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This paper studies the exact invariants and the adiabatic invariants for systems of generalized classical mechanics in the high-dimensional extended phase space and gives the relations between the invariants and the symmetries of the system s in the space. Based on the concept of high-order adiabatic invariant of mechan ical systems with the action of small disturbance, it presents the form of the h igh-order adiabatic invariant and the conditions for their existence,and establi shes the relationship between adiabatic invariant and symmetrical transformation . In the end of this paper, an example is given to illustrate the application of the results.

By means of the invariance of Hamilton’s action under infinitesimal transformation in the high-dimensional extended phase space, the symmetry and invariant of the generalized holonomic conservative and nonconservative mechanical systems and some related propositions are obtained. Finally one example is given.