Symmetries and conserved quantities for systems of generalized classical mechanics

In this paper, the symmetries and the conserved quantities for systems of generalized classical mechanics are studied. First, the generalized Noether's theorem and the generalized Noether's inverse theorem of the systems are given, which are based upon the invariant properties of the canonical action with respect to the action of the infinitesimal transformation of r-parameter finite group of transformation; second, the Lie symmetries and conserved quantities of the systems are studied in accordance with the Lie's theory of the invariance of differential equations under the transformation of infinitesimal groups; and finally, the inner connection between the two kinds of symmetries of systems is discussed.