Abstract: The important notions and results of the integral invariants of Poincaré and Cartan-Poincaré and the relationship between integral invariant and invariant form established first by E. Cartan in the classical mechanics are generalized to Hamilton mechanics on Kähler manifold, by the theory of modern geometry and advanced calculus, to get the corresponding wider and deeper results.

Key words: Kä, hler manifold, symplectic manifold, invariant form, integral invariant, vector field, form field, Lie derivative, exterior differential