Polhode

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.

The concept of polhode motion dates back to the 17th century, and Corollary 21 to Proposition 66 in Section 11, Book 1, of Isaac Newton's Principia. Later Leonhard Euler 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 months.

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 Léon 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 (pivot or end of an axis) + (path or way)—thus, polhode is the path of the pole.

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