  The problem of whether or not the equations of motion of a quantum system
determine the commutation relations was posed by E.P.Wigner in 1950. A similar
problem (known as "The Inverse Problem in the Calculus of Variations") was
posed in a classical setting as back as in 1887 by H.Helmoltz and has received
great attention also in recent times. The aim of this paper is to discuss how
these two apparently unrelated problems can actually be discussed in a somewhat
unified framework. After reviewing briefly the Inverse Problem and the
existence of alternative structures for classical systems, we discuss the
geometric structures that are intrinsically present in Quantum Mechanics,
starting from finite-level systems and then moving to a more general setting by
using the Weyl-Wigner approach, showing how this approach can accomodate in an
almost natural way the existence of alternative structures in Quantum Mechanics
as well.
