Variational formulation of ideal fluid flows according to gauge principle (original) (raw)

Variational formulation of the motion of an ideal fluid on the basis of gauge principle

Physica D: Nonlinear Phenomena, 2008

On the basis of gauge principle in the field theory, a new variational formulation is presented for flows of an ideal fluid. The fluid is defined thermodynamically by mass density and entropy density, and its flow fields are characterized by symmetries of translation and rotation. A structure of rotation symmetry is equipped with a Lagrangian ΛA including vorticity, in addition to Lagrangians of translation symmetry. From the action principle, Euler's equation of motion is derived. In addition, the equations of continuity and entropy are derived from the variations. Equations of conserved currents are deduced as the Noether theorem in the space of Lagrangian coordinate a. It is shown that, with the translation symmetry alone, there is freedom in the transformation between the Lagrangian a-space and Eulerian x-space. The Lagrangian ΛA provides non-trivial topology of vorticity field and yields a source term of the helicity. The vorticity equation is derived as an equation of the gauge field. Present formulation provides a basis on which the transformation between the a space and the x space is determined uniquely.

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

Fluid Dynamics Research, 2007

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.

Gauge principle for flows of an ideal fluid

Fluid Dynamics Research, 2003

A gauge principle is applied to ows of a compressible ideal uid. First, a free-ÿeld Lagrangian is deÿned with a constraint condition of continuity equation. The Lagrangian is invariant with respect to global SO(3) gauge transformations as well as Galilei transformation. From the variational principle, we obtain the equation of motion for a potential ow. Next, in order to satisfy local SO(3) gauge invariance, we deÿne a gauge ÿeld and a gauge-covariant derivative. Requiring the covariant derivative to be Galilei-invariant, it is found that the gauge ÿeld coincides with the vorticity and the covariant derivative is the material derivative for the velocity. Based on the gauge principle and the gauge-covariant derivative, the Euler's equation of motion is derived for a homentropic rotational ow. Noether's law associated with global SO(3) gauge invariance leads to the conservation of total angular momentum. This provides a gauge-theoretical ground for analogy between acoustic-wave and vortex interaction in uid dynamics and the electron-wave and magnetic-ÿeld interaction in quantum electrodynamics.

Gauge Theoretic Approach to Fluid Dynamics

The Hamiltonian dynamics of a compressible inviscid fluid is formulated as a gauge theory. The idea of gauge equivalence is exploited to unify the study of apparantly distinct physical problems and solutions of new models can be generated from known fluid velocity profiles.

Gauge theory of vortices in a fluid

International Journal of Engineering Science, 1991

A theory of general flow of an Euler fluid has been developed in the framework of the Yang-Mills gauge theory. The state of irrotational flow is taken as the reference state. The standard procedure of gauge theory with the one dimensional internal translation group as the gauge group leads to field equations of general fluid flow including dynamics of vortices. As an application, it is shown that the flow field of a vortex line acquires a core and singularity at the line is removed.

Lagrangian and Eulerian Representations of Fluid Flow

This essay considers the two major ways that the motion of a fluid continuum may be described, either by observing or predicting the trajectories of parcels that are carried about with the flow – which yields a Lagrangian or material representation of the flow — or by observing or predicting the fluid velocity at fixed points in space — which yields an Eulerian or field representation of the flow. Lagrangian methods are often the most efficient way to sample a fluid domain and most of the physical conservation laws begin with a Lagrangian perspective. Nevertheless, almost all of the theory in fluid dynamics is developed in Eulerian or field form. The premise of this essay is that it is helpful to understand both systems, and the transformation between systems is the central theme.

On the dissociation between potential vorticity conservation and symmetries

arXiv (Cornell University), 2018

Using a four-dimensional manifestly covariant formalism suitable for classical fluid dynamics, it is shown that the conservation of potential vorticity is not associated with any symmetry of the equations of motion but is instead a trivial conservation law of the second kind. The demonstration is provided in arbitrary coordinates and therefore applies to comoving (or label) coordinates. Since this is at odds with previous studies, which claimed that potential vorticity conservation is associated with a symmetry under particle-relabeling, a detailed discussion on relabeling transformations is also presented.