  We analyze the relation between the concept of auxiliary variables and the
Inverse problem of the calculus of variations to construct a Lagrangian from a
given set of equations of motion. The problem of the construction of a
consistent second order dynamics from a given first order dynamics is
investigated. At the level of equations of motion we find that this reduction
process is consistent provided that the mapping of the boundary data be taken
properly into account. At the level of the variational principle we analyze the
obstructions to construct a second order Lagrangian from a first order one and
give an explicit formal non-local Lagrangian that reproduce the second order
projected dynamics. Finally we apply our ideas to the so called
``Noncommutative classical dynamics''.
