  The modal analysis is revisited through the symplectic formalism, what leads
to two intertwined eigenproblems. Studying the properties of the solutions, we
prove that they form a canonical basis. The method is general and works even if
the Hamiltonian is not the sum of the potential and kinetic energies. On this
ground, we want to address the following problem: data being given in the form
of one or more structural evolutions, we want to construct an approximation of
the Hamiltonian from a covariant snapshot matrix and to perform a symplectic
decomposition. We prove the convergence properties of the method when the time
discretization is refined. If the data cloud is not enough rich, we extract the
principal component of the Hamiltonian corresponding to the leading modes,
allowing to perform a model order reduction for very high dimension models. The
method is illustrated by a numerical example.
