Gauge-Natural Parametrized Variational Problems, Vakonomic Field Theories and Relativistic Hydrodynamics of a Charged Fluid

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 Nöther theorem that links horizontal and gauge symmetries to the relative conservation laws (generalizing what Fernández, García 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.

Chetaev versus vakonomic prescriptions in constrained field theories with parametrized variational calculus

“…In [BFF07] we have shown that the coefficients E A , F µ A and F µν A are the components of two global morphisms…”

“…In particular we find that the non-holonomic theory seems to be non physical (every admissible section is a solution). Let us consider a region M of spacetime with metric g αβ (here we consider it to be fixed, but if we want to study the coupling with gravity the formalism can do it as well, see [BFF07]) filled with a barotropic and isoentropic fluid. The world lines of the fluid particles and its density describe completely the configuration of the system.…”

“…Nevertheless the vakonomic schema proved to give interesting results in other frameworks, such as optimal control theory (see for example [BBCM03]). In field theory, however, the situation is much less clear: both procedures have been generalized to provide field equations and Nöther currents in some cases (see [MPSW01,FGR04,BFF07] for vak and [BdLdDS02,VCdLMdD05,KV05] for NH), but it is still not evident which one should be better applied in concrete cases. Moreover no fundamental reason justify the NH method since d'Alembert principle cannot be formulated.…”

The Relative Energy of Homogeneous and Isotropic Universes From Variational Principles

“…Let us review here a variational formulation of relativistic hydrodynamics that was originally presented in detail in Ref. [5]. It is in practice equivalent to the ones of Refs.…”

A class of nonholonomic kinematic constraints in elasticity

“…Here, (y (i) ,ẏ (i) ) is a coordinate system on the ith factor of T S N . Note that there is no summation over the index i in (1), and that the summation over individual coordinates is implicit, as shown in the term between brackets.…”