Gauge principle and variational formulation for ideal fluids with reference to translation symmetry

Abstract

Following the gauge principle in the field theory of physics, a new variational formulation is presented for flows of an ideal fluid. In the present gauge-theoretical analysis, it is assumed that the field of fluid flow is characterized by a translation symmetry (group) and in addition that the fluid itself is a material in motion characterized thermodynamically by mass density and entropy (per unit mass). Local gauge transformation in the present case is local Galilean transformation (without rotation) which is a subgroup of a generalized local Galilean transformation group between non-inertial frames. In complying with the requirement of local gauge invariance of Lagrangians, a gauge-covariant derivative with respect to time is defined by introducing a gauge term. Galilean invariance requires that the covariant derivative should be the convective derivative, i.e. the so-called Lagrange derivative. Using this gauge-covariant operator, a free-field Lagrangian and Lagrangians associated with gauge fields are defined under the gauge symmetry. Euler's equation of motion is derived from the action principle. Simutaneously, the equation of continuity and equation of entropy conservation are derived from the variational principle. It is found that general solution thus obtained is equivalent to the classical Clebsch solution. If entropy of the fluid is non-uniform, the flow will be rotational. However, if the entropy is uniform throughout the space (i.e. homentropic), then the flow field reduces to that of a potential flow. Discussions are given on the issue. From the gauge invariance with respect to translational transformations, a differential conservation law of momentum is deduced as Noether's theorem.