  In this paper, we calculate H\"ormander index in the finite-dimensional case.
Then we use the result to give some iteration inequalities, and prove almost
existence of mean indices for given complete autonomous Hamiltonian system on
compact symplectic manifold with symplectic trivial tangent bundle and given
autonomous Hamiltonian system on regular compact energy hypersurface of
symplectic manifold with symplectic trivial tangent bundle.
