NONINTEGRABLE VELOCITIES AND NONHOLONOMIC COORDINATES

Euler's equations of rigid body motion for the Cartesian rotation rates omega(i) are normally derived directly from Newton's second law rather than from a Lagrangian formulation. The reason is that a set of f independent velocities omega(i) that are defined by linear transformations on time rates of change of group parameters is generally nonintegrable and therefore cannot be integrated to yield a set of f generalized coordinates. We analyze and answer the following related question: when can a particular parameterization of a continuous group be used as a set of generalized coordinates? An understanding of the distinction between holonomic and nonholonomic coordinates via elementary Lie theory paves the way toward a more qualitatively complete understanding of the idea of integrability of a Hamiltonian dynamical system.