Gauge principle and variational formulation for flows of an ideal fluid

Abstract

A gauge principle is applied to mass flows of an ideal compressible fluid subject to Galilei transformation. A free-field Lagrangian defined at the outset is invariant with respect to globalSO(3) gauge transformations as well as Galilei transformations. The action principle leads to the equation of potential flows under constraint of a continuity equation. However, the irrotational flow is not invariant with respect to localSO(3) gauge transformations. According to the gauge principle, a gauge-covariant derivative is defined by introducing a new gauge field. Galilei invariance of the derivative requires the gauge field to coincide with the vorticity, i.e. the curl of the velocity field. A full gauge-covariant variational formulation is proposed on the basis of the Hamilton's principle and an associated Lagrangian. By means of an isentropic material variation taking into account individual particle motion, the Euler's equation of motion is derived for isentropic flows by using the covariant derivative. Noether's law associated with globalSO(3) gauge invariance leads to the conservation of total angular momentum. In addition the Lagrangian has a local symmetry of particle permutation which results in local conservation law equivalent to the vorticity equation.