  When numerically integrating canonical Hamiltonian systems, the long-term
conservation of some of its invariants, among which the Hamiltonian function
itself, assumes a central role. The classical approach to this problem has led
to the definition of symplectic methods, among which we mention Gauss-Legendre
collocation formulae. Indeed, in the continuous setting, energy conservation is
derived from symplecticity via an infinite number of infinitesimal contact
transformations. However, this infinite process cannot be directly transferred
to the discrete setting. By following a different approach, in this paper we
describe a sequence of methods, sharing the same essential spectrum (and, then,
the same essential properties), which are energy preserving starting from a
certain element of the sequence on, i.e., after a finite number of steps.
