New non-Noether conserved quantities of mechanical system in phase space

This paper focuses on studying non-Noether conserved quantities of Liesymmetry and of form invariance for a mechanical system in phase spaceunder the general infinitesimal transformation of groups. We obtain a newnon-Noether conserved quantity of Lie symmetry of the system, and Hojman andMei's results are of special cases of our conclusion. We find acondition under which the form invariance of the system will lead to a Liesymmetry, and, further, obtain a new non-Noether conserved quantity of forminvariance of the system. An example is given finally to illustrate theseresults.