Gauge Theory of Ideal Fluid Flows and Variational Principle

Abstract

Gauge-covariant variational formulation is presented for flows of a compressible ideal fluid. Symmetry groups, i.e. the gauge groups, of fluid flows are a translation group and a rotation group. We propose a Galilei-invariant Lagrangian, and require its variation to be gauge-invariant (both global and local) with respect to the symmetry groups, and try to deduce the Euler's equation of motion for fluid flows from the gauge principle. The velocity field consistent with the first translation group is found to be irrotational, and corresponding equation of motion is that for potential flows.
Next, we consider the second gauge group, that is SO(3). It will be shown that the new gauge transformation introduces a rotational component in the velocity field (i.e. vorticity). In complying with the local gauge invariance, a gauge-covariant derivative is defined by introducing a new gauge field. Galilei invariance of the covariant derivative requires that the gauge field coincides with the vorticity. As a result, the covariant derivative of velocity is found to be the material derivative of velocity. Thus, the Euler's equation of motion for an ideal fluid is derived from the Hamilton's principle. From the Noether's theorem, the gauge symmetry with respect to the translation group results in the conservation law of total momentum, while the symmetry with respect to the rotation group results in the conservation of total angular momentum.