2017/04/13 by Martin Charron, Charron, Martin, Ayrton Zadra +1
Physics and Astronomy · #Cosmology and Gravitation Theories #Solar and Space Plasma Dynamics #Black Holes and Theoretical Physics
paper · pdf · doi:10.48550/arxiv.1704.04214
Using a manifestly invariant Lagrangian density based on Clebsch fields and\nsuitable for geophysical fluid dynamics, the conservation of mass, entropy,\nmomentum and energy, and the associated symmetries are investigated. In\ncontrast, it is shown that the conservation of Ertel's potential vorticity is\nnot associated with any symmetry of the equations of motion, but is instead a\ntrivial conservation law of the second kind. This is at odds with previous\nstudies which claimed that potential vorticity conservation relates to a\nsymmetry under particle-relabeling transformations. From the invariant\nLagrangian density, a canonical Hamiltonian formulation is obtained in which\nDirac constraints explicitly include the (possibly time-dependent) metric\ntensor. In this case, it is shown that all Dirac constraints are primary and of\nthe second class, which implies that no local gauge symmetry transformations of\nClebsch fields exist. Finally, the corresponding non-canonical Hamiltonian\nstructure with time-dependent strong constraints is derived using tensor\ncomponents. The existence of Casimir invariants is then investigated in\narbitrary coordinates for two choices of dynamical variables in phase space.\n