Tennis racket theorem
The tennis racket theorem or intermediate axis theorem, is a kinetic phenomenon of classical mechanics which describes the movement of a rigid body with three distinct principal moments of inertia.
It has also been dubbed the Dzhanibekov effect, after Soviet cosmonaut Vladimir Dzhanibekov, who noticed one of the theorem's logical consequences whilst in space in 1985.
[1] The effect was known for at least 150 years prior, having been described by Louis Poinsot in 1834[2][3] and included in standard physics textbooks such as Classical Mechanics by Herbert Goldstein throughout the 20th century.
The theorem describes the following effect: rotation of an object around its first and third principal axes is stable, whereas rotation around its second principal axis (or intermediate axis) is not.
This can be demonstrated by the following experiment: Hold a tennis racket at its handle, with its face being horizontal, and throw it in the air such that it performs a full rotation around its horizontal axis perpendicular to the handle (ê2 in the diagram), and then catch the handle.
By contrast, it is easy to throw the racket so that it will rotate around the handle axis (ê1) without accompanying half-rotation around another axis; it is also possible to make it rotate around the vertical axis perpendicular to the handle (ê3) without any accompanying half-rotation.
The experiment can be performed with any object that has three different moments of inertia, for instance with a (rectangular) book, remote control, or smartphone.
[4] The tennis racket theorem can be qualitatively analysed with the help of Euler's equations.
denote the object's principal moments of inertia, and we assume
The angular velocities around the object's three principal axes are
Consider the situation when the object is rotating around the axis with moment of inertia
To determine the nature of equilibrium, assume small initial angular velocities along the other two axes.
is being opposed and so rotation around this axis is stable for the object.
Similar reasoning gives that rotation around the axis with moment of inertia
Now apply the same analysis to the axis with moment of inertia
is not opposed (and therefore will grow) and so rotation around the second axis is unstable.
In particular, the motion of the body in free space (obtained by integrating
on the fixed ellipsoid of constant squared angular momentum.
, then the angular momentum ellipsoid's major axes are in ratios of
Thus the angular momentum ellipsoid is both flatter and sharper, as visible in the animation.
We can see that the curves evolve as follows: The tennis racket effect occurs when
An observer watching the body's motion in free space would see its angular momentum vector
viewed in the rigid body's reference frame are also mostly in the same direction.
When the body is not exactly rigid, but can flex and bend or contain liquid that sloshes around, it can dissipate energy through its internal degrees of freedom.
In this case, the body still has constant angular momentum, but its energy would decrease, until it reaches the minimal point.
As analyzed geometrically above, this happens when the body's angular velocity is exactly aligned with its axis of maximal moment of inertia.
This happened to Explorer 1, the first satellite launched by the United States in 1958.
The elongated body of the spacecraft had been designed to spin about its long (least-inertia) axis but refused to do so, and instead started precessing due to energy dissipation from flexible structural elements.
In general, celestial bodies large or small would converge to a constant rotation around its axis of maximal moment of inertia.
Whenever a celestial body is found in a complex rotational state, it is either due to a recent impact or tidal interaction, or is a fragment of a recently disrupted progenitor.
