Symmetry Properties of Hamiltonian Dynamical System

We derive generalized first Noether's theorem for regular and singular Lagrangian system respectively, the dynamics of which is described by Hamilton's canonical variables, derive generalized Noeth-er's identities for gauge-variant system in phase space, and deduce the strong and weak conservation laws for constrained Hamiltonian system in phase space. If Dirac conjecture is valid, along the trajectory of motion of constrained system, we obtain some relationship of Lagrangian multipliers in connection with secondary first class constraints. A preliminary application to Yang-Mill fields was given. The consistancy conditions of constraints can not be deduced from the properties of the gauge invariance, such consistency conditions do not hold in the strong validity sense. Dirac-Beramann's algorithm is consistent in the sense of weak validity.