Cauchy's invariants revisited. An alternative variational proof

TL;DR

This paper offers a clear, geometric variational proof of Cauchy's invariants for rotational flows, highlighting their fundamental role in conservation laws and flow analysis.

Contribution

It introduces a new variational approach based on geometric relabeling to derive Cauchy's invariants, emphasizing their importance in rotational flow conservation theorems.

Findings

Provides a simple proof of Cauchy's invariants using geometric relabeling.

Shows all key conservation theorems in rotational flows follow from CIs.

Highlights the central role of CIs in understanding rotational flow dynamics.

Abstract

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.