Unified Symmetry For The Mechanical Systems In Phase Space

The researches on the symmetry theory and the conservation laws of mechanical systems have important theoretical and practical significance. The modern methods to find conserved quantities are mainly the followings: Noether symmetry, Lie symmetry and Mei symmetry. Unified symmetry is a new symmetry. In the paper, we study unified symmetry in phase space. First, we study the unified symmetry of holonomic and nonholonomic system in phase space, give the criterion equation of unifed symmetry and get the Noether conserved quantity, Hojman conserved quantity and Mei conserved quantity of the two types of systems; Second, we study the unified symmetry of holonomic and nonholonomic system with variable mass in phase space, give the criterion equation of unifed symmetry and get the Noether conserved quantity, Hojman conserved quantity and Mei conserved quantity of the two types of systems; Finally, we study the perturbation to symmetry and exact and adiabatic invariants, give the determining equation of unified symmetry after being perturbed of Hamilton system and the perturbation to Lie symmetry of generalized Raitzin canonical equation of motion, get the Noether and Hojman adiabatic invariants of the two types of systems respectively.