Nonholonomic mapping theory and geometric formulation for rotation of a rigid body with one fixed point

Abstract

The method of nonholonomic mapping is adopted to construct a Riemann-Cartan space embedded in a known Riemann space. As a special case, Weitzenbock space is embedded in an Euclidean pace. By means of the nonholonomic mapping and d’Alembert-Lagrange principle a geodesic in a Riemann space is mapped to an autoparallel in a Riemann-Cartan space. The mapping theory is applied to the problem of rotation of a rigid body with a fixed point. It is proved that Euler equations for the rigid body are equations of geodesic in the Riemann configuration space described by Euler angles, whereas the equations in the pseudo-coordinate space corresponding to angular velocities of the rigid body are equations of autoparallel in the Riemann-Cartan space with constant torsion.