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The present paper is the first of two articles aimed at constructing$n$-degree-of-freedom Hamiltonian systems by an algebraic approach. Inmolecular spectroscopy, the construction of vibrational Hamiltonian forstrongly excited molecular systems by using an algebraic formalism requires theintroduction by hand the operators describing the change in energy by numerousquanta and it is tedious to predict in advance the total number of operatorsappearing in the development. The goal of the two articles is not only topropose in the local limit a systematic method for constructing a normalizedvibrational Hamiltonian for a strongly excited $n$-degree-of-freedom molecularsystem from the generators of the Lie algebra, the algebra of polynomialinvariants, but also to enumerate the number of independent operators neededfor the construction of the Hamiltonian developed in the base of thesegenerators up to the given order $N$. The first article introduces thetheoretical tools used in the both papers (section \ref{norm}), and presentsthe method of construction in case of absence of resonance (section\ref{const}). Finally, an application for a triatomic non-linear ClOH moleculeis considered in case close to the dissociation limit. (section \ref{Appli}).

Construction and counting of the number operators of an $n$-degree-of-freedom normalized non-resonant vibrational Hamiltonian