Variational principle for an incompressible flow

Abstract

This paper gives a general approach to the inverse problem of calculus of variations. The 2-D Euler equations of incompressible flow are used as an example to show how to derive a variational formulation. The paper begins with ideal Laplace equation for its potential flow without vorticity, which admits the Kelvin 1849 variational principle. The next step is to assume a small vorticity to obtain an approximate variational formulation, which is then amended by adding an additional unknown term for further determined, this process leads to the well-known semi-inverse method. Lagrange crisis is also introduced, and some methods to solve the crisis are discussed