  We prove that, given a time-independent Lagrangian defined in the first
tangent bundle of configuration space, every infinitesimal Noether symmetry
that is defined in the $n$-tangent bundle and is not vanishing on-shell, can be
written as a canonical symmetry in an enlarged phase space, up to constraints
that vanish on-shell. The proof is performed by the implementation of a change
of variables from the the $n$-tangent bundle of the Lagrangian theory to an
extension of the Hamiltonian formalism which is particularly suited for the
case when the Lagrangian is singular. This result proves the assertion that any
Noether symmetry can be canonically realized in an enlarged phase space. Then
we work out the regular case as a particular application of this ideas and
rederive the Noether identities in this framework. Finally we present an
example to illustrate our results.
