Clebsch Potentials in the Variational Principle for a Perfect Fluid

Abstract

Equations for a perfect fluid can be obtained by means of the variational principle both in the Lagrangian description and in the Eulerian one. It is known that we need additional fields somehow to describe a rotational isentropic flow in the latter description. We give a simple explanation for these fields; they are introduced to fix both ends of a pathline in the variational calculus. This restriction is imposed in the former description, and should be imposed in the latter description. It is also shown that we can derive a canonical Hamiltonian formulation for a perfect fluid by regarding the velocity field as the input in the framework of control theory.Comment: 15 page

Forms of action for perfect fluid in general relativity and mimetic gravity

“…e. the liquid moves as before along the geodesics. However, this motion is not potential now, and it can be quite arbitrary according Clebsch representation [17] for the speed, see details in [8,12]. We easily verify that, taking into account the equations of motion (22), corresponding to the action (21) the EMT expression is reduced to (9) as in the previous section.…”

“…The action (19) is a special case of the action used in [13], restricted to the case of an isentropic pressureless fluid. Thus all the six forms of the action (5), (11), (12), (14), (17), (19) describe the same system: a potentially moving perfect fluid without pressure, and they can be used as appropriate in particular tasks. In the next sections the obtained results will be generalized to the cases of a nonpotential motion and the presence of pressure.…”

Field theory of the Eulerian perfect fluid

“…Although some pioneers in [7,8,10] have found that the potentials in the Clebsch parametrization form canonical pairs, they really could not reduce the phase space {φ, ψ, α; ρ, s, β} down to the configuration space {φ, ψ, α}. In order to obtain the real action in the configuration space, we shall abandon the Lin constraint of the standard formalisms [3,6,7,8,12,11,10,15,16].…”

“…For instance, (I) the Clebsh parametrization includes extra degrees of freedom of the Lagrange multipliers (φ, ψ, and α) of added constraints, which Schutz (1970) expresses as "too many" and pointed out the necessity of action principle with a minimum number of variables [8]. (II) Physical interpretation of the Clebsch potentials themselves are still controversial; some identify them as the Lagrangian-coordinate variables [6,7,11], some relate them with the Chern-Simons theory [10,12]. Yet, no conclusive agreement on the nature of the potentials has been reached.…”

Manifestly invariant Lagrangians for geophysical fluids

“…While Seliger and Whitham () claim that five Clebsch potentials suffice to represent all possible types of fluid flows, Bretherton () argues that more than five are needed. Fukagawa and Fujitani () also claim that these additional potentials are necessary to fix both ends of the pathlines of fluid particles. The field formulation presented here can accomodate any odd number (equal to or larger than five) of Clebsch potentials.…”