Stochastic geometric mechanics with diffeomorphisms

TL;DR

This paper extends Noether's theorem to stochastic geometric mechanics, elucidating how broken symmetries influence fluid circulation in stochastic fluid models, particularly through the Kelvin-Noether theorem.

Contribution

It introduces a stochastic version of Noether's theorem applicable to geometric mechanics with broken symmetry, linking it to fluid circulation in SALT models.

Findings

Extended Noether's theorem to stochastic settings

Derived Kelvin-Noether theorem for SALT models

Clarified role of broken symmetry in fluid circulation

Abstract

Noether's celebrated theorem associating symmetry and conservation laws in classical field theory is adapted to allow for broken symmetry in geometric mechanics and is shown to play a central role in deriving and understanding the generation of fluid circulation via the Kelvin-Noether theorem for ideal fluids with stochastic advection by Lie transport (SALT).