The relativistic fluid is a highly successful model used to describe the dynamics of many-particle, relativistic systems. It takes as input basic physics from microscopic scales and yields as output predictions of bulk, macroscopic motion. By inverting the process, an understanding of bulk features can lead to insight into physics on the microscopic scale. Relativistic fluids have been used to model systems as “small” as heavy ions in collisions, and as large as the Universe itself, with “intermediate” sized objects like neutron stars being considered along the way. The purpose of this review is to discuss the mathematical and theoretical physics underpinnings of the relativistic (multiple) fluid model. We focus on the variational principle approach championed by Brandon Carter and his collaborators, in which a crucial element is to distinguish the momenta that are conjugate to the particle number density currents. This approach differs from the “standard” text-book derivation of the equations of motion from the divergence of the stress-energy tensor in that one explicitly obtains the relativistic Euler equation as an “integrability” condition on the relativistic vorticity. We discuss the conservation laws and the equations of motion in detail, and provide a number of (in our opinion) interesting and relevant applications of the general theory.

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Relativistic fluid dynamics: physics for many different scales

Coupling dark energy and dark matter through an effective fluid description is a very common procedure in cosmology; however, it always remains in comoving coordinates in the special FLRW space. We construct a consistent, general, and covariant formulation, where the interaction is a natural implication of the imperfectness of the fluids. This imperfectness makes difficult the final step towards a robust formulation of interacting fluids, namely the construction of a Lagrangian whose variation would give rise to the interacting equations. Nevertheless, we present a formal solution to this problem for a single fluid, through the introduction of an effective metric.

Covariantizing the interaction between dark energy and dark matter

We present a direct approach to the construction of Lagrangians for a large class of one-dimensional dynamical systems with a simple dependence (monomial or polynomial) on the velocity. We rederive and generalize some recent results and find Lagrangian formulations which seem to be new. Some of the considered systems (e.g. motions with the friction proportional to the velocity and to the square of the velocity) admit infinite families of different explicit Lagrangian formulations.

http://iopscience.iop.org/article/10.1088/1751-8113/43/17/175205/pdf

A direct approach to the construction of standard and non-standard Lagrangians for dissipative-like dynamical systems with variable coefficients

We present a direct approach to the construction of Lagrangians for a large class of one-dimensional dynamical systems with a simple dependence (monomial or polynomial) on the velocity. We rederive and generalize some recent results and find Lagrangian formulations which seem to be new. Some of the considered systems (e.g., motions with the friction proportional to the velocity and to the square of the velocity) admit infinite families of different Lagrangian formulations.

A direct approach to the construction of standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients

The Brans-Dicke-like field of scalar-tensor gravity can be described as an imperfect fluid in an approach in which the field equations are regarded as effective Einstein equations. After completing this approach, we recover, as a special case, the known effective fluid for a scalar coupled nonminimally to the Ricci curvature, and we describe the imperfect fluid equivalent of $f(R)$ gravity. A symmetry of electrovacuum Brans-Dicke gravity is translated into a symmetry of the corresponding effective fluid. The discussion is valid for any spacetime geometry.

Imperfect fluid description of modified gravities

A Lagrangian and Hamiltonian formulation can be given for a damped harmonic oscillator with damping linear in the velocity. The canonical momentum is not equal to the kinetic momentum, and the Hamiltonian is not equal to the energy. On the other hand, a pendulum accreting mass has the same Lagrangian and equation of motion. However, in this case the canonical momentum is equal to the kinetic momentum, and the Hamiltonian is equal to the energy. No ambiguity arises if the physical situation is kept in mind.

Stanislaw Sieniutycz

Papers

Lagrangians for dissipative systems