Conservation laws of classical mechanics in nonholonomic coordinates

Noether's theorem and inverse theorem for mechanical systems with gauge-invariant Lagrangians under symmetric infinitesimal transformations and whose motion is described by quasi-coordinates are established. The existence of first integrals is shown to depend on the existence of solutions to the Killing equations. Nonholonomic mechanical systems are analyzed, and their special properties are pointed out. The transformation which corresponds to the Kowalevski's first integral in rigid body dynamics is found. The nature of the energy integral in nonholonomic mechanical systems is shown, and a few new first integrals for nonconservative problems are obtained.