Lie symmetries and conserved quantities of the constraint mechanical systems on time scales

We introduce a new method to study Lie symmetries and conserved quantities of constraint mechanical systems which include Lagrangian systems, nonconservative systems and nonholonomic systems on time scales T. For the constraint mechanical systems on time scales, based on the transformation Lie group, we get a series of significant results including the variational principle of systems on time scales, the equations of motion, the determining equations, the structure equations, the restriction equations as well as the Lie theorems of the Lie symmetries of the systems on time scales. Furthermore, a set of new conserved quantities of the constraint mechanical systems on time scales are given. More significant is that this work unifies the theories of Lie symmetries of the two cases for the continuous and the discrete constraint mechanical systems by applying the time scales. And then taking the discrete ( T=ℤ) nonholonomic system for example, we derive the corresponding discrete Lie symmetry theory. Finally, two examples are designed to illustrate these results.