Particle Relabelling Symmetry Research Articles

On New Hydrodynamic Conservation Laws Related to the Particle Relabeling Symmetry

AbstractPure particle relabeling symmetry is a special orthogonal Lie group of transformations of the labeling coordinates of Lagrangian hydrodynamics. In reversible hydrodynamics it leaves the corresponding Lagrangian function invariant. Via Noether's theory it generates a conservation law not discussed before. It not only allows to generalize the concepts of helicity and enstrophy but also yields Ertel's and Hollmann's vorticity conservation theorems. This new conservation theorem may be regarded as the base of the whole vorticity theory. A generalization to irreversible motion of arbitrary fluid is discussed briefly. Some secondary consequences are: The baroclinic fluid is isotropic only if the isentropic surfaces are spherical (§ 3); for a stationary motion the vorticity conservation theorems are special reformulations of the Bernoulli equation (end of § 4). A scaling symmetry which is the superposition of a special particle relabeling symmetry and a certain scaling of time and distance from center of gravitation is also discussed. The physical meaning of the corresponding conservation law is not completely investigated.

Point Symmetries for Lagrangian Fluid Dynamics

AbstractAlthough it is very difficult to find the whole symmetry group of the Lagrangian hydrodynamic equations of a Newtonian fluid, we are able to present all of the point symmetries of the dynamics of a non‐viscous and non‐heat conducting perfect gas in the Lagrangian (co‐moving) frame of reference. The latter possesses the structure of a Hamiltonian dynamics for which one can discuss not only Lie, but also Noether symmetries. New results are found for some kind of particle relabeling symmetry. Consequences of these symmetries, as conservation laws and invariant solutions, are discussed elsewhere. Most of the calculations have been carried out with the computer algebraic system REDUCE.

A general method for finding extremal states of Hamiltonian dynamical systems, with applications to perfect fluids

In addition to the Hamiltonian functional itself, non-canonical Hamiltonian dynamical systems generally possess integral invariants known as ‘Casimir functionals’. In the case of the Euler equations for a perfect fluid, the Casimir functionals correspond to the vortex topology, whose invariance derives from the particle-relabelling symmetry of the underlying Lagrangian equations of motion. In a recent paper, Vallis, Carnevale & Young (1989) have presented algorithms for finding steady states of the Euler equations that represent extrema of energy subject to given vortex topology, and are therefore stable. The purpose of this note is to point out a very general method for modifying any Hamiltonian dynamical system into an algorithm that is analogous to those of Vallis etal. in that it will systematically increase or decrease the energy of the system while preserving all of the Casimir invariants. By incorporating momentum into the extremization procedure, the algorithm is able to find steadily-translating as well as steady stable states. The method is applied to a variety of perfect-fluid systems, including Euler flow as well as compressible and incompressible stratified flow.