Polhode
Encyclopedia
The details of a spinning body may impose restrictions on the motion of its angular velocity
vector, ω. The curve produced by the angular velocity vector on the inertia ellipsoid, is known as the polhode, coined from Greek meaning "path of the pole". The surface created by the angular velocity vector is termed the body cone.
derived a set of equations
that described the dynamics of rigid bodies
in torque-free motion. In particular, Euler and his contemporaries Jean d’Alembert
, Louis Lagrange
, and others noticed small variations in latitude
due to wobbling of the Earth
around its polar
spin axis
. A portion of this wobble (later to be called the Earth’s polhode motion) was due to the natural, torque
-free behavior of the rotating Earth. Incorrectly assuming that the Earth was a completely rigid body
, they calculated the period
of Earth’s polhode wobble to be about 9–10 month
s.
During the mid 19th century, Louis Poinsot
developed a geometric interpretation
of the physics of rotating bodies that provided a visual counterpart to Euler’s algebraic equations. Poinsot was a contemporary of Leon Foucault
, who invented the gyroscope
and whose pendulum experiments
provided incontrovertible evidence the Earth rotates. In the fashion of the day, Poinsot coined the terms polhode and its counterpart, herpolhode
, to describe this wobble in the motion of rotating rigid bodies. Poinsot derived these terms from the ancient Greek
πόλος (pivot or end of an axis) + ὁδός (path or way)—thus, polhode is the path
of the pole
.
Poinsot’s geometric interpretation of Earth’s polhode motion was still based on the incorrect assumption that the Earth was a completely rigid rotating body. It was not until 1891 that the American astronomer, Seth Carlo Chandler
, made measurement
s showing that there was a periodic motion of 14 months in the Earth’s wobble and suggesting that this was the polhode motion. Initially, Chandler’s measurement, now referred to as the “Chandler wobble
,” was dismissed because it was significantly greater than the long-accepted 9–10 month period calculated by Euler, Poinsot, et al. and because Chandler was unable convincingly to explain this discrepancy. However, within months, another American astronomer, Simon Newcomb
, realized that Chandler was correct and provided a plausible reason for Chandler’s measurements. Newcomb realized that the Earth’s mass
is partly rigid and partly elastic
, and that the elastic component has no effect on the Earth’s polhode period, because the elastic part of the Earth’s mass stretches so that it is always symmetrical
about the Earth’s spin axis. The rigid part of the Earth’s mass is not symmetrically distributed, and this is what causes the Chandler Wobble, or more precisely, the Earth’s polhode path.
body inherently has three principal axes through its center of mass
, and each of these axes has a corresponding moment of inertia
. The moment of inertia about an axis is a measurement of how difficult it is to accelerate
the body about that axis. The closer the concentration of mass to the axis, the smaller the torque required to get it spinning at the same rate
about that axis.
The moment of inertia of a body depends on the mass distribution of the body and on the arbitrarily selected axis about which the moment of inertia is defined. The moments of inertia about two of the principal axes are the maximum and minimum moments of inertia of the body about any axis. The third is perpendicular to the other two and has a moment of inertia somewhere between the maximum and minimum.
If energy
is dissipated while an object is rotating, this will cause the polhode motion about the axis of maximum inertia (also called the major principal axis) to damp out
or stabilize, with the polhode path becoming a smaller and smaller ellipse
or circle
, closing in
on the axis.
A body is never stable
when spinning about the intermediate principal axis, and dissipated energy will cause the polhode to start migrating to the object’s axis of maximum inertia
. The transition point between two stable axes of rotation is called the separatrix along which the angular velocity passes through the axis of intermediate inertia.
Rotation about the axis of minimum inertia (also called the minor principal axis) is also stable, but given enough time, any perturbations due to energy dissipation or torques would cause the polhode path to expand, in larger and larger ellipses or circles, and eventually migrate through the separatrix and its axis of intermediate inertia to its axis of maximum inertia.
It is important to note that these changes in the orientation
of the body as it spins may not be due to external torques, but rather result from energy dissipated
internally as the body is spinning. Even if angular momentum is conserved
(no external torques), internal energy can be dissipated during rotation if the body is not perfectly rigid, and any rotating body will continue to change its orientation until it has stabilized around its axis of maximum inertia, where the amount of energy corresponding to its angular momentum
is least.
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
vector, ω. The curve produced by the angular velocity vector on the inertia ellipsoid, is known as the polhode, coined from Greek meaning "path of the pole". The surface created by the angular velocity vector is termed the body cone.
History
The concept of polhode motion dates back to the 18th century, when Leonhard EulerLeonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
derived a set of equations
Newton-Euler equations
The Newton–Euler equations describe the combined translational and rotational dynamics of a rigid body.With respect to a coordinate frame whose origin coincides with the body's center of mass, they can be expressed in matrix form as:...
that described the dynamics of rigid bodies
Rigid body dynamics
In physics, rigid body dynamics is the study of the motion of rigid bodies. Unlike particles, which move only in three degrees of freedom , rigid bodies occupy space and have geometrical properties, such as a center of mass, moments of inertia, etc., that characterize motion in six degrees of...
in torque-free motion. In particular, Euler and his contemporaries Jean d’Alembert
Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert was a French mathematician, mechanician, physicist, philosopher, and music theorist. He was also co-editor with Denis Diderot of the Encyclopédie...
, Louis Lagrange
Joseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
, and others noticed small variations in latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
due to wobbling of the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
around its polar
Geographical pole
A geographical pole is either of the two points—the north pole and the south pole—on the surface of a rotating planet where the axis of rotation meets the surface of the body...
spin axis
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
. A portion of this wobble (later to be called the Earth’s polhode motion) was due to the natural, torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
-free behavior of the rotating Earth. Incorrectly assuming that the Earth was a completely rigid body
Rigid body
In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...
, they calculated the period
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
of Earth’s polhode wobble to be about 9–10 month
Month
A month is a unit of time, used with calendars, which was first used and invented in Mesopotamia, as a natural period related to the motion of the Moon; month and Moon are cognates. The traditional concept arose with the cycle of moon phases; such months are synodic months and last approximately...
s.
During the mid 19th century, Louis Poinsot
Louis Poinsot
Louis Poinsot 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 couple.-Life:...
developed a geometric interpretation
Poinsot's construction
In classical mechanics, Poinsot's construction 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 physics of rotating bodies that provided a visual counterpart to Euler’s algebraic equations. Poinsot was a contemporary of Leon Foucault
Léon Foucault
Jean Bernard Léon Foucault was a French physicist best known for the invention of the Foucault pendulum, a device demonstrating the effect of the Earth's rotation...
, who invented the gyroscope
Gyroscope
A gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum. In essence, a mechanical gyroscope is a spinning wheel or disk whose axle is free to take any orientation...
and whose pendulum experiments
Foucault pendulum
The Foucault pendulum , or Foucault's pendulum, named after the French physicist Léon Foucault, is a simple device conceived as an experiment to demonstrate the rotation of the Earth. While it had long been known that the Earth rotated, the introduction of the Foucault pendulum in 1851 was the...
provided incontrovertible evidence the Earth rotates. In the fashion of the day, Poinsot coined the terms polhode and its counterpart, herpolhode
Herpolhode
A herpolhode is the curve traced out by the endpoint of the angular velocity vector ω of a rigid rotor, a rotating rigid body. The endpoint of the angular velocity moves in a plane in absolute space, called the invariable plane, that is orthogonal to the angular momentum vector L...
, to describe this wobble in the motion of rotating rigid bodies. Poinsot derived these terms from the ancient Greek
Ancient Greek
Ancient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...
πόλος (pivot or end of an axis) + ὁδός (path or way)—thus, polhode is the path
Path (topology)
In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...
of the pole
Geographical pole
A geographical pole is either of the two points—the north pole and the south pole—on the surface of a rotating planet where the axis of rotation meets the surface of the body...
.
Poinsot’s geometric interpretation of Earth’s polhode motion was still based on the incorrect assumption that the Earth was a completely rigid rotating body. It was not until 1891 that the American astronomer, Seth Carlo Chandler
Seth Carlo Chandler
Seth Carlo Chandler, Jr. was an American astronomer.He was born in Boston, Massachusetts. During his last year in high school he performed mathematical computations for Benjamin Peirce, of the Harvard College Observatory.After graduating, he became the assistant of Benjamin A. Gould...
, made measurement
Measurement
Measurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...
s showing that there was a periodic motion of 14 months in the Earth’s wobble and suggesting that this was the polhode motion. Initially, Chandler’s measurement, now referred to as the “Chandler wobble
Chandler wobble
The Chandler wobble is a small motion in the Earth's axis of rotation relative to the Earth's surface, which was discovered by American astronomer Seth Carlo Chandler in 1891. It amounts to on the Earth's surface and has a period of 433 days...
,” was dismissed because it was significantly greater than the long-accepted 9–10 month period calculated by Euler, Poinsot, et al. and because Chandler was unable convincingly to explain this discrepancy. However, within months, another American astronomer, Simon Newcomb
Simon Newcomb
Simon Newcomb was a Canadian-American astronomer and mathematician. Though he had little conventional schooling, he made important contributions to timekeeping as well as writing on economics and statistics and authoring a science fiction novel.-Early life:Simon Newcomb was born in the town of...
, realized that Chandler was correct and provided a plausible reason for Chandler’s measurements. Newcomb realized that the Earth’s mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
is partly rigid and partly elastic
Elasticity (physics)
In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....
, and that the elastic component has no effect on the Earth’s polhode period, because the elastic part of the Earth’s mass stretches so that it is always symmetrical
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
about the Earth’s spin axis. The rigid part of the Earth’s mass is not symmetrically distributed, and this is what causes the Chandler Wobble, or more precisely, the Earth’s polhode path.
Description
Every solidSolid
Solid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
body inherently has three principal axes through its center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
, and each of these axes has a corresponding moment of inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...
. The moment of inertia about an axis is a measurement of how difficult it is to accelerate
Angular acceleration
Angular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared , and is usually denoted by the Greek letter alpha .- Mathematical definition :...
the body about that axis. The closer the concentration of mass to the axis, the smaller the torque required to get it spinning at the same rate
Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
about that axis.
The moment of inertia of a body depends on the mass distribution of the body and on the arbitrarily selected axis about which the moment of inertia is defined. The moments of inertia about two of the principal axes are the maximum and minimum moments of inertia of the body about any axis. The third is perpendicular to the other two and has a moment of inertia somewhere between the maximum and minimum.
If energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
is dissipated while an object is rotating, this will cause the polhode motion about the axis of maximum inertia (also called the major principal axis) to damp out
Damping
In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...
or stabilize, with the polhode path becoming a smaller and smaller ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
or circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
, closing in
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
on the axis.
A body is never stable
Directional stability
Directional stability is stability of a moving body or vehicle about an axis which is perpendicular to its direction of motion. Stability of a vehicle concerns itself with the tendency of a vehicle to return to its original direction in relation to the oncoming medium when disturbed away from...
when spinning about the intermediate principal axis, and dissipated energy will cause the polhode to start migrating to the object’s axis of maximum inertia
Inertia
Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...
. The transition point between two stable axes of rotation is called the separatrix along which the angular velocity passes through the axis of intermediate inertia.
Rotation about the axis of minimum inertia (also called the minor principal axis) is also stable, but given enough time, any perturbations due to energy dissipation or torques would cause the polhode path to expand, in larger and larger ellipses or circles, and eventually migrate through the separatrix and its axis of intermediate inertia to its axis of maximum inertia.
It is important to note that these changes in the orientation
Orientation (rigid body)
In geometry the orientation, angular position, or attitude of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it is in....
of the body as it spins may not be due to external torques, but rather result from energy dissipated
Dissipation
In physics, dissipation embodies the concept of a dynamical system where important mechanical models, such as waves or oscillations, lose energy over time, typically from friction or turbulence. The lost energy converts into heat, which raises the temperature of the system. Such systems are called...
internally as the body is spinning. Even if angular momentum is conserved
Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
(no external torques), internal energy can be dissipated during rotation if the body is not perfectly rigid, and any rotating body will continue to change its orientation until it has stabilized around its axis of maximum inertia, where the amount of energy corresponding to its angular momentum
Rotational energy
The rotational energy or angular kinetic energy is the kinetic energy due to the rotation of an object and is part of its total kinetic energy...
is least.