- Poinsot's ellipsoid
In

classical mechanics ,**Poinsot's construction**is a geometrical method for visualizing the torque-free motion of a rotatingrigid body , that is, the motion of a rigid body on which no external forces are acting. This motion has four constants: thekinetic energy of the body and the three components of theangular momentum , expressed with respect to a laboratory frame. Theangular velocity vector $oldsymbolomega$ of therigid 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 $oldsymbolomega$. If the rigid rotor is symmetric (has two equalmoments of inertia ), the vector $oldsymbolomega$ describes a cone (and its endpoint a circle). This is the torque-freeprecession of the rotation axis of the rotor.**Kinetic energy constraint**In the absence of applied torques, the kinetic energy $T$ is conserved so $frac\{dT\}\{dt\}\; =\; 0$.

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

:$T\; =\; frac\{1\}\{2\}\; oldsymbolomega\; cdot\; mathbf\{I\}\; cdot\; oldsymbolomega\; =\; frac\{1\}\{2\}\; I\_\{1\}\; omega\_\{1\}^\{2\}\; +\; frac\{1\}\{2\}\; I\_\{2\}\; omega\_\{2\}^\{2\}\; +\; frac\{1\}\{2\}\; I\_\{3\}\; omega\_\{3\}^\{2\}$

where $omega\_\{k\}$ are the components of the

angular velocity vector $oldsymbolomega$ 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-dimensionalangular velocity vector $oldsymbolomega$; in the principal axis frame, it must lie on anellipsoid , called**inertia ellipsoid**.:$T\; =frac\{1\}\{2\}\; I\_\{1\}\; omega\_\{1\}^\{2\}\; +\; frac\{1\}\{2\}\; I\_\{2\}\; omega\_\{2\}^\{2\}\; +\; frac\{1\}\{2\}\; I\_\{3\}\; omega\_\{3\}^\{2\}$

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 $oldsymbolomega$ is called the(Greek for "pole path") and is generally circular orpolhode taco -shaped.**Angular momentum constraint**In the absence of applied torques, the angular momentum vector $mathbf\{L\}$ is conserved in an

inertial reference frame $frac\{dmathbf\{L\{dt\}\; =\; 0$.The angular momentum vector $mathbf\{L\}$ can also be expressedin terms of the moment of inertia tensor $mathbf\{I\}$ and the angular velocity vector $oldsymbolomega$

:$mathbf\{L\}\; =\; mathbf\{I\}\; cdot\; oldsymbolomega$

which leads to the equation

:$T\; =\; frac\{1\}\{2\}\; oldsymbolomega\; cdot\; mathbf\{L\}$

showing that the angular velocity vector $oldsymbolomega$has a constant component in the direction of the angular momentum vector $mathbf\{L\}$. This imposes a second constraint on the vector $oldsymbolomega$; 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 $oldsymbolomega$ on the invariable plane is called the**herpolhode**(Greek for "serpentine path"). The etymology of "herpolhode" is obscure, since most herpolhodes are simple circles, not sinusoidal; Poinsot may have been thinking of theouroboros , the symbol of endless cycling.**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 $oldsymbolomega$ equals the angular momentum vector $mathbf\{L\}$

:$frac\{dT\}\{doldsymbolomega\}\; =\; mathbf\{I\}\; cdot\; oldsymbolomega\; =\; mathbf\{L\}$

Hence, the normal vector to the kinetic-energy ellipsoid at $oldsymbolomega$ 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 $oldsymbolomega$ 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 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:$L^\{2\}\; =\; L\_\{1\}^\{2\}\; +\; L\_\{2\}^\{2\}\; +\; L\_\{3\}^\{2\}$

:$T\; =\; frac\{L\_\{1\}^\{2\{2I\_\{1\; +\; frac\{L\_\{2\}^\{2\{2I\_\{2\; +\; frac\{L\_\{3\}^\{2\{2I\_\{3$

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 anellipsoid, 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 solutionsfor $mathbf\{L\}$.This construction differs from Poinsot's construction because it considersthe angular momentum vector $mathbf\{L\}$ rather than the angular velocity vector $oldsymbolomega$. 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

**ee also***

polhode

*precession

*Principal axes

*Moment of inertia

*Tait-Bryan rotations

*Euler angles

*Wikimedia Foundation.
2010.*

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