather than the angular velocity vector ${\displaystyle {\boldsymbol {\omega }}}$. 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

External links

• Poinsot construction in stereo 3D simulation - online and free.

See also

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

Summary

DescriptionBell observers experiment2.png	English: A thought experiment related to one discussed by J. S. Bell (see Bell's spaceship paradox). Two observers in Minkowski spacetime accelerate with constant magnitude ${\displaystyle k}$ acceleration for proper time ${\displaystyle \sigma }$ (acceleration and elapsed time measured by the observers themselves, not some inertial observer). They are comoving and inertial before and after this phase. As Bell pointed out, in Minkowski geometry, the length of the spacelike line segment A′B″ turns out to be greater than the length of the spacelike line segment AB. Русский: Мысленный эксперимент, связанный с рассмотрением Дж. С. Беллом (см. Парадокс Белла). Два наблюдателя в пространстве-времени Минковского ускоряются с постоянной величиной ускорения ${\displaystyle k}$ за собственное время ${\displaystyle \sigma }$ (ускорение и прошедшее время измеряются самими наблюдателями, а не сторонним инерциальным наблюдателем). Они являются сопутствующими и инерциальными до и после фазы ускорения. Как отметил Белл, в геометрии Минковского длина пространственноподобного отрезка A′B″ "оказывается больше длины пространственноподобного отрезка AB.
Date	10 June 2006
Source	This png image was converted from a jpeg image made using Maple by User:Hillman.
Author	Hillman (talk) (Uploads)

This image was uploaded with an opaque background where it should have been transparent. If possible, please upload a PNG or SVG version of this image, derived from a non-JPEG source so that it has an alpha channel and no compression artifacts (or with existing artifacts removed). If it is not possible to obtain a cleaner version then consider creating a new, derivative, image with the background removed, while leaving this image alone; use templates such as {{Retouched picture}} and {{Derivative versions}} as appropriate; then ignore the rest of these requests. After doing so, please: Find all pages in all Wikimedia projects that use this image, and replace the old image with the new image in each. Mark this image as having been superseded by adding one of these templates. For more information, see Wikipedia:Preparing images for upload. Deutsch ∙ English ∙ español ∙ français ∙ italiano ∙ português ∙ sicilianu ∙ македонски ∙ മലയാളം ∙ 日本語 ∙ +/−

Licensing

Hillman at the English-language Wikipedia, the copyright holder of this work, hereby publishes it under the following license:

	This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. Subject to disclaimers.
	Attribution: Hillman at the English-language Wikipedia
	You are free: to share – to copy, distribute and transmit the work to remix – to adapt the work Under the following conditions: attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
	This licensing tag was added to this file as part of the GFDL licensing update.http://creativecommons.org/licenses/by-sa/3.0/CC BY-SA 3.0Creative Commons Attribution-Share Alike 3.0truetrue

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License. Subject to disclaimers.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

Original upload log

• 2017-06-11 09:15 Stigmatella aurantiaca 320×320× (13178 bytes) Aligned A' and B'. Cleaned up the more objectionable jpeg artifacts. Strengthened the primes.
• 2017-06-09 19:16 Stigmatella aurantiaca 320×320× (27504 bytes) Trimmed excess whitespace, otherwise unchanged from the version that Chris Hillman uploaded.
• 2006-05-10 18:17 Hillman 400×400× (29544 bytes)
• 2006-05-10 18:08 Hillman 400×400× (45996 bytes) A thought experiment related to one discussed by J. S. Bell (see [[Bell's spaceship paradox]]). Two observers in [[Minkowski spacetime]] accelerate with constant magnitude <math>k</math> acceleration for proper time <math>sigma</math> (acceleration and e