A Lagrangian Approach to the Inhomogeneous Incompressible Euler Equation

In this paper, we study the inhomogeneous incompressible Euler equation (IIE in short) from a Lagrangian perspective. We establish a geodesic description of this equation and discuss the associated geometric structures. We also find the derivation of IIE from the Hamilton-Pontryagin action principle and derive the corresponding Lagrangian formulation. A byproduct is a new vorticity formulation of IIE. We also prove the Lagrangian analyticity of IIE using our Lagrangian representation formula.