Solution to Euler’s Gyrodynamics—I

Abstract

A little used parameterization of the three-dimensional rotation group is taken as basis in deriving an easily integrable kinematic relation (a 4-vector linear differential equation) for the attitude rate, in terms of the present attitude and angular velocity of one reference frame relative to another. If the angular velocity is known and well behaved one obtains the exact solution from an iteration procedure explained in detail. The formal solution to a large class of rigid-body problems is thus implied; a particular one being that of an axially symmetric rocket with variable thrust vector and constant moment-of-inertia tensor which as such generalizes Jacobi’s torque-free case.