On the representatation of orientation space

TLDR

The study investigates new variable sets aimed at improving numerical stability and efficiency in computer simulations of rigid classical molecules. The authors identify variables that yield the simplest Euclidean expressions for the orientation space metric tensor. They show that the Lattman metric is a scalar multiple of the Jacobi metric, that Euler’s quaternion parameters produce a Euclidean orientation metric, and that these simplifications eliminate singularities and improve accuracy and efficiency in numerical and Monte Carlo simulations of rigid bodies.

Abstract

New sets of variables are studied which should lead to significantly improved numerical stability and efficiency for computer stimulation studies of rigid classical molecules. The search for these new variables is made by finding variables which lead to the simplest (i.e. euclidean) expressions for the metric tensor of orientation space. It is shown that the intuitively defined Lattman metric [1] is a scalar multiple of the metric tensor which arises naturally from the Jacobi formulation of the action principle for spherical tops. It is then shown that Euler's quaternion parameters lead to a euclidean form for the orientation metric. These parameters lead to many associated simplifications in the equations of motion of classical rigid bodies including the removal of singularities and spurious behaviour near θ = 0. It is felt that these benefits will translate into increased accuracy and efficiency both for numerical integration of the equations of motion and for performing Monte Carlo integrations of phase space.