  An iterative Noether scheme, advocated by Deser, is used to introduce gauge
invariant couplings to nonrelativistic matter with global symmetries related to
usual charge conservation and dipole conservation recently discussed in
fractonic theories. No reference to any gauge principle, fractonic or
otherwise, is required. A dual description is found where the theory is defined
either in terms of the usual vector gauge field or, alternatively, in terms of
higher derivatives of a symmetric tensor field. A connection between these two
descriptions is obtained by providing an explicit map between the vector and
tensor fields. This method yields a novel `minimal' prescription for the tensor
fields which is identified with the usual minimal prescription for the vector
fields by using the mapping. It also spells out the structure of the pure gauge
field action in both formulations. Extension of the abelian $U(1)$ invariance
to the nonabelian $SU(N)$ invariance is done for the standard formulation.
