Cauchy's invariants revisited. An alternative variational proof

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.