  In classical mechanics, we can describe the dynamics of a given system using
either the Lagrangian formalism or the Hamiltonian formalism, the choice of
either one being determined by whether one wants to deal with a second degree
differential equation or a pair of first degree ones. For the former approach,
we know that the Euler-Lagrange equation of motion remains invariant under
additive total derivative with respect to time of any function of coordinates
and time in the Lagrangian function, whereas the latter one is invariant under
canonical transformations. In this short paper we address the question whether
the transformation that leaves the Euler-Lagrange equation of motion invariant
is also a canonical transformation and show that it is not.
