NONINTEGRABLE VELOCITIES AND NONHOLONOMIC COORDINATES

Euler's equations of rigid body motion for the Cartesian rotation rate s omega(i) are normally derived directly from Newton's second law rath er than from a Lagrangian formulation. The reason is that a set of f i ndependent velocities omega(i) that are defined by linear transformati ons on time rates of change of group parameters is generally nonintegr able and therefore cannot be integrated to yield a set of f generalize d coordinates. We analyze and answer the following related question: w hen 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 the ory paves the way toward a more qualitatively complete understanding o f the idea of integrability of a Hamiltonian dynamical system.