  In this paper, we prove that the two well-known natural normalizations of
Hamiltonian functions on the symplectic manifold $(M,\omega)$ canonically
relates the action spectra of different normalized Hamiltonians on {\it
arbitrary} symplectic manifolds $(M,\omega)$. The natural class of normalized
Hamiltonians consists of those whose mean value is zero for the closed
manifold, and those which are compactly supported in $\text{Int} M$ for the
open manifold. We also study the effect of the action spectrum under the
$\pi_1$ of Hamiltonian diffeomorphism group. This forms a foundational basis
for our study of spectral invariants of the Hamiltonian diffeomorphism in
[Oh4].
