HAMILTONIAN AND LAGRANGIAN FORMULATIONS OF RIGID-BODY ROTATIONAL-DYNAMICS BASED ON THE EULER PARAMETERS

The orientation of an arbitrary rigid body is specified in terms of th e set of four Euler parameters. A corresponding set of four generalize d angular momentum variables is derived and then used to replace the u sual three-component angular velocity vector in specifying the rate of change of orientation of the body with respect to an inertial frame. Through the use of Euler's equations, this leads naturally to a formul ation of rigid body rotational dynamics in terms of a system of eight first-order coupled differential equations involving the four Euler pa rameters and the four conjugate momenta. These equations, expressed in matrix form, are recognizable as Hamilton's canonical equations, and exhibit a remarkable symmetry. A set of four second-order coupled diff erential equations for the Euler parameters is also derived, and the e quations are recognizable as Lagrange's equations. The sets of differe ntial equations are solved numerically for a representative case of to rque-free motion and the results are presented graphically.