It is shown that the conservation of potential vorticity density in geophysical fluid dynamics is a trivial law of the second kind, i.e. one for which the equations of motion are not required. Noether's first theorem, which leads to conservation laws only when the equations of motion are satisfied, is therefore irrelevant to associate the conservation of potential vorticity density with a symmetry, in particular particle-relabeling. The demonstration is provided in arbitrary coordinates and applies to comoving (or label) coordinates. Since this contradicts previous studies, a discussion on relabeling transformations is presented. Moreover, a canonical Hamiltonian formulation with weak phase-space Dirac constraints is obtained. It is shown that all Dirac constraints are of the second class, which implies that no infinite-dimensional symmetries exist and that Noether's second theorem does not apply to the governing equations. Therefore, the equations of motion do not admit a symmetry associated with the conservation of potential vorticity density via Noether's two theorems.