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.

Angular kinetic energy constraint

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.

Angular momentum constraint

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]

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

Zdroj:https://en.wikipedia.org?pojem=Tumbling_(rigid_body)
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