Gauge-natural parameterized variational problems, vakonomic field theories and relativistic hydrodynamics of a charged fluid

TL;DR

This paper develops a gauge-natural variational framework for field theories with constrained variations, applying it to relativistic charged fluid hydrodynamics, simplifying computations and linking symmetries to conservation laws.

Contribution

It introduces gauge-natural parametrized variational problems with a new N"other theorem, and applies the theory to charged fluid hydrodynamics, providing a more efficient variational formulation.

Findings

Established a gauge-covariant variational framework for constrained field theories.

Derived a new N"other theorem linking symmetries to conservation laws in this context.

Presented a simplified variational formulation for charged fluid hydrodynamics.

Abstract

Variational principles for field theories where variations of fields are restricted along a parametrization are considered. In particular, gauge-natural parametrized variational problems are defined as those in which both the Lagrangian and the parametrization are gauge covariant and some further conditions is satisfied in order to formulate a N\"other theorem that links horizontal and gauge symmetries to the relative conservation laws (generalizing what Fern{\'a}ndez, Garc{\'{\i}}a and Rodrigo did in some recent papers). The case of vakonomic constraints in field theory is also studied within the framework of parametrized variational problems, defining and comparing two different concepts of criticality of a section, one arising directly from the vakonomic schema, the other making use of an adapted parametrization. The general theory is then applied to the case of hydrodynamics of a charged fluid coupled with its gravitational and electromagnetic field. A variational formulation including conserved currents and superpotentials is given that turns out to be computationally much easier than the standard one.