Year: 2026
Type: preprint
Abstract: This paper presents a self-contained, vector-only formulation of gyroscope precession about a fixed pivot -- gravity-driven, not the torque-free Euler-Poinsot case. The derivation of its governing equation uses vector integrals and cross products of intrinsic vectors, with no component decomposition; no tensor or dyadic formalism, matrix or Euler angle enters. Inertia about the center of mass enters through two mass integrals alone: the polar second moment of mass $J_0$ (isotropic) and the vector operator $\vec F$ (the rest, including all non-isotropic inertia). Using material time derivatives referenced to the inertial observer, the torque about the pivot is assembled once from the complete acceleration of each mass element, giving an exact, unified vector equation for steady precession. The balance of the gravity torque against this inertia torque reproduces the classical steady-precession relation of the heavy symmetric top exactly at non-vertical tilts, on both the slow and fast branches, with the horizontal-shaft case exact rather than approximate. Closed forms are given for the disk, sphere, and circular and rectangular toruses. The constant tilt angle of dynamic equilibrium is explained as torque equalization: the two torques about the pivot balance exactly at that tilt and unequally on either side of it, treated throughout as a geometric function of the tilt under constant precession and self-spin rates, and never as a function of time. Informationally equivalent to the classical description, the work contributes, in order, an interpretation (the classical balance read in torque vectors throughout) and the vector method as its tool. The exact relation predates this work: Klein and Sommerfeld stated it in 1897, Brand in 1930. Begun by the author in 1975 and re-verified symbolically, the work is intended for undergraduate readers, with every step of the derivation stated.
Source: arXiv (Cornell University)
Author: Hariton, Dan I.
Language: English
Subfield: Control and Systems Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208773385