  The inverse problem of calculus of variations and $s$-equivalence are
re-examined by using results obtained from non-commutative geometry ideas. The
role played by the structure of the modified Poisson brackets is discussed in a
general context and it is argued that classical $s$-equivalent systems may be
non-equivalent at the quantum mechanical level. This last fact is explicitly
discussed comparing different approaches to deal with the Nair-Polychronakos
oscillator.
