  The purpose of this paper is two-fold. First, to provide a straightforward
proof of the Cauchy's invariants (CIs) from the particle relabeling symmetry of
the action functional for rotational barotropic flows, using pure geometric
relabeling. Second, to emphasize the central role of CIs in the study of
rotational flows. The first goal is achieved by introducing a new variational
approach for derivivg invariants from symmetries, which, when applied to the
Lagrangian action of rotational flows, leads to a simple and natural proof of
CIs. The second goal is clarified by showing that all important conservation
theorems of rotational flows, either local or integral, are simple consequences
of CIs.
