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

TL;DR

This work constructs a Lagrangian formulation for generalized higher-derivative perfect-fluid equations that are invariant under the $\ell$-conformal Galilei group for half-integer $\ell$, using a Clebsch-type parametrization that requires decomposing only the highest velocity component $\upsilon^{(2n)}_i$ as $\upsilon^{(2n)}_i=\partial_i\theta+\alpha\partial_i\beta+\sum_{k=0}^{n-1}(-1)^{k+1}\upsilon_j^{(k)}\partial_i\upsilon_j^{(2n-k-1)}$. The resulting Lagrangian $L=-\int dx\rho(\partial_0\theta+\alpha\partial_0\beta+\sum_{k=0}^{n-1}(-1)^{k+1}\upsilon_i^{(k)}\partial_0\upsilon_i^{(2n-k-1)})-H$ yields the continuity and generalized Euler equations upon variation, while Dirac constraint analysis for $\ell=\tfrac{3}{2}$ shows all constraints are second-class and reproduces the non-canonical Hamiltonian structure via Dirac brackets, clarifying the canonical pairs. This work thus provides a coherent Lagrangian framework that aligns with the known Hamiltonian description and opens avenues for thermodynamic interpretation, supersymmetric extensions, and fluid/gravity applications.

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.