Lagrangian formulation for perfect fluid equations with the l-conformal Galilei symmetry

TL;DR

This paper develops a Lagrangian formulation for perfect fluid equations invariant under the $ ext{l}$-conformal Galilei group, generalizing the Euler fluid equations and analyzing the transition to Hamiltonian form.

Contribution

It introduces a new Lagrangian approach based on Clebsch parametrization for fluids with $ ext{l}$-conformal Galilei symmetry, extending previous models.

Findings

Reproduces Euler fluid equations for $ ext{l}=rac{1}{2}$

Provides a detailed transition from Lagrangian to Hamiltonian formulation

Generalizes fluid symmetry to $ ext{l}$-conformal Galilei group

Abstract

Lagrangian formulation for perfect fluid equations which hold invariant under the $\ell$-conformal Galilei group with half-integer $\ell$ is proposed. It is based on a Clebsch-type parametrization and reproduces Lagrangian description of the Euler fluid equations for $\ell=\frac12$. The transition from the Lagrangian formulation to the Hamiltonian one is analyzed in detail.