  The present paper is the first of two articles aimed at constructing
$n$-degree-of-freedom Hamiltonian systems by an algebraic approach. In
molecular spectroscopy, the construction of vibrational Hamiltonian for
strongly excited molecular systems by using an algebraic formalism requires the
introduction by hand the operators describing the change in energy by numerous
quanta and it is tedious to predict in advance the total number of operators
appearing in the development. The goal of the two articles is not only to
propose in the local limit a systematic method for constructing a normalized
vibrational Hamiltonian for a strongly excited $n$-degree-of-freedom molecular
system from the generators of the Lie algebra, the algebra of polynomial
invariants, but also to enumerate the number of independent operators needed
for the construction of the Hamiltonian developed in the base of these
generators up to the given order $N$. The first article introduces the
theoretical tools used in the both papers (section \ref{norm}), and presents
the method of construction in case of absence of resonance (section
\ref{const}). Finally, an application for a triatomic non-linear ClOH molecule
is considered in case close to the dissociation limit. (section \ref{Appli}).
