In
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, Poinsot's construction (after
Louis Poinsot
Louis Poinsot (3 January 1777 – 5 December 1859) was a French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a co ...
) is a geometrical method for visualizing the torque-free motion of a rotating
rigid body
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
, that is, the motion of a rigid body on which no external forces are acting. This motion has four constants: the
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its a ...
of the body and the three components of the
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed sy ...
, expressed with respect to an inertial laboratory frame. The
angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
vector
of the
rigid rotor
In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special ri ...
is ''not constant'', but satisfies
Euler's equations. Without explicitly solving these equations,
Louis Poinsot
Louis Poinsot (3 January 1777 – 5 December 1859) was a French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a co ...
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
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In o ...
of the rotation axis of the rotor.
Angular kinetic energy constraint
The law of
conservation of energy
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means tha ...
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
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
vector
, and the
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
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
vector
; in the principal axis frame, it must lie on the
ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...
defined by the above equation, called the inertia ellipsoid.
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
A taco (, , ) is a traditional Mexican food consisting of a small hand-sized corn- or wheat-based tortilla topped with a filling. The tortilla is then folded around the filling and eaten by hand. A taco can be made with a variety of filli ...
-shaped.
Angular momentum constraint
The law of
conservation of angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syste ...
states that in the absence of applied torques, the angular momentum vector
is conserved in an
inertial reference frame
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleratio ...
, 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).
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
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gra ...
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 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
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
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. These two surfaces intersect in two curves shaped like the edge of a
taco
A taco (, , ) is a traditional Mexican food consisting of a small hand-sized corn- or wheat-based tortilla topped with a filling. The tortilla is then folded around the filling and eaten by hand. A taco can be made with a variety of filli ...
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.
The orientation of the body in space thus has two degrees of freedom. Firstly, some point on the "taco edge" has to align with
which is a constant vector in absolute space. Secondly, with the vector in the body frame that goes through this point fixed, the body can have any amount of rotation around that vector. So in principle, the body's orientation is some point on a
toroidal 2-manifold inside the 3-manifold of all orientations. In general, the object will follow a non-periodic path on this torus, but it may follow a periodic path. The time taken for
to complete one cycle around its track in the body frame is constant, but after a cycle the body will have rotated by an amount that my not be a rational number of degrees, in which case the orientation will not be periodic, but
almost periodic.
In general a torus is almost determined by three parameters: the ratios of the second and third moments of inertia to the highest of the three moments of inertia, and the ratio
relating the angular momentum to the energy times the highest moment of inertia. But for any such a set of parameters there are two tori, because there are two "tacos" (corresponding to two polhodes). A set of 180° rotations carries any orientation of one torus into an orientation of the other with the opposite point aligned with the angular momentum vector. If the angular momentum is exactly aligned with a principal axes, the torus degenerates into a single loop. If exactly two moments of inertia are equal (a so-called symmetric body), then in addition to tori there will be an infinite number of loops, and if all three moments of inertia are equal, there will be loops but no tori. If the three moments of inertia are all different and
but the intermediate axis is not aligned with the angular momentum, then the orientation will be some point on a topological
open annulus.
Because of all this, when the angular velocity vector (or the angular momentum vector) is not close to the axis of highest or lowest inertia, the body "tumbles". Most moons rotate more or less around their axis of greatest inertia (due to viscous effects), but
Hyperion
Hyperion may refer to:
Greek mythology
* Hyperion (Titan), one of the twelve Titans
* ''Hyperion'', a byname of the Sun, Helios
* Hyperion of Troy or Yperion, son of King Priam
Science
* Hyperion (moon), a moon of the planet Saturn
* ''Hyp ...
(a moon of Saturn), two
moons of Pluto and
many other small bodies of the Solar System have tumbling rotations.
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. If the alignment with the intermediate axis is not perfect then
will eventually move off this point along one of the four tracks that depart from this point, and head to the opposite point. This corresponds to
moving to its antipode 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 case
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
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has c ...
(the shape of an American football), or rotation of an
oblate spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has ...
(the shape of a flattened sphere). 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
In astronomy, axial precession is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's rotational axis. In the absence of precession, the astronomical body's orbit would show axial parallelism. In particu ...
of the rotation of a planet (the case of an oblate spheroid.)
Applications
One of the applications of Poinsot's construction is in visualizing the rotation of a spacecraft in orbit.
[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.]
See also
*
Polhode
*
Precession
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In o ...
*
Principal axes
*
Rigid body dynamics
*
Moment of inertia
The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...
*
Tait–Bryan rotations
*
Euler angles
*
MacCullagh ellipsoid
*
Rattleback
A rattleback is a semi-ellipsoidal top which will rotate on its axis in a preferred direction. If spun in the opposite direction, it becomes unstable, "rattles" to a stop and reverses its spin to the preferred direction.
This spin-reversal appea ...
References
Sources
* Poinsot (1834) ''Theorie Nouvelle de la Rotation des Corps'', Bachelier, Paris.
* Landau LD and Lifshitz EM (1976) ''Mechanics'', 3rd. ed., Pergamon Press. (hardcover) and (softcover).
* Goldstein H. (1980) ''Classical Mechanics'', 2nd. ed., Addison-Wesley.
* Symon KR. (1971) ''Mechanics'', 3rd. ed., Addison-Wesley. {{ISBN, 0-201-07392-7
External links
Poinsot construction in stereo 3D simulation - online and free
Rigid bodies