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On New Hydrodynamic Conservation Laws Related to the Particle Relabeling Symmetry
Author(s) -
Albert H.F.
Publication year - 1997
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19970771216
Subject(s) - noether's theorem , conservation law , physics , classical mechanics , symmetry (geometry) , vorticity , isotropy , enstrophy , mathematical physics , gravitation , mathematics , vortex , quantum mechanics , geometry , mechanics , lagrangian
Pure 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.