New conserved quantities of Noether--Mei symmetry of mechanical system in phase space

This paper studies two new types of conserved quantities deduced by Noether--Mei symmetry of mechanical system in phase space. The definition and criterion of Noether--Mei symmetry for the system are given. A coordination function is introduced, and the conditions under which the Noether--Mei symmetry leads to the two types of conserved quantities and the forms of the two types of conserved quantities are obtained. An illustrative example is given. The coordination function can be selected according to the demand for finding the gauge function, and the choice of the coordination function has multiformity, so more conserved quantities deduced from Noether--Mei symmetry of mechanical system can be obtained.